Research ArticleFrequency Diverse Array Radar Crameacuter-Rao Lower Bounds forEstimating Direction Range and Velocity
Yongbing Wang1 Wen-Qin Wang12 and Huaizong Shao1
1 School of Communication and Information Engineering University of Electronic Science and Technology of ChinaChengdu 611731 China
2Department of Electrical and Electronic Engineering Imperial College London London SW7 2AZUK UK
Correspondence should be addressed to Yongbing Wang wangyongbinghi163com
Received 27 January 2014 Accepted 19 February 2014 Published 8 April 2014
Academic Editor Frankie KitWing Chan
Copyright copy 2014 Yongbing Wang et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Different from phased-array radar frequency diverse array (FDA) radar offers range-dependent beampattern and thus providesnew application potentials But there is a fundamental question what estimation performance can achieve for an FDA radar Inthis paper we derive FDA radar Cramer-Rao lower bounds (CRLBs) for estimating direction range (time delay) and velocity(Doppler shift) Two different data models including pre- and postmatched filtering are investigated separately As the FDA radarhas range-angle coupling we use a simple transmit subaperturing strategy which divides the whole array into two subarrays eachuses a distinct frequency increment Assuming temporally white Gaussian noise and linear frequency modulated transmit signalextensive simulation examples are performed When compared to conventional phased-array radar FDA can yield better CRLBsfor estimating the direction range and velocity Moreover the impacts of the element number and frequency increment are alsoanalyzed Simulation results show that the CRLBs decrease with the increase of the elements number and frequency increment
1 Introduction
The main task of a radar is to detect the existence oftargets and estimate their unknown parameters for examplerange velocity and direction of arrival (DOA) [1ndash4] All theelements in a phased-array antenna transmit the same signalat a fixed carrier frequency By controlling the linear phaseprogression across elements the beam can be steered to thedesired direction precisely But the beam steering is fixed inan angle for all the ranges and cannot mitigate undesirablerange-dependent interferences For this reason the range andangle information of targets cannot be directly obtained for aconventional phased-array radar which offers only the angleinformation However in some applications it is desired thatthe beam pointing can be steerable to the range of interest
Recently frequency diverse array (FDA) is proposed in[5ndash7] as amethod to achieve range-dependent beamformingwhich can be used to suppress range-dependent interfer-ences and clutter [8] or improve detection performance
[9] The most important difference of FDA antenna from aconventional phased-array antenna is that a small amountof frequency increment compared to the carrier frequencyis used across the array elements The periodicity of FDAbeampattern in range angle and time is studied in [10] In[11] the FDA is investigated from a simulation perspectiveand a low-cost FDA is designedThe FDA using chirp (linearfrequency modulation) waveforms with different startingfrequencies is analyzed in [12] to characterize the associatedrange-dependent beampattern The application of FDA insynthetic aperture radar (SAR) is investigated in [13 14]Additional studies to exploit the range-dependent beampat-tern characteristics have been reported in [15]
FDA offers a range-angle-dependent beampattern whichis of great importance as this provides a potential for range-angle localization of targets Therefore this paper considersa fundamental question what estimation performance canbe achieved for an FDA radar Referring to conventionalphased-array range radar the Cramer-Rao lower bound
Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 830869 15 pageshttpdxdoiorg1011552014830869
2 International Journal of Antennas and Propagation
(CRLB) expressions for estimating velocity direction andDoppler shift are derived in [16 17] By assuming a narrow-band signal model prior to matched filtering the CRLBs forestimating the time delay direction and Doppler shift with aphased-array radar are detailed in [18 19] Furthermore theCRLB for jointly estimating the attributes of a moving targetusing MIMO radar is derived in [20ndash22]
Different from the above literatures this paper derivesthe CRLBs for jointly estimating the range direction andDoppler shift with an FDA radar Two typical data mod-els namely pre- and postmatched filtering are analyzedseparately for static and moving targets respectively Sincethe range and angle of targets cannot be estimated directlydue to the range-angle coupling we use a simple transmitsubaperturing strategy for the FDA radar This methoddivides the whole array elements into two subarrays eachuses a different frequency increment In doing so the rangeand angle of targets can be solely estimated The corre-sponding CRLBs for estimating direction range and veloc-ity are derived Furthermore the impacts of the elementsnumber and frequency increment on the CRLBs are alsoinvestigated
The rest of the paper is organized as follows The FDAis briefly introduced in Section 2 In Section 3 we considerthe data model after matched filtering and derive the CRLBexpressions for estimating the range and direction In orderto decouple the range-angle coupling response we divide the
whole array into two subarrays The corresponding CRLBsare derived In Section 4 the data model prior to matchedfiltering is investigated and the CRLB expressions for estimat-ing the direction range and Doppler shift are derived wheretemporally white Gaussian noise and chirp pulse signal areassumed Finally extensive simulation results are provided inSection 5 This paper is concluded in Section 6
2 Frequency Diverse Array (FDA)
In conventional phased-array radars it is assumed that anidentical waveform is radiated from each element excludingthe amplitudes and phases Different from phased-arrayradars FDA elements can be either excited by the samewaveform or different waveforms For simplicity and withoutloss of generality we consider a uniform linear array (ULA)FDA and assume that the waveforms radiated from eachantenna element are identical but with a frequency incrementof Δ119891HzThat is the radiation frequency of the119898th elementis
119891119898= 119891
0+ (119898 minus 1) sdot Δ119891 119898 = 0 1 119872 minus 1 (1)
where 1198910is the radar carrier frequency and119872 is the number
of the elementsSupposing a narrowband transmit signal and taking the
first element as the reference for the array the steering vectoris given by the following equation [11]
a (120579 119903) = [1 119890minus119895((21205871198910119889 sin 120579119888)+((2120587Δ119891sdot119889 sin 120579)119888)minus(2120587119903Δ119891119888))
sdot sdot sdot 119890minus119895(((21205871198910(119872minus1)119889 sin 120579)119888)+((2120587sdot(119872minus1)
2Δ119891sdot119889 sin 120579)119888)minus((2120587119903(119872minus1)Δ119891)119888))]
119879
(2)
where 120579 is the direction 119903 is the range 119889 is the elementspacing 119888 is the speed of light and 119879 is the transpose Since1198910
≫ Δ119891 and 119903 ≫ (119872 minus 1)119889 sin(120579) in an amplitude
sense an approximation can be taken by ignoring the phaseterm 2120587 sdot (119898 minus 1)
2
Δ119891 sdot 119889 sin(120579)119888 because it has ignorableimpacts In this case the steering vector can be simplifiedto
a (120579 119903) asymp [1 119890minus119895((21205871198910119889 sin(120579)119888)minus((2120587sdotΔ119891sdot119903)119888))
sdot sdot sdot 119890minus119895(((21205871198910(119898minus1)119889 sin(120579))119888)minus((2120587sdot(119898minus1)Δ119891sdot119903)119888))
]119879
(3)
The FDA characteristics can be summarized as follows[7ndash12] (1) If the frequency offset Δ119891 is fixed the beamdirection will vary as a function of the range 119903 that is it isa range- dependent beam (2) If the range is 119903 fixed the beamdirection will vary as a function of Δ119891 This means that theFDA is frequency-increment-dependent (3) If the frequencyincrement across the array is not applied (ie Δ119891 = 0)the corresponding FDA is just a conventional ULA phasedarray
3 CRLB with Signal Model afterMatched Filtering
31 Data Models Suppose that a target is located at (120579 119903)After matched filtering the baseband equivalent of thecomplex valued signals at the receiver can be expressed as
y = 1205730a (120579 119903) + n = (120595) + n (4)
International Journal of Antennas and Propagation 3
where1205730is a constant for a given target (120595) = 120573
0a(120579 119903)with
120595 = [120579 119903]119879 and n is a zero-mean complex Gaussian white
noise with spatial covariance
119864 [nnH] = Rn = 120590
2
119899I119872 (5)
where 119864[sdot] is the expectation operator H is the conjugatetranspose Rn is the spatial noise covariance matrix 1205902
119899is the
noise power and I119872is an119872times119872 identity matrix
32 CRLBs of Angle and Range Estimations In the followingwe discuss the CRLBs for several different cases separately
321 Range Is Known and Angle Is Unknown In this case120595 = 120579 The corresponding Fisher information matrix (FIM)is I
120595120595[23]
I120595120595
= 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) (6)
where Resdot is the real part of the signal 120595119894is the 119894th element
of 120595 and the119863120595119894(120595) is [24]
119863120595119894(120595) =
120597 (120595)
120597120595119894
(7)
Under the signal model (4) the FDA FIM with respect to 120579 isexpressed as
119868120579120579FDA
= 81205872
1198892cos2 (120579)
sdot SNRsum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(8)
where SNR is the signal-to-noise ratio (SNR) Accordinglythe CRLB is
CRLB120579120579FDA
= 119868minus1
120579120579FDA (9)
where minus1 denotes the inverse matrix In particular whenΔ119891 = 0 it is simplified to
119868120579120579phased-array
= 8SNR1205872
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(10)
Proof See Appendix A
It can be easily proved that
CRLB120579120579FDA
lt CRLB120579120579phased-array
(11)
That is to say the FDA radar has better CRLB for angleestimation than phased-array radar
322 Angle Is Known and Range Is Unknown As phased-array radar has range-independent beam here we onlycalculate the range estimation for FDA radar The FIM withrespect to 119903 can be expressed as
119868119903119903FDA
= 2SNR4120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(12)
The corresponding CRLB is
CRLB119903119903FDA
= 119868minus1
119903119903FDA (13)
Proof See Appendix B
323 Both Angle and Range Are Unknown In this case the 120579and 119903 CRLBs can be similarly determined as
CRLB120579120579FDA
= (1198882
sdot (
119872
sum
119898=1
(119898 minus 1)2
))
times (2SNR sdot 41205872
1198892
sdot (Δ119891)2cos2 (120579)
times (
119872
sum
119898=1
(119898 minus 1)4
119872
sum
119898=1
(119898 minus 1)2
minus(
119872
sum
119898=1
(119898 minus 1)3
)
2
))
minus1
CRLB119903119903FDA
= (1198884
sdot (sum
119872
119898=1(119898 minus 1)
2
1205822
+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888))
times (2SNR sdot 41205872
sdot (Δ119891)4
times (
119872
sum
119898=1
(119898 minus 1)4
119872
sum
119898=1
(119898 minus 1)2
minus(
119872
sum
119898=1
(119898minus1)3
)
2
))
minus1
(14)
Proof See Appendix C
When the targetrsquos range and angle are estimated jointlythe range and angle CRLBs will be significantly degradeddue to the range-angle coupling (which will be furtherinvestigated in Section 5) Consequently the range and angle
4 International Journal of Antennas and Propagation
d
1 2 N
middot middot middot
1 2 N
middot middot middot
Subarray 1 Subarray 2
Figure 1 Illustration of transmit subaperturing FDA radar
of targets cannot be estimated directly by a standard FDAradar To overcome this problem we present a transmitsubaperturing method for the FDA radar
324 CRLBs of Transmit Subaperturing FDA Radar Todecouple the range and angle peaks and estimate both therange and angle of target we divide the whole array into twoequal subarrays [25 26] Suppose that the number of elementsis 119872 each subarray has 119873 elements namely 119872 = 2119873The first subarray uses the frequency increment of Δ119891
1 and
the second subarray uses the frequency increment of Δ1198912
as shown in Figure 1 The resulting system is referred to astransmit subaperturing FDA (TS-FDA) radar
Taking the first element of the first subarray as thereference the new steering vector can be represented as thefollowing equation
b (120579 119903) = [1 119890minus1198951205931 sdot sdot sdot 119890
minus119895(119873minus1)1205931 119890minus119895120593119897 119890
minus119895(1205932+120593119873) sdot sdot sdot 119890minus119895(119873minus1)1205932+120593119873]
119879
(15)
where
1205931= (
2120587119889 sin (120579)120582
) minus (2120587119903 sdot Δ119891
1
119888) (16a)
120593119873= (
2120587119873119889 sin (120579)120582
) (16b)
1205932= (
2120587119889 sin (120579)120582
) minus (2120587119903 sdot Δ119891
2
119888) (16c)
The corresponding FIM can be derived as
ITS-FDA = 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) = [
119868120579120579
119868120579119903
119868119903120579
119868119903119903
] (17)
where
119868120579120579
= 2
10038161003816100381610038161205730
1003816100381610038161003816
2
1205902
119899
41205872
1198892cos2 (120579)1205822
2119873
sum
119898=1
(119898 minus 1)2
(18a)
119868119903119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2
1205902
119899
41205872
(Δ1198912
1+ Δ119891
2
2)
1198882
119873
sum
119898=1
(119898 minus 1)2
(18b)
119868120579119903= 119868
119903120579= minus
210038161003816100381610038161205730
1003816100381610038161003816
2
1205902
119899
41205872
119889 cos (120579)120582119888
times [(Δ1198911+ Δ119891
2)
119873
sum
119898=1
(119898 minus 1)2
+ Δ1198912119873
119873
sum
119898=1
(119898 minus 1)]
(19)
The angle and range CRLBs are the first two diagonalelements of the inverse of the FIM
CRLB120579120579TS-FDA
= [Iminus1
TS-FDA]11
(20a)
CRLB119903119903TS-FDA
= [Iminus1
TS-FDA]22
(20b)
where [sdot]119894119895is the element at the 119894th row and 119895th column of the
matrix
4 CRLB with Signal Model prior toMatched Filtering
41 Data Models Unlike the first data model the data modelprior to matched filtering allows to estimate the Doppler shift(velocity) Suppose an 119872-element antenna array receives atime delayed and Doppler-shifted echo of the transmittedsignal 119904(119905) exp(119895Ω
119888119905) where Ω
119888is the center frequency
Knowing the time delay and Doppler shift Ω119863(assuming
a target with constant radial velocity) the range and radialcomponent of velocity can be determined by 119903 = 1198881205912
and V = Ω119863119888(2Ω
119888) We denote the continuous-time signal
119904(119905) as 119904[119897] = 119904(119897 sdot Δ119905) where Δ119905 is the sampling intervalCorrespondingly the time delay and Doppler shift in thesampled signal domain are 119897
120591= 120591Δ119905 and 119908
119863= Ω
119863sdot Δ119905
respectively [27]After converting to baseband and sampling the received
signal at time 119897 sdot Δ119905 becomes
y [119899] = 120573 sdot a (120579 119903) sdot 119904 [119897 minus 119897120591]exp (119895119908
119863119897) + n [119897] 119897 = 1 119871
(21)
where 120573 is complex amplitude of the signal and n[119897] is theadditive noise [28]
We assume that the snapshots taken at 119897 = 1 119871 coverthe whole of a coherent processing interval (CPI) Thereforethe time duration of the CPI is 119879CPI = 119871 sdot Δ119905 Under thedata model of (21) the complex amplitude 120573 is assumed tobe an unknown deterministic constant during the CPI Tomodel the Doppler effect with a frequency shift the radialcomponent of the target velocity needs to be much smallerthan the propagation speed (ie V119888 ≪ 1) Then the time-bandwidth product of the complex envelope should be largerthan 1 In addition it is assumed that the propagation time ofthe signal across the array is much smaller than the reciprocal
International Journal of Antennas and Propagation 5
of the signal bandwidth which is the narrowband arrayassumption in array processing
Define the vector of unknown target parameters as120581 = [Re120573 Imag120573 120579 120578119879
] Stacking all samples into a singlevector (21) can be rewritten as
z = 120573 sdot 120601 (120578) otimes a (120579 119903) + n = (120581) + n (22)
where otimes denotes the Kronecker product (120581) =
120573 sdot 120601(120578) otimes a(120579 119903)
120601 (120578) = [119904 [1 minus2119903
119888 sdot Δ119905] exp (119895119908
1198631)
119904 [2 minus2119903
119888 sdot Δ119905] exp (119895119908
1198632)
119904 [119871 minus2119903
119888 sdot Δ119905] exp (119895119908
119863119871)]
119879
(23)
The noise n is assumed to be zero-mean Gaussian spatiallyand temporally correlated with spatiotemporal covariance[29]
119864 [nnH] = Cn otimes Rn (24)
where Cn is the temporal noise covariance matrixUnder the above data model the signal and noise param-
eters are disjoint and satisfy the space-time separability [30]
42 CRLBs of Angle Range and Doppler Shift Suppose thereceived signal is completely covered by the observations119897 = 1 119871 and the sampling is dense (ie Δ119905 rarr 0) TheRn is assumed to be constant in the frequency band 119891 isin
(minus1(2Δ119905) 1(2Δ119905)) where 119891 denotes the frequency in thecontinuous-time domain The vector of Doppler shift andtime delay in the continuous-time domain is defined as
120578 = [119903 Ω119863]119879
(25)
For Cn = I119871and Rn = 120590
2
119899I119872
with I119871being an 119871 times 119871
identity matrix We define the signal power s = 120601H 120601 The
CRLBs expression for 120579 and 120578 follows fromAppendix D as thefollowing
CRLB120579120578120579120578FDA
=
1205902
119899
210038161003816100381610038161205731003816100381610038161003816
2s
times
[[[[[[[[[[[
[
12058721198892cos2 (120579)119872(119872
2minus 1)
[
[
1
31205822
+
(Δ119891)2[(2119872minus 1) (8119872
2minus 3119872minus 11)]
451198882(119872+ 1)
+
2Δ119891 (119872minus 1)
3119888120582
]
]
minus1205872119889Δ119891 cos (120579)119872(119872
2minus 1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 0
minus1205872119889Δ119891 cos (120579)119872(119872
2minus 1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 1205872(Δ119891)2119872(119872
2minus 1)
31198882
+
12058511
sImag 12058512
s
0
Imag 12058512
s12058522
s
]]]]]]]]]]]
]
minus1
(26)
where
12058511
= int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905
minus1
s[int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905]
2
(27a)
12058522
= int
infin
minusinfin
1199052
|119904 (119905 minus 120591)|2
119889119905
minus1
s[int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
2
(27b)
12058512
= int
infin
minusinfin
119905119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905
minus [1
sint
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905
sdot int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905]
(28)
The terms 12058511
and 12058522
are proportional to the rootmean squared (RMS) bandwidth and RMS duration of 119904(119905)respectively Note that the decoupling is a consequence ofthe assumed space-time separability of signal and noisemodels and the assumption of the complex amplitude 120573 as anunknown deterministic constant In addition since the signaland additive noise parameters are disjoint the above CRLBexpressions hold regardless of whether the spatiotemporalnoise covariance Cn otimes Rn is known or unknown
43 CRLBs When Chirp Pulse Signal Is Employed In thissection we derive the CRLB expressions for the rangeDoppler shift and direction estimation when 119904(119905) is a chirppulse signal with a large time-bandwidth product [31 32] Itcan be expressed as
119904 (119905) =
119875minus1
sum
119901=0
1199040(119905 minus 119901119879
119901) (29)
6 International Journal of Antennas and Propagation
where 119879119901is the pulse repetition interval 119875 is the number of
chirp pulses and 1199040(119905) is expressed as [12]
1199040(119905) = exp [119895120587 119861
1198790
(119905 minus1
21198790)
2
] sdot [ℎ (119905) minus ℎ (119905 minus 1198790)]
(30)
where1198790is the chirp pulse duration119861 is the chirp bandwidth
and ℎ(119905) is the Heaviside step function
Assume the time-bandwidth product of the pulse is1198790sdot 119861 ≫ 1 Using the signal given in (29) and (30) in
continuous-time domain we obtain the signal power s =
1198751198790 120585
11= 1198754120587
2
1198612
11987903119888
2 Imag12058512 = minus(119875120587119861119879
2
03119888) and
12058522
= (119875MT3
012)[1 + (119879
0119879
119877)2
(1198752
minus 1)] Thus the CRLBexpressions of 120579 and 120578 for FDA are derived as the followingequation
CRLB120579120578120579120578FDA
=
1
2SNR
sdot
[[[[[[[[[[[[
[
12058721198892cos2(120579)119872(119872
2minus 1)
[
[
1
31205822+
(Δ119891)2[(2119872minus 1) (8119872
2minus3119872minus11)]
451198882119872(119872+ 1)
+
2Δ119891 (119872minus 1)
3119888120582
]
]
minus1205872119889Δ119891 cos(120579)119872(119872
2minus1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 0
minus1205872119889Δ119891 cos(120579)119872(119872
2minus 1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 1205872(Δ119891)2119872(119872
2minus1)
31198882
+
412058721198612119872
31198882
minus
1198721205871198611198790
3119888
0 minus
1198721205871198611198790
3119888
1198721198792
0
12
[1+(
119879119877
1198790
)
2
(1198752minus1)]
]]]]]]]]]]]]
]
minus1
(31)
Specially for the phased array (ie Δ119891 = 0) the CRLB is
CRLB120579120578120579120578phased-array
=1
2SNR
times
[[[[[[[[
[
1205872
1198892cos2 (120579)119872(119872
2
minus 1)
312058220 0
04120587
2
1198612
119872
31198882minus119872120587119861119879
0
3119888
0 minus119872120587119861119879
0
3119888
1198721198792
0
12[1 + (
119879119877
1198790
)
2
(1198752
minus 1)]
]]]]]]]]
]
minus1
(32)
When only one pulse (119875 = 1) is used the CRLBin (32) will be infinite because the model is not identifi-able and the range and Doppler shift cannot be uniquelyestimated [18] since phase-array radar has no rangeidentity capability In contrast (31) does not have sucha problem
According to the previous discussion the angle and rangeof targets cannot be estimated jointly due to the range-anglecoupling Using the similar transmit subaperturing approachpresented in Section 3 and steering vector b(120579 119903) (15) theCRLBs of angle range and Doppler shift are derived as inthe following equation
CRLB120579120578120579120578TS-FDA =
1
2SNR
times
[[[[[[[[[[[[[[[[[[[
[
412058721198892cos2 (120579)1205822
2119873
sum
119898=1
(119898 minus 1)2
minus
41205872119889 cos (120579)120582119888
[[[[
[
(Δ1198911 + Δ1198912)
119873
sum
119898=1
(119898 minus 1)2
+119873Δ1198912
119873
sum
119898=1
(119898 minus 1)
]]]]
]
0
minus
41205872119889 cos (120579)120582119888
[[[[
[
(Δ1198911 + Δ1198912)
119873
sum
119898=1
(119898 minus 1)2
+119873Δ1198912
119873
sum
119898=1
(119898 minus 1)
]]]]
]
(Δ1198912
1+ Δ1198912
2)
1198882
119873
sum
119898=1
(119898 minus 1)2+
412058721198612119872
31198882
minus
1198721205871198611198790
3119888
0 minus
1198721205871198611198790
3119888
1198721198792
0
12
[1 + (
119879119877
1198790
)
2
(1198752minus 1)]
]]]]]]]]]]]]]]]]]]]
]
minus1
(33)
International Journal of Antennas and Propagation 7
20151050
10minus2
10minus3
10minus47001 7002 7003
10minus248203
10minus248201
10minus248199
CRLB
of a
ngle
estim
atio
n (d
eg)
SNR (dB)
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
Figure 2 CRLB for estimating angle versus SNR when the range isknown
5 Simulation and Verification
In this section we consider several numerical examples thatcompare the CRLBs in different signal and noise modelsConsider an X-band FDA radar with the carrier frequency1198910
= 10GHz We assume a ULA of 119872 elements usedfor transmitting The array elements are spaced half of thewavelength apart from each other namely 119889 = 1205822 Onetarget of interest is supposed to reflect a plane wave thatimpinges on the array from direction of angle 120579 = 30
∘Under the signal model after matched filtering Figures
2 and 3 compare the CRLBs according to (9) and (13)respectively It can be noticed that the CRLBs are improvedwhen a larger number of elements are employed Howeverit has no significant difference when different frequencyincrements are used This is because sum
119872
119898=1(119898 minus 1)
2
1205822
≫
(Δ119891)2
sum119872
119898=1(119898 minus 1)
4
1198882
+ 2Δ119891sum119872
119898=1(119898 minus 1)
3
120582119888 thus thefrequency increment has a small impact on the CRLBsIn [33] a frequency offset selection strategy is derived toprecisely steer the beam toward a fixed range with a desiredangle
Figure 4 shows the CRLBs of angle and range whenboth the angle and range are unknown The CRLBs aresignificantly degraded due to the range-angle coupling Con-sequently the range and angle of targets cannot be estimateddirectly by the FDA radar However the CRLBs decreaseas the increase of the number of elements and frequencyincrement still holds Moreover generally more elementsmean that better CRLBs performance can be achieved for theFDA radar
To overcome the problem that the range and angle oftargets cannot be estimated directly by the FDA radar weuse the transmit subaperturing strategy on the transmitfrequency increments Figure 5 shows the corresponding
CRLB
of r
ange
estim
atio
n (m
)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
102
101
10020151050
SNR (dB)
Figure 3 CRLB for estimating range versus SNR when the angle isknown
CRLBs where 119872 = 32 is assumed It can be noticed that toobtain a lower CRLB the Δ119891
1and Δ119891
2should have inverse
signs that is one is passive and the other is negative Onereason is that in this case the FDA radar has a wider systembandwidth Figure 6 shows the CRLBs of angle and rangewhen 119872 = 20 is employed It can be noticed that theCRLBs performance improves with the increase of the sensornumber
Under the data model prior to matched filtering wesuppose the following signal parameters bandwidth 119861 =
10MHz repetition period 119879119901
= 1ms and pulse duration1198790
= 250 120583s In this case the approximate expressionsgiven in (31) are valid because the transmitted has a largetime-bandwidth product (119879
0sdot 119861 = 2500 ≫ 1) Figure 7
shows the CRLBs for direction range and Doppler shift asa function of SNR Note that when SNR = minus10 dB and119872 = 32 are employed we can get CRLB
119903119903FDA= 653m
and CRLBΩ119863Ω119863FDA
= 136937 rads that corresponds to theCRLB for velocity is 327 cms (since V = Ω
119863119888(2Ω
119888) and
Ω119888= 2120587119891
119888) Since (Δ119891)2 sum119872
119898=1(119898 minus 1)
4
1198882
≪ 1 the frequencyincrement has a small impact on the CRLBs In additionobserve that the CRLB for 120579 and 120578 is block-diagonal (see(32)) and therefore decoupled that is CRLB
120578120578FDAremains the
same whether or not 120579 is known and similarly CRLB120579120579FDA
is the same whether or not 120578 is known The decouplingis a consequence of the assumed space-time separability ofsignal and noise models and the assumption of the complexamplitude 120573 as an unknown deterministic constant
Figure 8 shows that the CRLBs versus SNR for differentcombinations of Δ119891
1and Δ119891
2 Comparing Figures 8 and
7 the CRLBs have been significantly improved Likewisecomparing Figures 8 and 9 theCRLBs performance improvesas the number of elements increases
8 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
104
103
102
101
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
108
107
106
105
10420151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
Figure 4 Both angle and range are unknown
6 Conclusion
In this paper we derive the CRLB to jointly estimate theattributes of a moving target using FDA radar and computethe corresponding CRLB expressions First we briefly intro-duce the FDA concept and make a summary on the FDAcharacteristics Then we consider two different data modelsnamely pre- and postmatched filtering Under differentsignal and noise models we compute the CRLB expressionsfor estimating the range direction and Doppler shift Wedemonstrate that the FDA radar beamforming is coupledin range and angle and that the targetrsquos range and anglecannot be estimated directly by the FDA radar To overcome
20151050
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
CRLB
of r
ange
estim
atio
n (m
)102
101
100
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
Figure 5 CRLBs of TS-FDA radar with119872 = 32
this problem this paper proposes a transmit subaperturingstrategy for the FDA radar In doing so the range and angle oftargets are estimated from the transmit-receive beamformingoutput Moreover we also specialize the CRLB results tothe case of temporally white noise and a chirp pulse signalExtensive simulation results verify the correctness of thederived CRLBs It is shown that the CRLBs decrease with theincrease of the number of elements and frequency incrementThe CRLBs can be further improved through three aspectsincreasing the number of elements enhancing the systembandwidth by employing a larger frequency increment andusing transmit subaperturing strategy with more subarrays
International Journal of Antennas and Propagation 9
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
CRLB
of r
ange
estim
atio
n (m
)
103
102
101
10020151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
Figure 6 CRLBs of TS-FDA radar with119872 = 20
Appendices
A Derive the CRLB for Angle WhenRange Is Known
To derive the CRLB we start with a well-known expressionfor the FIM under the data model in Section 3 We define thespatial noise covariance matrix as Rn = 120590
2
119899I119872
and signal-to-noise ratio (SNR) as SNR = |120573
0|2
1205902
119899 Suppose the target range
is known the FIM of 120579 is
119868120579120579
= 2Re 119863H120579(120595) (Rminus1
n )119863120579(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
(A1)
For a phased-array radar there is (120595) = 1205730a(120579) We then
have
120597a (120579)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579) (A2)
whereD = diag[0 1 119872 minus 1] and
120597aH (120579)
120597120579
120597a (120579)120597120579
=4120587
2
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A3)
The FIM of the phased-array radar is
119868120579120579phased-array
= 2SNR41205872
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A4)
Similarly for the FDA radar there is (120595) = 1205730a(120579 119903)The
derivation of a(120579 119903) with respect to 120579 is
120597a (120579 119903)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579 119903)
minus 1198952120587119889Δ119891 cos (120579)
119888
times diag [0 1 (119872 minus 1)2
] a (120579 119903)
120597aH (120579 119903)
120597120579
120597a (120579 119903)120597120579
= 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822
+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A5)
Accordingly the FIM of 120579 for the FDA can be expressedas
119868120579120579FDA
= 2SNR sdot 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A6)
10 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
(c) CRLB for Doppler shift versus SNR
Figure 7 General CRLB results of FDA radar
B Derive the CRLB for Range WhenAngle Is Known
Under the data model in Section 3 when the direction 120579 isknown the parameter to be estimated is 119903 The FIM of 119903 is
119868119903119903FDA
= 2Re 119863H119903(120595) (Rminus1
n )119863119903(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597aH (120579 119903)
120597119903Rminus1
n120597a (120579 119903)
120597119903
(B1)
The derivation of a(120579 119903) with respect to 119903 for FDA is
120597a (120579 119903)120597119903
= 1198952120587Δ119891
119888Da (120579 119903)
120597aH (120579 119903)
120597119903
120597a (120579 119903)120597119903
=4120587
2
Δ1198912
1198882
119872
sum
119898=1
(119898 minus 1)2
(B2)
The FIM of 119903 is thus given by
119868119903119903FDA
= 2SNR4120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(B3)
International Journal of Antennas and Propagation 11
C Derive the CRLB for Range and Angle
Under the data model in Section 3 both the angle and rangeare unknown The range and angle of targets are estimatedjointly The FIM for parameters 120579 and 119903 can be expressed as
IFDA = 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) = [
119868120579120579
119868120579119903
119868119903120579
119868119903119903
] (C1)
We then have
IFDA = 210038161003816100381610038161205730
1003816100381610038161003816
2
times
[[[[
[
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
]]]]
]
(C2)
Since (120597H(120595)120597119903)Rminus1
n (120597(120595)120597120579) = (120597H(120595)
120597120579)Rminus1
n (120597(120595)120597119903) then 119868120579119903= 119868
119903120579 We can get
119868120579120579
= 210038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
= 81205872
1198892cos2 (120579)
times SNRsum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
119868120579119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
= minus81205872
119889Δ119891 cos (120579)
times SNR[sum
119872
119898=1(119898 minus 1)
2
120582119888+Δ119891sum
119872
119898=1(119898 minus 1)
3
1198882]
119868119903119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
= SNR8120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(C3)
Since CRLB120579120579FDA
= [Iminus1
FDA]11 CRLB119903119903FDA
= [Iminus1
FDA]22 Theexpressions for the CRLB of angle and range estimationsgiven in (14) can be obtained by substituting (C3) into (C2)
D General CRLB Results
Under the data model in Section 4 we derive the continuousCRLB expressions for temporally and spatially white noiseand denote SNR = |120573|
2
s1205902
119899with s being the power We
also start with FIM
I120581119894120581119895
= 2Re 119863H120581119894
(120581) (Cminus1
n otimes Rminus1
n )119863120581119895(120581) (D1)
where 120581119894is the 119894th element of 120581 and 119863
120581119894(120595) = 120597(120581)120597120581
119894
Consider
IFDA =
[[[[[
[
119868120573120573
119868119879
120579120573119868119879
120578120573
119868120579120573
119868120579120579
119868119879
120578120579
119868120578120573
119868120578120579
119868120578120578
]]]]]
]
(D2)
For clarity we rewrite Fisherrsquos information matrix I as
IFDA = [A UV B] (D3)
where
V = U119879
(D4a)
V = [119868120579120573
119868120578120573
] (D4b)
B = [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] (D4c)
According to the matrix inversion lemma the inversematrix of IFDA is
Iminus1
FDA = [
[
(A minus UBminus1V)minus1
minusAminus1U(B minus VAminus1U)minus1
minusBminus1V(A minus UBminus1V)minus1
(B minus VAminus1U)minus1
]
]
(D5)
where the submatrix (A minus UBminus1V)minus1 corresponds to theCRLBof120573which is nuisance and (B minus VAminus1U)minus1 correspondsto the CRLB of 120579 and 120578 = [119903 Ω
119863]119879 which are of interest
CRLB120579120578120579120578FDA
= (B minus VAminus1U)minus1
= [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] minus [119868120579120573
119868120578120573
] 119868minus1
120573120573[119868
119879
120579120573119868119879
120578120573]
minus1
(D6)
where
119868120573120573
= 2 sdot[[[
[
s sdot 1198721205902
119899
0
0s sdot 1198721205902
119899
]]]
]
(D7a)
119868120579120573
= 2 sdot Re[1 119895] otimess1205902
119899
sdot 120573lowast
1198601 (D7b)
119868120578120573
= 2 sdot Re[1 119895] otimes120573
lowast
1205902
119908
sdot [119872 sdot 1198603+ s sdot 119860
4] (D7c)
119868120578120578
= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
[119872 sdot 1198606+ 119860
H3119860
4+ 119860
3119860
H4+ s sdot 119860
7]
(D7d)
12 International Journal of Antennas and Propagation
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHz
70001 70001 70001 70002100286514
100286515
100286515
kHz
(c) CRLB for Doppler shift versus SNR
Figure 8 General CRLB results of TS-FDA radar with119872 = 32
119868120579120579minus (119868
120579120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198602minus119860
1119860
H1
119872) (D8a)
119868120578120579minus (119868
120578120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198605minus119860
4119860
H1
119872) (D8b)
119868120578120578
minus (119868120578120573119868119879
120578120573)
1205902
119899
s119872
= 2
10038161003816100381610038161205731003816100381610038161003816
2
s1205902
119899
(119872119860
6
s+ 119860
7minus119872119860
3119860
H3
s2minus119860
4119860
H4
119872)
(D8c)
1198601= minus119895(
2120587119889 cos (120579)sum119872
119898=1(119898 minus 1)
120582
+2120587119889Δ119891 cos (120579)sum119872
119898=1(119898 minus 1)
2
119888)
(D9a)
1198602= 4120587
2
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(D9b)
International Journal of Antennas and Propagation 13
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(c) CRLB for Doppler shift versus SNR
Figure 9 General CRLB results of TS-FDA radar with119872 = 20
1198603= [int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905 int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
119879
(D9c)
1198604= [
2120587Δ119891sum119872
119898=1(119898 minus 1)
1198880]
119879
(D9d)
1198605= [minus4120587
2119889Δ119891 cos(120579) [
sum119872
119898=1(119898 minus 1)
2
12120582119888
+
Δ119891sum119872
119898=1(119898 minus 1)
3
121198882
] 0]
119879
(D9e)
1198606=[[
[
int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905 int
infin
minusinfin
119905119904H(119905 minus 120591)
119889119904 (119905 minus 120591)
119889119905
119889119905
int
infin
minusinfin
119905119904 (119905 minus 120591)
119889119904(119905 minus 120591)H
119889119905
119889119905 int
infin
minusinfin
1199052|119904 (119905 minus 120591)|
2119889119905
]]
]
(D9f)
1198607=[[
[
41205872
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
0
0 0
]]
]
(D9g)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework described in this paper was supported in part by theNational Natural Science Foundation of China under Grant
14 International Journal of Antennas and Propagation
41101317 the Program for New Century Excellent Talentsin University under Grant NCET-12-0095 Sichuan ProvinceScience Fund for Distinguished Young Scholars under Grant2013JQ0003 and Fundamental Research Fund for theCentralUniversities
References
[1] F Bandiera M Mancino and G Ricci ldquoLocalization strategiesfor multiple point-like radar targetsrdquo IEEE Transactions onSignal Processing vol 60 no 12 pp 6708ndash6712 2012
[2] D R Fuhrmann J P Browning and M Rangaswamy ldquoSignal-ing strategies for the hybrid MIMO phased-array radarrdquo IEEEJournal on Selected Topics in Signal Processing vol 4 no 1 pp66ndash78 2010
[3] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011
[4] M Compagnoni P Bestagini F Antonacci A Sarti and STubaro ldquoLocalization of acoustic sources through the fittingof propagation cones using multiple independent arraysrdquo IEEETransactions on Audio Speech and Language Processing vol 20no 7 pp 1964ndash1975 2012
[5] P Antonik M C Wicks H D Griffiths and C J BakerldquoFrequency diverse array radarsrdquo in Proceedings of the IEEERadar Conference (RADAR rsquo06) pp 215ndash217 Verona NY USAApril 2006
[6] P Antonik M CWicks H D Griffiths and C J Baker ldquoMulti-mission multi-mode waveform diversityrdquo in Proceedings of theIEEE Radar Conference (RADAR rsquo06) pp 580ndash582 Verona NYUSA April 2006
[7] P Antonik H D Griffiths and C J Baker ldquoRange depen-dent beamforming using element level waveform diversityrdquo inProceedings of the International Waveform Diversity and DesignConference pp 1ndash4 Las Vegas Nev USA January 2006
[8] P Baizert T B HaleM A Temple andM CWicks ldquoForward-looking radar GMTI benefits using a linear frequency diversearrayrdquo Electronics Letters vol 42 no 22 pp 1311ndash1312 2006
[9] B W Jung R S Adve and J Chun ldquoFrequency diversity inmultistatic radarsrdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo08) pp 1ndash6 Rome Italy May 2008
[10] M Secmen S Demir A Hizal and T Eker ldquoFrequencydiverse array antenna with periodic time modulated pattern inrange and anglerdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo07) pp 427ndash430 Boston Mass USA April 2007
[11] J Huang K F Tong and C J Baker ldquoFrequency diversearray with beam scanning featurerdquo in Proceedings of the IEEEAntennas and Propagation Conference (AP-S rsquo08) pp 1ndash4 SanDiego Calif USA July 2008
[12] T Higgins and S D Blunt ldquoAnalysis of range-angle coupledbeamforming with frequency-diverse chirpsrdquo in Proceedings ofthe International Waveform Diversity and Design Conference(WDD rsquo09) pp 140ndash144 Orlando Fla USA February 2009
[13] J Farooq M A Temple and M A Saville ldquoApplication of fre-quency diverse arrays to synthetic aperture radar imagingrdquo inProceedings of the International Conference on Electromagneticsin Advanced Applications (ICEAA rsquo07) pp 447ndash449 TorinoItaly September 2007
[14] J Farooq M A Temple and M A Saville ldquoExploiting fre-quency diverse array processing to improve SAR image reso-lutionrdquo in Proceedings of the IEEE Radar Conference (RADARrsquo08) pp 1ndash5 Rome Italy May 2008
[15] W QWang ldquoPhased-MIMO radar with frequency diversity forrangedependent beamformingrdquo IEEE Sensors Journal vol 13no 8 pp 1320ndash1328 2013
[16] A L Swindlehurst and P Stoica ldquoMaximum likelihood meth-ods in radar array signal processingrdquo Proceedings of the IEEEvol 86 no 2 pp 421ndash441 1998
[17] J Ward ldquoCramer-Rao bounds for target angle and Dopplerestimation with space-time adaptive processing radarrdquo in Pro-ceedings of the 29th Asilomar Conference on Signals Systems andComputers pp 1198ndash1202 Pacific Grove Calif USA November1995
[18] A Dogandzic and A Nehorai ldquoCramer-Rao bounds for esti-mating range velocity and direction with an active arrayrdquo IEEETransactions on Signal Processing vol 49 no 6 pp 1122ndash11372001
[19] A Dogandzic and A Nehorai ldquoEstimating range velocity anddirection with a radar arrayrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo99) pp 2773ndash2776 Phoenix Ariz USA March 1999
[20] J J Zhang G Maalouli A P Suppappola and D MorrellldquoCramer-Rao lower bounds for the joint estimation of targetattributes using MIMO radarrdquo in Proceedings of the Interna-tional Waveform Diversity and Design Conference (WDD rsquo09)pp 103ndash107 Orlando Fla USA February 2009
[21] A Hassanien S A Vorobyov and A B Gershman ldquoMovingtarget parameters estimation in noncoherent MIMO radarsystemsrdquo IEEE Transactions on Signal Processing vol 60 no 5pp 2354ndash2361 2012
[22] Q He R S Blum and A M Haimovich ldquoNoncoherent MIMOradar for location and velocity estimation more antennasmeans better performancerdquo IEEE Transactions on Signal Pro-cessing vol 58 no 7 pp 3661ndash3680 2010
[23] S M Kay Fundamentals of Statistical Signal Processing Estima-tion Theory vol 1 Pearson London UK 2nd edition 2011
[24] H L van TreesOptimum Array Processing JohnWiley amp SonsNew York NY USA 2002
[25] D Wilcox and M Sellathurai ldquoOn MIMO radar subarrayedtransmit beamformingrdquo IEEE Transactions on Signal Processingvol 60 no 4 pp 2076ndash2081 2012
[26] WQWang andH Z Shao ldquoRange-angle localization of targetsby a double-pulse frequency diverse array radarrdquo IEEE Journalon Selected Topics in Signal Processing vol 8 no 1 pp 106ndash1142014
[27] S Gogineni and A Nehorai ldquoTarget estimation using sparsemodeling for distributed MIMO radarrdquo IEEE Transactions onSignal Processing vol 59 no 11 pp 5315ndash5325 2011
[28] T Li andA Nehorai ldquoMaximum likelihood direction finding inspatially colored noise fields using sparse sensor arraysrdquo IEEETransactions on Signal Processing vol 59 no 3 pp 1048ndash10622011
[29] S Sen ldquoOFDMradar space-time adaptive processing by exploit-ing spatio-temporal sparsityrdquo IEEE Transactions on SignalProcessing vol 61 no 1 pp 118ndash130 2013
[30] J J Blanz A Papathanassiou M Haardt I Furio and P WBaier ldquoSmart antennas for combined DOA and joint channelestimation in time-slotted CDMA mobile radio systems withjoint detectionrdquo IEEE Transactions onVehicular Technology vol49 no 2 pp 293ndash306 2000
[31] P N Pathirana S C K Herath and A V Savkin ldquoMultitargettracking via space transformations using a single frequencycontinuous wave radarrdquo IEEE Transactions on Signal Processingvol 60 no 10 pp 5217ndash5229 2012
International Journal of Antennas and Propagation 15
[32] P H Leong T D Abhayapala and T A Lamahewa ldquoMultipletarget localization using wideband echo chirp signalsrdquo IEEETransactions on Signal Processing vol 61 no 16 pp 4077ndash40892013
[33] L Zhuang X Liu and W Yu ldquoPrecisely beam steering forfrequency diverse arrays based on frequency offset selectionrdquoin Proceedings of the International Radar Conference (RADARrsquo09) pp 1ndash4 Bordeaux France December 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
2 International Journal of Antennas and Propagation
(CRLB) expressions for estimating velocity direction andDoppler shift are derived in [16 17] By assuming a narrow-band signal model prior to matched filtering the CRLBs forestimating the time delay direction and Doppler shift with aphased-array radar are detailed in [18 19] Furthermore theCRLB for jointly estimating the attributes of a moving targetusing MIMO radar is derived in [20ndash22]
Different from the above literatures this paper derivesthe CRLBs for jointly estimating the range direction andDoppler shift with an FDA radar Two typical data mod-els namely pre- and postmatched filtering are analyzedseparately for static and moving targets respectively Sincethe range and angle of targets cannot be estimated directlydue to the range-angle coupling we use a simple transmitsubaperturing strategy for the FDA radar This methoddivides the whole array elements into two subarrays eachuses a different frequency increment In doing so the rangeand angle of targets can be solely estimated The corre-sponding CRLBs for estimating direction range and veloc-ity are derived Furthermore the impacts of the elementsnumber and frequency increment on the CRLBs are alsoinvestigated
The rest of the paper is organized as follows The FDAis briefly introduced in Section 2 In Section 3 we considerthe data model after matched filtering and derive the CRLBexpressions for estimating the range and direction In orderto decouple the range-angle coupling response we divide the
whole array into two subarrays The corresponding CRLBsare derived In Section 4 the data model prior to matchedfiltering is investigated and the CRLB expressions for estimat-ing the direction range and Doppler shift are derived wheretemporally white Gaussian noise and chirp pulse signal areassumed Finally extensive simulation results are provided inSection 5 This paper is concluded in Section 6
2 Frequency Diverse Array (FDA)
In conventional phased-array radars it is assumed that anidentical waveform is radiated from each element excludingthe amplitudes and phases Different from phased-arrayradars FDA elements can be either excited by the samewaveform or different waveforms For simplicity and withoutloss of generality we consider a uniform linear array (ULA)FDA and assume that the waveforms radiated from eachantenna element are identical but with a frequency incrementof Δ119891HzThat is the radiation frequency of the119898th elementis
119891119898= 119891
0+ (119898 minus 1) sdot Δ119891 119898 = 0 1 119872 minus 1 (1)
where 1198910is the radar carrier frequency and119872 is the number
of the elementsSupposing a narrowband transmit signal and taking the
first element as the reference for the array the steering vectoris given by the following equation [11]
a (120579 119903) = [1 119890minus119895((21205871198910119889 sin 120579119888)+((2120587Δ119891sdot119889 sin 120579)119888)minus(2120587119903Δ119891119888))
sdot sdot sdot 119890minus119895(((21205871198910(119872minus1)119889 sin 120579)119888)+((2120587sdot(119872minus1)
2Δ119891sdot119889 sin 120579)119888)minus((2120587119903(119872minus1)Δ119891)119888))]
119879
(2)
where 120579 is the direction 119903 is the range 119889 is the elementspacing 119888 is the speed of light and 119879 is the transpose Since1198910
≫ Δ119891 and 119903 ≫ (119872 minus 1)119889 sin(120579) in an amplitude
sense an approximation can be taken by ignoring the phaseterm 2120587 sdot (119898 minus 1)
2
Δ119891 sdot 119889 sin(120579)119888 because it has ignorableimpacts In this case the steering vector can be simplifiedto
a (120579 119903) asymp [1 119890minus119895((21205871198910119889 sin(120579)119888)minus((2120587sdotΔ119891sdot119903)119888))
sdot sdot sdot 119890minus119895(((21205871198910(119898minus1)119889 sin(120579))119888)minus((2120587sdot(119898minus1)Δ119891sdot119903)119888))
]119879
(3)
The FDA characteristics can be summarized as follows[7ndash12] (1) If the frequency offset Δ119891 is fixed the beamdirection will vary as a function of the range 119903 that is it isa range- dependent beam (2) If the range is 119903 fixed the beamdirection will vary as a function of Δ119891 This means that theFDA is frequency-increment-dependent (3) If the frequencyincrement across the array is not applied (ie Δ119891 = 0)the corresponding FDA is just a conventional ULA phasedarray
3 CRLB with Signal Model afterMatched Filtering
31 Data Models Suppose that a target is located at (120579 119903)After matched filtering the baseband equivalent of thecomplex valued signals at the receiver can be expressed as
y = 1205730a (120579 119903) + n = (120595) + n (4)
International Journal of Antennas and Propagation 3
where1205730is a constant for a given target (120595) = 120573
0a(120579 119903)with
120595 = [120579 119903]119879 and n is a zero-mean complex Gaussian white
noise with spatial covariance
119864 [nnH] = Rn = 120590
2
119899I119872 (5)
where 119864[sdot] is the expectation operator H is the conjugatetranspose Rn is the spatial noise covariance matrix 1205902
119899is the
noise power and I119872is an119872times119872 identity matrix
32 CRLBs of Angle and Range Estimations In the followingwe discuss the CRLBs for several different cases separately
321 Range Is Known and Angle Is Unknown In this case120595 = 120579 The corresponding Fisher information matrix (FIM)is I
120595120595[23]
I120595120595
= 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) (6)
where Resdot is the real part of the signal 120595119894is the 119894th element
of 120595 and the119863120595119894(120595) is [24]
119863120595119894(120595) =
120597 (120595)
120597120595119894
(7)
Under the signal model (4) the FDA FIM with respect to 120579 isexpressed as
119868120579120579FDA
= 81205872
1198892cos2 (120579)
sdot SNRsum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(8)
where SNR is the signal-to-noise ratio (SNR) Accordinglythe CRLB is
CRLB120579120579FDA
= 119868minus1
120579120579FDA (9)
where minus1 denotes the inverse matrix In particular whenΔ119891 = 0 it is simplified to
119868120579120579phased-array
= 8SNR1205872
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(10)
Proof See Appendix A
It can be easily proved that
CRLB120579120579FDA
lt CRLB120579120579phased-array
(11)
That is to say the FDA radar has better CRLB for angleestimation than phased-array radar
322 Angle Is Known and Range Is Unknown As phased-array radar has range-independent beam here we onlycalculate the range estimation for FDA radar The FIM withrespect to 119903 can be expressed as
119868119903119903FDA
= 2SNR4120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(12)
The corresponding CRLB is
CRLB119903119903FDA
= 119868minus1
119903119903FDA (13)
Proof See Appendix B
323 Both Angle and Range Are Unknown In this case the 120579and 119903 CRLBs can be similarly determined as
CRLB120579120579FDA
= (1198882
sdot (
119872
sum
119898=1
(119898 minus 1)2
))
times (2SNR sdot 41205872
1198892
sdot (Δ119891)2cos2 (120579)
times (
119872
sum
119898=1
(119898 minus 1)4
119872
sum
119898=1
(119898 minus 1)2
minus(
119872
sum
119898=1
(119898 minus 1)3
)
2
))
minus1
CRLB119903119903FDA
= (1198884
sdot (sum
119872
119898=1(119898 minus 1)
2
1205822
+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888))
times (2SNR sdot 41205872
sdot (Δ119891)4
times (
119872
sum
119898=1
(119898 minus 1)4
119872
sum
119898=1
(119898 minus 1)2
minus(
119872
sum
119898=1
(119898minus1)3
)
2
))
minus1
(14)
Proof See Appendix C
When the targetrsquos range and angle are estimated jointlythe range and angle CRLBs will be significantly degradeddue to the range-angle coupling (which will be furtherinvestigated in Section 5) Consequently the range and angle
4 International Journal of Antennas and Propagation
d
1 2 N
middot middot middot
1 2 N
middot middot middot
Subarray 1 Subarray 2
Figure 1 Illustration of transmit subaperturing FDA radar
of targets cannot be estimated directly by a standard FDAradar To overcome this problem we present a transmitsubaperturing method for the FDA radar
324 CRLBs of Transmit Subaperturing FDA Radar Todecouple the range and angle peaks and estimate both therange and angle of target we divide the whole array into twoequal subarrays [25 26] Suppose that the number of elementsis 119872 each subarray has 119873 elements namely 119872 = 2119873The first subarray uses the frequency increment of Δ119891
1 and
the second subarray uses the frequency increment of Δ1198912
as shown in Figure 1 The resulting system is referred to astransmit subaperturing FDA (TS-FDA) radar
Taking the first element of the first subarray as thereference the new steering vector can be represented as thefollowing equation
b (120579 119903) = [1 119890minus1198951205931 sdot sdot sdot 119890
minus119895(119873minus1)1205931 119890minus119895120593119897 119890
minus119895(1205932+120593119873) sdot sdot sdot 119890minus119895(119873minus1)1205932+120593119873]
119879
(15)
where
1205931= (
2120587119889 sin (120579)120582
) minus (2120587119903 sdot Δ119891
1
119888) (16a)
120593119873= (
2120587119873119889 sin (120579)120582
) (16b)
1205932= (
2120587119889 sin (120579)120582
) minus (2120587119903 sdot Δ119891
2
119888) (16c)
The corresponding FIM can be derived as
ITS-FDA = 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) = [
119868120579120579
119868120579119903
119868119903120579
119868119903119903
] (17)
where
119868120579120579
= 2
10038161003816100381610038161205730
1003816100381610038161003816
2
1205902
119899
41205872
1198892cos2 (120579)1205822
2119873
sum
119898=1
(119898 minus 1)2
(18a)
119868119903119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2
1205902
119899
41205872
(Δ1198912
1+ Δ119891
2
2)
1198882
119873
sum
119898=1
(119898 minus 1)2
(18b)
119868120579119903= 119868
119903120579= minus
210038161003816100381610038161205730
1003816100381610038161003816
2
1205902
119899
41205872
119889 cos (120579)120582119888
times [(Δ1198911+ Δ119891
2)
119873
sum
119898=1
(119898 minus 1)2
+ Δ1198912119873
119873
sum
119898=1
(119898 minus 1)]
(19)
The angle and range CRLBs are the first two diagonalelements of the inverse of the FIM
CRLB120579120579TS-FDA
= [Iminus1
TS-FDA]11
(20a)
CRLB119903119903TS-FDA
= [Iminus1
TS-FDA]22
(20b)
where [sdot]119894119895is the element at the 119894th row and 119895th column of the
matrix
4 CRLB with Signal Model prior toMatched Filtering
41 Data Models Unlike the first data model the data modelprior to matched filtering allows to estimate the Doppler shift(velocity) Suppose an 119872-element antenna array receives atime delayed and Doppler-shifted echo of the transmittedsignal 119904(119905) exp(119895Ω
119888119905) where Ω
119888is the center frequency
Knowing the time delay and Doppler shift Ω119863(assuming
a target with constant radial velocity) the range and radialcomponent of velocity can be determined by 119903 = 1198881205912
and V = Ω119863119888(2Ω
119888) We denote the continuous-time signal
119904(119905) as 119904[119897] = 119904(119897 sdot Δ119905) where Δ119905 is the sampling intervalCorrespondingly the time delay and Doppler shift in thesampled signal domain are 119897
120591= 120591Δ119905 and 119908
119863= Ω
119863sdot Δ119905
respectively [27]After converting to baseband and sampling the received
signal at time 119897 sdot Δ119905 becomes
y [119899] = 120573 sdot a (120579 119903) sdot 119904 [119897 minus 119897120591]exp (119895119908
119863119897) + n [119897] 119897 = 1 119871
(21)
where 120573 is complex amplitude of the signal and n[119897] is theadditive noise [28]
We assume that the snapshots taken at 119897 = 1 119871 coverthe whole of a coherent processing interval (CPI) Thereforethe time duration of the CPI is 119879CPI = 119871 sdot Δ119905 Under thedata model of (21) the complex amplitude 120573 is assumed tobe an unknown deterministic constant during the CPI Tomodel the Doppler effect with a frequency shift the radialcomponent of the target velocity needs to be much smallerthan the propagation speed (ie V119888 ≪ 1) Then the time-bandwidth product of the complex envelope should be largerthan 1 In addition it is assumed that the propagation time ofthe signal across the array is much smaller than the reciprocal
International Journal of Antennas and Propagation 5
of the signal bandwidth which is the narrowband arrayassumption in array processing
Define the vector of unknown target parameters as120581 = [Re120573 Imag120573 120579 120578119879
] Stacking all samples into a singlevector (21) can be rewritten as
z = 120573 sdot 120601 (120578) otimes a (120579 119903) + n = (120581) + n (22)
where otimes denotes the Kronecker product (120581) =
120573 sdot 120601(120578) otimes a(120579 119903)
120601 (120578) = [119904 [1 minus2119903
119888 sdot Δ119905] exp (119895119908
1198631)
119904 [2 minus2119903
119888 sdot Δ119905] exp (119895119908
1198632)
119904 [119871 minus2119903
119888 sdot Δ119905] exp (119895119908
119863119871)]
119879
(23)
The noise n is assumed to be zero-mean Gaussian spatiallyand temporally correlated with spatiotemporal covariance[29]
119864 [nnH] = Cn otimes Rn (24)
where Cn is the temporal noise covariance matrixUnder the above data model the signal and noise param-
eters are disjoint and satisfy the space-time separability [30]
42 CRLBs of Angle Range and Doppler Shift Suppose thereceived signal is completely covered by the observations119897 = 1 119871 and the sampling is dense (ie Δ119905 rarr 0) TheRn is assumed to be constant in the frequency band 119891 isin
(minus1(2Δ119905) 1(2Δ119905)) where 119891 denotes the frequency in thecontinuous-time domain The vector of Doppler shift andtime delay in the continuous-time domain is defined as
120578 = [119903 Ω119863]119879
(25)
For Cn = I119871and Rn = 120590
2
119899I119872
with I119871being an 119871 times 119871
identity matrix We define the signal power s = 120601H 120601 The
CRLBs expression for 120579 and 120578 follows fromAppendix D as thefollowing
CRLB120579120578120579120578FDA
=
1205902
119899
210038161003816100381610038161205731003816100381610038161003816
2s
times
[[[[[[[[[[[
[
12058721198892cos2 (120579)119872(119872
2minus 1)
[
[
1
31205822
+
(Δ119891)2[(2119872minus 1) (8119872
2minus 3119872minus 11)]
451198882(119872+ 1)
+
2Δ119891 (119872minus 1)
3119888120582
]
]
minus1205872119889Δ119891 cos (120579)119872(119872
2minus 1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 0
minus1205872119889Δ119891 cos (120579)119872(119872
2minus 1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 1205872(Δ119891)2119872(119872
2minus 1)
31198882
+
12058511
sImag 12058512
s
0
Imag 12058512
s12058522
s
]]]]]]]]]]]
]
minus1
(26)
where
12058511
= int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905
minus1
s[int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905]
2
(27a)
12058522
= int
infin
minusinfin
1199052
|119904 (119905 minus 120591)|2
119889119905
minus1
s[int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
2
(27b)
12058512
= int
infin
minusinfin
119905119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905
minus [1
sint
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905
sdot int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905]
(28)
The terms 12058511
and 12058522
are proportional to the rootmean squared (RMS) bandwidth and RMS duration of 119904(119905)respectively Note that the decoupling is a consequence ofthe assumed space-time separability of signal and noisemodels and the assumption of the complex amplitude 120573 as anunknown deterministic constant In addition since the signaland additive noise parameters are disjoint the above CRLBexpressions hold regardless of whether the spatiotemporalnoise covariance Cn otimes Rn is known or unknown
43 CRLBs When Chirp Pulse Signal Is Employed In thissection we derive the CRLB expressions for the rangeDoppler shift and direction estimation when 119904(119905) is a chirppulse signal with a large time-bandwidth product [31 32] Itcan be expressed as
119904 (119905) =
119875minus1
sum
119901=0
1199040(119905 minus 119901119879
119901) (29)
6 International Journal of Antennas and Propagation
where 119879119901is the pulse repetition interval 119875 is the number of
chirp pulses and 1199040(119905) is expressed as [12]
1199040(119905) = exp [119895120587 119861
1198790
(119905 minus1
21198790)
2
] sdot [ℎ (119905) minus ℎ (119905 minus 1198790)]
(30)
where1198790is the chirp pulse duration119861 is the chirp bandwidth
and ℎ(119905) is the Heaviside step function
Assume the time-bandwidth product of the pulse is1198790sdot 119861 ≫ 1 Using the signal given in (29) and (30) in
continuous-time domain we obtain the signal power s =
1198751198790 120585
11= 1198754120587
2
1198612
11987903119888
2 Imag12058512 = minus(119875120587119861119879
2
03119888) and
12058522
= (119875MT3
012)[1 + (119879
0119879
119877)2
(1198752
minus 1)] Thus the CRLBexpressions of 120579 and 120578 for FDA are derived as the followingequation
CRLB120579120578120579120578FDA
=
1
2SNR
sdot
[[[[[[[[[[[[
[
12058721198892cos2(120579)119872(119872
2minus 1)
[
[
1
31205822+
(Δ119891)2[(2119872minus 1) (8119872
2minus3119872minus11)]
451198882119872(119872+ 1)
+
2Δ119891 (119872minus 1)
3119888120582
]
]
minus1205872119889Δ119891 cos(120579)119872(119872
2minus1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 0
minus1205872119889Δ119891 cos(120579)119872(119872
2minus 1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 1205872(Δ119891)2119872(119872
2minus1)
31198882
+
412058721198612119872
31198882
minus
1198721205871198611198790
3119888
0 minus
1198721205871198611198790
3119888
1198721198792
0
12
[1+(
119879119877
1198790
)
2
(1198752minus1)]
]]]]]]]]]]]]
]
minus1
(31)
Specially for the phased array (ie Δ119891 = 0) the CRLB is
CRLB120579120578120579120578phased-array
=1
2SNR
times
[[[[[[[[
[
1205872
1198892cos2 (120579)119872(119872
2
minus 1)
312058220 0
04120587
2
1198612
119872
31198882minus119872120587119861119879
0
3119888
0 minus119872120587119861119879
0
3119888
1198721198792
0
12[1 + (
119879119877
1198790
)
2
(1198752
minus 1)]
]]]]]]]]
]
minus1
(32)
When only one pulse (119875 = 1) is used the CRLBin (32) will be infinite because the model is not identifi-able and the range and Doppler shift cannot be uniquelyestimated [18] since phase-array radar has no rangeidentity capability In contrast (31) does not have sucha problem
According to the previous discussion the angle and rangeof targets cannot be estimated jointly due to the range-anglecoupling Using the similar transmit subaperturing approachpresented in Section 3 and steering vector b(120579 119903) (15) theCRLBs of angle range and Doppler shift are derived as inthe following equation
CRLB120579120578120579120578TS-FDA =
1
2SNR
times
[[[[[[[[[[[[[[[[[[[
[
412058721198892cos2 (120579)1205822
2119873
sum
119898=1
(119898 minus 1)2
minus
41205872119889 cos (120579)120582119888
[[[[
[
(Δ1198911 + Δ1198912)
119873
sum
119898=1
(119898 minus 1)2
+119873Δ1198912
119873
sum
119898=1
(119898 minus 1)
]]]]
]
0
minus
41205872119889 cos (120579)120582119888
[[[[
[
(Δ1198911 + Δ1198912)
119873
sum
119898=1
(119898 minus 1)2
+119873Δ1198912
119873
sum
119898=1
(119898 minus 1)
]]]]
]
(Δ1198912
1+ Δ1198912
2)
1198882
119873
sum
119898=1
(119898 minus 1)2+
412058721198612119872
31198882
minus
1198721205871198611198790
3119888
0 minus
1198721205871198611198790
3119888
1198721198792
0
12
[1 + (
119879119877
1198790
)
2
(1198752minus 1)]
]]]]]]]]]]]]]]]]]]]
]
minus1
(33)
International Journal of Antennas and Propagation 7
20151050
10minus2
10minus3
10minus47001 7002 7003
10minus248203
10minus248201
10minus248199
CRLB
of a
ngle
estim
atio
n (d
eg)
SNR (dB)
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
Figure 2 CRLB for estimating angle versus SNR when the range isknown
5 Simulation and Verification
In this section we consider several numerical examples thatcompare the CRLBs in different signal and noise modelsConsider an X-band FDA radar with the carrier frequency1198910
= 10GHz We assume a ULA of 119872 elements usedfor transmitting The array elements are spaced half of thewavelength apart from each other namely 119889 = 1205822 Onetarget of interest is supposed to reflect a plane wave thatimpinges on the array from direction of angle 120579 = 30
∘Under the signal model after matched filtering Figures
2 and 3 compare the CRLBs according to (9) and (13)respectively It can be noticed that the CRLBs are improvedwhen a larger number of elements are employed Howeverit has no significant difference when different frequencyincrements are used This is because sum
119872
119898=1(119898 minus 1)
2
1205822
≫
(Δ119891)2
sum119872
119898=1(119898 minus 1)
4
1198882
+ 2Δ119891sum119872
119898=1(119898 minus 1)
3
120582119888 thus thefrequency increment has a small impact on the CRLBsIn [33] a frequency offset selection strategy is derived toprecisely steer the beam toward a fixed range with a desiredangle
Figure 4 shows the CRLBs of angle and range whenboth the angle and range are unknown The CRLBs aresignificantly degraded due to the range-angle coupling Con-sequently the range and angle of targets cannot be estimateddirectly by the FDA radar However the CRLBs decreaseas the increase of the number of elements and frequencyincrement still holds Moreover generally more elementsmean that better CRLBs performance can be achieved for theFDA radar
To overcome the problem that the range and angle oftargets cannot be estimated directly by the FDA radar weuse the transmit subaperturing strategy on the transmitfrequency increments Figure 5 shows the corresponding
CRLB
of r
ange
estim
atio
n (m
)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
102
101
10020151050
SNR (dB)
Figure 3 CRLB for estimating range versus SNR when the angle isknown
CRLBs where 119872 = 32 is assumed It can be noticed that toobtain a lower CRLB the Δ119891
1and Δ119891
2should have inverse
signs that is one is passive and the other is negative Onereason is that in this case the FDA radar has a wider systembandwidth Figure 6 shows the CRLBs of angle and rangewhen 119872 = 20 is employed It can be noticed that theCRLBs performance improves with the increase of the sensornumber
Under the data model prior to matched filtering wesuppose the following signal parameters bandwidth 119861 =
10MHz repetition period 119879119901
= 1ms and pulse duration1198790
= 250 120583s In this case the approximate expressionsgiven in (31) are valid because the transmitted has a largetime-bandwidth product (119879
0sdot 119861 = 2500 ≫ 1) Figure 7
shows the CRLBs for direction range and Doppler shift asa function of SNR Note that when SNR = minus10 dB and119872 = 32 are employed we can get CRLB
119903119903FDA= 653m
and CRLBΩ119863Ω119863FDA
= 136937 rads that corresponds to theCRLB for velocity is 327 cms (since V = Ω
119863119888(2Ω
119888) and
Ω119888= 2120587119891
119888) Since (Δ119891)2 sum119872
119898=1(119898 minus 1)
4
1198882
≪ 1 the frequencyincrement has a small impact on the CRLBs In additionobserve that the CRLB for 120579 and 120578 is block-diagonal (see(32)) and therefore decoupled that is CRLB
120578120578FDAremains the
same whether or not 120579 is known and similarly CRLB120579120579FDA
is the same whether or not 120578 is known The decouplingis a consequence of the assumed space-time separability ofsignal and noise models and the assumption of the complexamplitude 120573 as an unknown deterministic constant
Figure 8 shows that the CRLBs versus SNR for differentcombinations of Δ119891
1and Δ119891
2 Comparing Figures 8 and
7 the CRLBs have been significantly improved Likewisecomparing Figures 8 and 9 theCRLBs performance improvesas the number of elements increases
8 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
104
103
102
101
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
108
107
106
105
10420151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
Figure 4 Both angle and range are unknown
6 Conclusion
In this paper we derive the CRLB to jointly estimate theattributes of a moving target using FDA radar and computethe corresponding CRLB expressions First we briefly intro-duce the FDA concept and make a summary on the FDAcharacteristics Then we consider two different data modelsnamely pre- and postmatched filtering Under differentsignal and noise models we compute the CRLB expressionsfor estimating the range direction and Doppler shift Wedemonstrate that the FDA radar beamforming is coupledin range and angle and that the targetrsquos range and anglecannot be estimated directly by the FDA radar To overcome
20151050
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
CRLB
of r
ange
estim
atio
n (m
)102
101
100
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
Figure 5 CRLBs of TS-FDA radar with119872 = 32
this problem this paper proposes a transmit subaperturingstrategy for the FDA radar In doing so the range and angle oftargets are estimated from the transmit-receive beamformingoutput Moreover we also specialize the CRLB results tothe case of temporally white noise and a chirp pulse signalExtensive simulation results verify the correctness of thederived CRLBs It is shown that the CRLBs decrease with theincrease of the number of elements and frequency incrementThe CRLBs can be further improved through three aspectsincreasing the number of elements enhancing the systembandwidth by employing a larger frequency increment andusing transmit subaperturing strategy with more subarrays
International Journal of Antennas and Propagation 9
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
CRLB
of r
ange
estim
atio
n (m
)
103
102
101
10020151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
Figure 6 CRLBs of TS-FDA radar with119872 = 20
Appendices
A Derive the CRLB for Angle WhenRange Is Known
To derive the CRLB we start with a well-known expressionfor the FIM under the data model in Section 3 We define thespatial noise covariance matrix as Rn = 120590
2
119899I119872
and signal-to-noise ratio (SNR) as SNR = |120573
0|2
1205902
119899 Suppose the target range
is known the FIM of 120579 is
119868120579120579
= 2Re 119863H120579(120595) (Rminus1
n )119863120579(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
(A1)
For a phased-array radar there is (120595) = 1205730a(120579) We then
have
120597a (120579)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579) (A2)
whereD = diag[0 1 119872 minus 1] and
120597aH (120579)
120597120579
120597a (120579)120597120579
=4120587
2
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A3)
The FIM of the phased-array radar is
119868120579120579phased-array
= 2SNR41205872
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A4)
Similarly for the FDA radar there is (120595) = 1205730a(120579 119903)The
derivation of a(120579 119903) with respect to 120579 is
120597a (120579 119903)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579 119903)
minus 1198952120587119889Δ119891 cos (120579)
119888
times diag [0 1 (119872 minus 1)2
] a (120579 119903)
120597aH (120579 119903)
120597120579
120597a (120579 119903)120597120579
= 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822
+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A5)
Accordingly the FIM of 120579 for the FDA can be expressedas
119868120579120579FDA
= 2SNR sdot 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A6)
10 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
(c) CRLB for Doppler shift versus SNR
Figure 7 General CRLB results of FDA radar
B Derive the CRLB for Range WhenAngle Is Known
Under the data model in Section 3 when the direction 120579 isknown the parameter to be estimated is 119903 The FIM of 119903 is
119868119903119903FDA
= 2Re 119863H119903(120595) (Rminus1
n )119863119903(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597aH (120579 119903)
120597119903Rminus1
n120597a (120579 119903)
120597119903
(B1)
The derivation of a(120579 119903) with respect to 119903 for FDA is
120597a (120579 119903)120597119903
= 1198952120587Δ119891
119888Da (120579 119903)
120597aH (120579 119903)
120597119903
120597a (120579 119903)120597119903
=4120587
2
Δ1198912
1198882
119872
sum
119898=1
(119898 minus 1)2
(B2)
The FIM of 119903 is thus given by
119868119903119903FDA
= 2SNR4120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(B3)
International Journal of Antennas and Propagation 11
C Derive the CRLB for Range and Angle
Under the data model in Section 3 both the angle and rangeare unknown The range and angle of targets are estimatedjointly The FIM for parameters 120579 and 119903 can be expressed as
IFDA = 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) = [
119868120579120579
119868120579119903
119868119903120579
119868119903119903
] (C1)
We then have
IFDA = 210038161003816100381610038161205730
1003816100381610038161003816
2
times
[[[[
[
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
]]]]
]
(C2)
Since (120597H(120595)120597119903)Rminus1
n (120597(120595)120597120579) = (120597H(120595)
120597120579)Rminus1
n (120597(120595)120597119903) then 119868120579119903= 119868
119903120579 We can get
119868120579120579
= 210038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
= 81205872
1198892cos2 (120579)
times SNRsum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
119868120579119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
= minus81205872
119889Δ119891 cos (120579)
times SNR[sum
119872
119898=1(119898 minus 1)
2
120582119888+Δ119891sum
119872
119898=1(119898 minus 1)
3
1198882]
119868119903119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
= SNR8120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(C3)
Since CRLB120579120579FDA
= [Iminus1
FDA]11 CRLB119903119903FDA
= [Iminus1
FDA]22 Theexpressions for the CRLB of angle and range estimationsgiven in (14) can be obtained by substituting (C3) into (C2)
D General CRLB Results
Under the data model in Section 4 we derive the continuousCRLB expressions for temporally and spatially white noiseand denote SNR = |120573|
2
s1205902
119899with s being the power We
also start with FIM
I120581119894120581119895
= 2Re 119863H120581119894
(120581) (Cminus1
n otimes Rminus1
n )119863120581119895(120581) (D1)
where 120581119894is the 119894th element of 120581 and 119863
120581119894(120595) = 120597(120581)120597120581
119894
Consider
IFDA =
[[[[[
[
119868120573120573
119868119879
120579120573119868119879
120578120573
119868120579120573
119868120579120579
119868119879
120578120579
119868120578120573
119868120578120579
119868120578120578
]]]]]
]
(D2)
For clarity we rewrite Fisherrsquos information matrix I as
IFDA = [A UV B] (D3)
where
V = U119879
(D4a)
V = [119868120579120573
119868120578120573
] (D4b)
B = [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] (D4c)
According to the matrix inversion lemma the inversematrix of IFDA is
Iminus1
FDA = [
[
(A minus UBminus1V)minus1
minusAminus1U(B minus VAminus1U)minus1
minusBminus1V(A minus UBminus1V)minus1
(B minus VAminus1U)minus1
]
]
(D5)
where the submatrix (A minus UBminus1V)minus1 corresponds to theCRLBof120573which is nuisance and (B minus VAminus1U)minus1 correspondsto the CRLB of 120579 and 120578 = [119903 Ω
119863]119879 which are of interest
CRLB120579120578120579120578FDA
= (B minus VAminus1U)minus1
= [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] minus [119868120579120573
119868120578120573
] 119868minus1
120573120573[119868
119879
120579120573119868119879
120578120573]
minus1
(D6)
where
119868120573120573
= 2 sdot[[[
[
s sdot 1198721205902
119899
0
0s sdot 1198721205902
119899
]]]
]
(D7a)
119868120579120573
= 2 sdot Re[1 119895] otimess1205902
119899
sdot 120573lowast
1198601 (D7b)
119868120578120573
= 2 sdot Re[1 119895] otimes120573
lowast
1205902
119908
sdot [119872 sdot 1198603+ s sdot 119860
4] (D7c)
119868120578120578
= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
[119872 sdot 1198606+ 119860
H3119860
4+ 119860
3119860
H4+ s sdot 119860
7]
(D7d)
12 International Journal of Antennas and Propagation
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHz
70001 70001 70001 70002100286514
100286515
100286515
kHz
(c) CRLB for Doppler shift versus SNR
Figure 8 General CRLB results of TS-FDA radar with119872 = 32
119868120579120579minus (119868
120579120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198602minus119860
1119860
H1
119872) (D8a)
119868120578120579minus (119868
120578120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198605minus119860
4119860
H1
119872) (D8b)
119868120578120578
minus (119868120578120573119868119879
120578120573)
1205902
119899
s119872
= 2
10038161003816100381610038161205731003816100381610038161003816
2
s1205902
119899
(119872119860
6
s+ 119860
7minus119872119860
3119860
H3
s2minus119860
4119860
H4
119872)
(D8c)
1198601= minus119895(
2120587119889 cos (120579)sum119872
119898=1(119898 minus 1)
120582
+2120587119889Δ119891 cos (120579)sum119872
119898=1(119898 minus 1)
2
119888)
(D9a)
1198602= 4120587
2
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(D9b)
International Journal of Antennas and Propagation 13
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(c) CRLB for Doppler shift versus SNR
Figure 9 General CRLB results of TS-FDA radar with119872 = 20
1198603= [int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905 int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
119879
(D9c)
1198604= [
2120587Δ119891sum119872
119898=1(119898 minus 1)
1198880]
119879
(D9d)
1198605= [minus4120587
2119889Δ119891 cos(120579) [
sum119872
119898=1(119898 minus 1)
2
12120582119888
+
Δ119891sum119872
119898=1(119898 minus 1)
3
121198882
] 0]
119879
(D9e)
1198606=[[
[
int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905 int
infin
minusinfin
119905119904H(119905 minus 120591)
119889119904 (119905 minus 120591)
119889119905
119889119905
int
infin
minusinfin
119905119904 (119905 minus 120591)
119889119904(119905 minus 120591)H
119889119905
119889119905 int
infin
minusinfin
1199052|119904 (119905 minus 120591)|
2119889119905
]]
]
(D9f)
1198607=[[
[
41205872
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
0
0 0
]]
]
(D9g)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework described in this paper was supported in part by theNational Natural Science Foundation of China under Grant
14 International Journal of Antennas and Propagation
41101317 the Program for New Century Excellent Talentsin University under Grant NCET-12-0095 Sichuan ProvinceScience Fund for Distinguished Young Scholars under Grant2013JQ0003 and Fundamental Research Fund for theCentralUniversities
References
[1] F Bandiera M Mancino and G Ricci ldquoLocalization strategiesfor multiple point-like radar targetsrdquo IEEE Transactions onSignal Processing vol 60 no 12 pp 6708ndash6712 2012
[2] D R Fuhrmann J P Browning and M Rangaswamy ldquoSignal-ing strategies for the hybrid MIMO phased-array radarrdquo IEEEJournal on Selected Topics in Signal Processing vol 4 no 1 pp66ndash78 2010
[3] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011
[4] M Compagnoni P Bestagini F Antonacci A Sarti and STubaro ldquoLocalization of acoustic sources through the fittingof propagation cones using multiple independent arraysrdquo IEEETransactions on Audio Speech and Language Processing vol 20no 7 pp 1964ndash1975 2012
[5] P Antonik M C Wicks H D Griffiths and C J BakerldquoFrequency diverse array radarsrdquo in Proceedings of the IEEERadar Conference (RADAR rsquo06) pp 215ndash217 Verona NY USAApril 2006
[6] P Antonik M CWicks H D Griffiths and C J Baker ldquoMulti-mission multi-mode waveform diversityrdquo in Proceedings of theIEEE Radar Conference (RADAR rsquo06) pp 580ndash582 Verona NYUSA April 2006
[7] P Antonik H D Griffiths and C J Baker ldquoRange depen-dent beamforming using element level waveform diversityrdquo inProceedings of the International Waveform Diversity and DesignConference pp 1ndash4 Las Vegas Nev USA January 2006
[8] P Baizert T B HaleM A Temple andM CWicks ldquoForward-looking radar GMTI benefits using a linear frequency diversearrayrdquo Electronics Letters vol 42 no 22 pp 1311ndash1312 2006
[9] B W Jung R S Adve and J Chun ldquoFrequency diversity inmultistatic radarsrdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo08) pp 1ndash6 Rome Italy May 2008
[10] M Secmen S Demir A Hizal and T Eker ldquoFrequencydiverse array antenna with periodic time modulated pattern inrange and anglerdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo07) pp 427ndash430 Boston Mass USA April 2007
[11] J Huang K F Tong and C J Baker ldquoFrequency diversearray with beam scanning featurerdquo in Proceedings of the IEEEAntennas and Propagation Conference (AP-S rsquo08) pp 1ndash4 SanDiego Calif USA July 2008
[12] T Higgins and S D Blunt ldquoAnalysis of range-angle coupledbeamforming with frequency-diverse chirpsrdquo in Proceedings ofthe International Waveform Diversity and Design Conference(WDD rsquo09) pp 140ndash144 Orlando Fla USA February 2009
[13] J Farooq M A Temple and M A Saville ldquoApplication of fre-quency diverse arrays to synthetic aperture radar imagingrdquo inProceedings of the International Conference on Electromagneticsin Advanced Applications (ICEAA rsquo07) pp 447ndash449 TorinoItaly September 2007
[14] J Farooq M A Temple and M A Saville ldquoExploiting fre-quency diverse array processing to improve SAR image reso-lutionrdquo in Proceedings of the IEEE Radar Conference (RADARrsquo08) pp 1ndash5 Rome Italy May 2008
[15] W QWang ldquoPhased-MIMO radar with frequency diversity forrangedependent beamformingrdquo IEEE Sensors Journal vol 13no 8 pp 1320ndash1328 2013
[16] A L Swindlehurst and P Stoica ldquoMaximum likelihood meth-ods in radar array signal processingrdquo Proceedings of the IEEEvol 86 no 2 pp 421ndash441 1998
[17] J Ward ldquoCramer-Rao bounds for target angle and Dopplerestimation with space-time adaptive processing radarrdquo in Pro-ceedings of the 29th Asilomar Conference on Signals Systems andComputers pp 1198ndash1202 Pacific Grove Calif USA November1995
[18] A Dogandzic and A Nehorai ldquoCramer-Rao bounds for esti-mating range velocity and direction with an active arrayrdquo IEEETransactions on Signal Processing vol 49 no 6 pp 1122ndash11372001
[19] A Dogandzic and A Nehorai ldquoEstimating range velocity anddirection with a radar arrayrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo99) pp 2773ndash2776 Phoenix Ariz USA March 1999
[20] J J Zhang G Maalouli A P Suppappola and D MorrellldquoCramer-Rao lower bounds for the joint estimation of targetattributes using MIMO radarrdquo in Proceedings of the Interna-tional Waveform Diversity and Design Conference (WDD rsquo09)pp 103ndash107 Orlando Fla USA February 2009
[21] A Hassanien S A Vorobyov and A B Gershman ldquoMovingtarget parameters estimation in noncoherent MIMO radarsystemsrdquo IEEE Transactions on Signal Processing vol 60 no 5pp 2354ndash2361 2012
[22] Q He R S Blum and A M Haimovich ldquoNoncoherent MIMOradar for location and velocity estimation more antennasmeans better performancerdquo IEEE Transactions on Signal Pro-cessing vol 58 no 7 pp 3661ndash3680 2010
[23] S M Kay Fundamentals of Statistical Signal Processing Estima-tion Theory vol 1 Pearson London UK 2nd edition 2011
[24] H L van TreesOptimum Array Processing JohnWiley amp SonsNew York NY USA 2002
[25] D Wilcox and M Sellathurai ldquoOn MIMO radar subarrayedtransmit beamformingrdquo IEEE Transactions on Signal Processingvol 60 no 4 pp 2076ndash2081 2012
[26] WQWang andH Z Shao ldquoRange-angle localization of targetsby a double-pulse frequency diverse array radarrdquo IEEE Journalon Selected Topics in Signal Processing vol 8 no 1 pp 106ndash1142014
[27] S Gogineni and A Nehorai ldquoTarget estimation using sparsemodeling for distributed MIMO radarrdquo IEEE Transactions onSignal Processing vol 59 no 11 pp 5315ndash5325 2011
[28] T Li andA Nehorai ldquoMaximum likelihood direction finding inspatially colored noise fields using sparse sensor arraysrdquo IEEETransactions on Signal Processing vol 59 no 3 pp 1048ndash10622011
[29] S Sen ldquoOFDMradar space-time adaptive processing by exploit-ing spatio-temporal sparsityrdquo IEEE Transactions on SignalProcessing vol 61 no 1 pp 118ndash130 2013
[30] J J Blanz A Papathanassiou M Haardt I Furio and P WBaier ldquoSmart antennas for combined DOA and joint channelestimation in time-slotted CDMA mobile radio systems withjoint detectionrdquo IEEE Transactions onVehicular Technology vol49 no 2 pp 293ndash306 2000
[31] P N Pathirana S C K Herath and A V Savkin ldquoMultitargettracking via space transformations using a single frequencycontinuous wave radarrdquo IEEE Transactions on Signal Processingvol 60 no 10 pp 5217ndash5229 2012
International Journal of Antennas and Propagation 15
[32] P H Leong T D Abhayapala and T A Lamahewa ldquoMultipletarget localization using wideband echo chirp signalsrdquo IEEETransactions on Signal Processing vol 61 no 16 pp 4077ndash40892013
[33] L Zhuang X Liu and W Yu ldquoPrecisely beam steering forfrequency diverse arrays based on frequency offset selectionrdquoin Proceedings of the International Radar Conference (RADARrsquo09) pp 1ndash4 Bordeaux France December 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 3
where1205730is a constant for a given target (120595) = 120573
0a(120579 119903)with
120595 = [120579 119903]119879 and n is a zero-mean complex Gaussian white
noise with spatial covariance
119864 [nnH] = Rn = 120590
2
119899I119872 (5)
where 119864[sdot] is the expectation operator H is the conjugatetranspose Rn is the spatial noise covariance matrix 1205902
119899is the
noise power and I119872is an119872times119872 identity matrix
32 CRLBs of Angle and Range Estimations In the followingwe discuss the CRLBs for several different cases separately
321 Range Is Known and Angle Is Unknown In this case120595 = 120579 The corresponding Fisher information matrix (FIM)is I
120595120595[23]
I120595120595
= 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) (6)
where Resdot is the real part of the signal 120595119894is the 119894th element
of 120595 and the119863120595119894(120595) is [24]
119863120595119894(120595) =
120597 (120595)
120597120595119894
(7)
Under the signal model (4) the FDA FIM with respect to 120579 isexpressed as
119868120579120579FDA
= 81205872
1198892cos2 (120579)
sdot SNRsum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(8)
where SNR is the signal-to-noise ratio (SNR) Accordinglythe CRLB is
CRLB120579120579FDA
= 119868minus1
120579120579FDA (9)
where minus1 denotes the inverse matrix In particular whenΔ119891 = 0 it is simplified to
119868120579120579phased-array
= 8SNR1205872
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(10)
Proof See Appendix A
It can be easily proved that
CRLB120579120579FDA
lt CRLB120579120579phased-array
(11)
That is to say the FDA radar has better CRLB for angleestimation than phased-array radar
322 Angle Is Known and Range Is Unknown As phased-array radar has range-independent beam here we onlycalculate the range estimation for FDA radar The FIM withrespect to 119903 can be expressed as
119868119903119903FDA
= 2SNR4120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(12)
The corresponding CRLB is
CRLB119903119903FDA
= 119868minus1
119903119903FDA (13)
Proof See Appendix B
323 Both Angle and Range Are Unknown In this case the 120579and 119903 CRLBs can be similarly determined as
CRLB120579120579FDA
= (1198882
sdot (
119872
sum
119898=1
(119898 minus 1)2
))
times (2SNR sdot 41205872
1198892
sdot (Δ119891)2cos2 (120579)
times (
119872
sum
119898=1
(119898 minus 1)4
119872
sum
119898=1
(119898 minus 1)2
minus(
119872
sum
119898=1
(119898 minus 1)3
)
2
))
minus1
CRLB119903119903FDA
= (1198884
sdot (sum
119872
119898=1(119898 minus 1)
2
1205822
+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888))
times (2SNR sdot 41205872
sdot (Δ119891)4
times (
119872
sum
119898=1
(119898 minus 1)4
119872
sum
119898=1
(119898 minus 1)2
minus(
119872
sum
119898=1
(119898minus1)3
)
2
))
minus1
(14)
Proof See Appendix C
When the targetrsquos range and angle are estimated jointlythe range and angle CRLBs will be significantly degradeddue to the range-angle coupling (which will be furtherinvestigated in Section 5) Consequently the range and angle
4 International Journal of Antennas and Propagation
d
1 2 N
middot middot middot
1 2 N
middot middot middot
Subarray 1 Subarray 2
Figure 1 Illustration of transmit subaperturing FDA radar
of targets cannot be estimated directly by a standard FDAradar To overcome this problem we present a transmitsubaperturing method for the FDA radar
324 CRLBs of Transmit Subaperturing FDA Radar Todecouple the range and angle peaks and estimate both therange and angle of target we divide the whole array into twoequal subarrays [25 26] Suppose that the number of elementsis 119872 each subarray has 119873 elements namely 119872 = 2119873The first subarray uses the frequency increment of Δ119891
1 and
the second subarray uses the frequency increment of Δ1198912
as shown in Figure 1 The resulting system is referred to astransmit subaperturing FDA (TS-FDA) radar
Taking the first element of the first subarray as thereference the new steering vector can be represented as thefollowing equation
b (120579 119903) = [1 119890minus1198951205931 sdot sdot sdot 119890
minus119895(119873minus1)1205931 119890minus119895120593119897 119890
minus119895(1205932+120593119873) sdot sdot sdot 119890minus119895(119873minus1)1205932+120593119873]
119879
(15)
where
1205931= (
2120587119889 sin (120579)120582
) minus (2120587119903 sdot Δ119891
1
119888) (16a)
120593119873= (
2120587119873119889 sin (120579)120582
) (16b)
1205932= (
2120587119889 sin (120579)120582
) minus (2120587119903 sdot Δ119891
2
119888) (16c)
The corresponding FIM can be derived as
ITS-FDA = 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) = [
119868120579120579
119868120579119903
119868119903120579
119868119903119903
] (17)
where
119868120579120579
= 2
10038161003816100381610038161205730
1003816100381610038161003816
2
1205902
119899
41205872
1198892cos2 (120579)1205822
2119873
sum
119898=1
(119898 minus 1)2
(18a)
119868119903119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2
1205902
119899
41205872
(Δ1198912
1+ Δ119891
2
2)
1198882
119873
sum
119898=1
(119898 minus 1)2
(18b)
119868120579119903= 119868
119903120579= minus
210038161003816100381610038161205730
1003816100381610038161003816
2
1205902
119899
41205872
119889 cos (120579)120582119888
times [(Δ1198911+ Δ119891
2)
119873
sum
119898=1
(119898 minus 1)2
+ Δ1198912119873
119873
sum
119898=1
(119898 minus 1)]
(19)
The angle and range CRLBs are the first two diagonalelements of the inverse of the FIM
CRLB120579120579TS-FDA
= [Iminus1
TS-FDA]11
(20a)
CRLB119903119903TS-FDA
= [Iminus1
TS-FDA]22
(20b)
where [sdot]119894119895is the element at the 119894th row and 119895th column of the
matrix
4 CRLB with Signal Model prior toMatched Filtering
41 Data Models Unlike the first data model the data modelprior to matched filtering allows to estimate the Doppler shift(velocity) Suppose an 119872-element antenna array receives atime delayed and Doppler-shifted echo of the transmittedsignal 119904(119905) exp(119895Ω
119888119905) where Ω
119888is the center frequency
Knowing the time delay and Doppler shift Ω119863(assuming
a target with constant radial velocity) the range and radialcomponent of velocity can be determined by 119903 = 1198881205912
and V = Ω119863119888(2Ω
119888) We denote the continuous-time signal
119904(119905) as 119904[119897] = 119904(119897 sdot Δ119905) where Δ119905 is the sampling intervalCorrespondingly the time delay and Doppler shift in thesampled signal domain are 119897
120591= 120591Δ119905 and 119908
119863= Ω
119863sdot Δ119905
respectively [27]After converting to baseband and sampling the received
signal at time 119897 sdot Δ119905 becomes
y [119899] = 120573 sdot a (120579 119903) sdot 119904 [119897 minus 119897120591]exp (119895119908
119863119897) + n [119897] 119897 = 1 119871
(21)
where 120573 is complex amplitude of the signal and n[119897] is theadditive noise [28]
We assume that the snapshots taken at 119897 = 1 119871 coverthe whole of a coherent processing interval (CPI) Thereforethe time duration of the CPI is 119879CPI = 119871 sdot Δ119905 Under thedata model of (21) the complex amplitude 120573 is assumed tobe an unknown deterministic constant during the CPI Tomodel the Doppler effect with a frequency shift the radialcomponent of the target velocity needs to be much smallerthan the propagation speed (ie V119888 ≪ 1) Then the time-bandwidth product of the complex envelope should be largerthan 1 In addition it is assumed that the propagation time ofthe signal across the array is much smaller than the reciprocal
International Journal of Antennas and Propagation 5
of the signal bandwidth which is the narrowband arrayassumption in array processing
Define the vector of unknown target parameters as120581 = [Re120573 Imag120573 120579 120578119879
] Stacking all samples into a singlevector (21) can be rewritten as
z = 120573 sdot 120601 (120578) otimes a (120579 119903) + n = (120581) + n (22)
where otimes denotes the Kronecker product (120581) =
120573 sdot 120601(120578) otimes a(120579 119903)
120601 (120578) = [119904 [1 minus2119903
119888 sdot Δ119905] exp (119895119908
1198631)
119904 [2 minus2119903
119888 sdot Δ119905] exp (119895119908
1198632)
119904 [119871 minus2119903
119888 sdot Δ119905] exp (119895119908
119863119871)]
119879
(23)
The noise n is assumed to be zero-mean Gaussian spatiallyand temporally correlated with spatiotemporal covariance[29]
119864 [nnH] = Cn otimes Rn (24)
where Cn is the temporal noise covariance matrixUnder the above data model the signal and noise param-
eters are disjoint and satisfy the space-time separability [30]
42 CRLBs of Angle Range and Doppler Shift Suppose thereceived signal is completely covered by the observations119897 = 1 119871 and the sampling is dense (ie Δ119905 rarr 0) TheRn is assumed to be constant in the frequency band 119891 isin
(minus1(2Δ119905) 1(2Δ119905)) where 119891 denotes the frequency in thecontinuous-time domain The vector of Doppler shift andtime delay in the continuous-time domain is defined as
120578 = [119903 Ω119863]119879
(25)
For Cn = I119871and Rn = 120590
2
119899I119872
with I119871being an 119871 times 119871
identity matrix We define the signal power s = 120601H 120601 The
CRLBs expression for 120579 and 120578 follows fromAppendix D as thefollowing
CRLB120579120578120579120578FDA
=
1205902
119899
210038161003816100381610038161205731003816100381610038161003816
2s
times
[[[[[[[[[[[
[
12058721198892cos2 (120579)119872(119872
2minus 1)
[
[
1
31205822
+
(Δ119891)2[(2119872minus 1) (8119872
2minus 3119872minus 11)]
451198882(119872+ 1)
+
2Δ119891 (119872minus 1)
3119888120582
]
]
minus1205872119889Δ119891 cos (120579)119872(119872
2minus 1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 0
minus1205872119889Δ119891 cos (120579)119872(119872
2minus 1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 1205872(Δ119891)2119872(119872
2minus 1)
31198882
+
12058511
sImag 12058512
s
0
Imag 12058512
s12058522
s
]]]]]]]]]]]
]
minus1
(26)
where
12058511
= int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905
minus1
s[int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905]
2
(27a)
12058522
= int
infin
minusinfin
1199052
|119904 (119905 minus 120591)|2
119889119905
minus1
s[int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
2
(27b)
12058512
= int
infin
minusinfin
119905119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905
minus [1
sint
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905
sdot int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905]
(28)
The terms 12058511
and 12058522
are proportional to the rootmean squared (RMS) bandwidth and RMS duration of 119904(119905)respectively Note that the decoupling is a consequence ofthe assumed space-time separability of signal and noisemodels and the assumption of the complex amplitude 120573 as anunknown deterministic constant In addition since the signaland additive noise parameters are disjoint the above CRLBexpressions hold regardless of whether the spatiotemporalnoise covariance Cn otimes Rn is known or unknown
43 CRLBs When Chirp Pulse Signal Is Employed In thissection we derive the CRLB expressions for the rangeDoppler shift and direction estimation when 119904(119905) is a chirppulse signal with a large time-bandwidth product [31 32] Itcan be expressed as
119904 (119905) =
119875minus1
sum
119901=0
1199040(119905 minus 119901119879
119901) (29)
6 International Journal of Antennas and Propagation
where 119879119901is the pulse repetition interval 119875 is the number of
chirp pulses and 1199040(119905) is expressed as [12]
1199040(119905) = exp [119895120587 119861
1198790
(119905 minus1
21198790)
2
] sdot [ℎ (119905) minus ℎ (119905 minus 1198790)]
(30)
where1198790is the chirp pulse duration119861 is the chirp bandwidth
and ℎ(119905) is the Heaviside step function
Assume the time-bandwidth product of the pulse is1198790sdot 119861 ≫ 1 Using the signal given in (29) and (30) in
continuous-time domain we obtain the signal power s =
1198751198790 120585
11= 1198754120587
2
1198612
11987903119888
2 Imag12058512 = minus(119875120587119861119879
2
03119888) and
12058522
= (119875MT3
012)[1 + (119879
0119879
119877)2
(1198752
minus 1)] Thus the CRLBexpressions of 120579 and 120578 for FDA are derived as the followingequation
CRLB120579120578120579120578FDA
=
1
2SNR
sdot
[[[[[[[[[[[[
[
12058721198892cos2(120579)119872(119872
2minus 1)
[
[
1
31205822+
(Δ119891)2[(2119872minus 1) (8119872
2minus3119872minus11)]
451198882119872(119872+ 1)
+
2Δ119891 (119872minus 1)
3119888120582
]
]
minus1205872119889Δ119891 cos(120579)119872(119872
2minus1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 0
minus1205872119889Δ119891 cos(120579)119872(119872
2minus 1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 1205872(Δ119891)2119872(119872
2minus1)
31198882
+
412058721198612119872
31198882
minus
1198721205871198611198790
3119888
0 minus
1198721205871198611198790
3119888
1198721198792
0
12
[1+(
119879119877
1198790
)
2
(1198752minus1)]
]]]]]]]]]]]]
]
minus1
(31)
Specially for the phased array (ie Δ119891 = 0) the CRLB is
CRLB120579120578120579120578phased-array
=1
2SNR
times
[[[[[[[[
[
1205872
1198892cos2 (120579)119872(119872
2
minus 1)
312058220 0
04120587
2
1198612
119872
31198882minus119872120587119861119879
0
3119888
0 minus119872120587119861119879
0
3119888
1198721198792
0
12[1 + (
119879119877
1198790
)
2
(1198752
minus 1)]
]]]]]]]]
]
minus1
(32)
When only one pulse (119875 = 1) is used the CRLBin (32) will be infinite because the model is not identifi-able and the range and Doppler shift cannot be uniquelyestimated [18] since phase-array radar has no rangeidentity capability In contrast (31) does not have sucha problem
According to the previous discussion the angle and rangeof targets cannot be estimated jointly due to the range-anglecoupling Using the similar transmit subaperturing approachpresented in Section 3 and steering vector b(120579 119903) (15) theCRLBs of angle range and Doppler shift are derived as inthe following equation
CRLB120579120578120579120578TS-FDA =
1
2SNR
times
[[[[[[[[[[[[[[[[[[[
[
412058721198892cos2 (120579)1205822
2119873
sum
119898=1
(119898 minus 1)2
minus
41205872119889 cos (120579)120582119888
[[[[
[
(Δ1198911 + Δ1198912)
119873
sum
119898=1
(119898 minus 1)2
+119873Δ1198912
119873
sum
119898=1
(119898 minus 1)
]]]]
]
0
minus
41205872119889 cos (120579)120582119888
[[[[
[
(Δ1198911 + Δ1198912)
119873
sum
119898=1
(119898 minus 1)2
+119873Δ1198912
119873
sum
119898=1
(119898 minus 1)
]]]]
]
(Δ1198912
1+ Δ1198912
2)
1198882
119873
sum
119898=1
(119898 minus 1)2+
412058721198612119872
31198882
minus
1198721205871198611198790
3119888
0 minus
1198721205871198611198790
3119888
1198721198792
0
12
[1 + (
119879119877
1198790
)
2
(1198752minus 1)]
]]]]]]]]]]]]]]]]]]]
]
minus1
(33)
International Journal of Antennas and Propagation 7
20151050
10minus2
10minus3
10minus47001 7002 7003
10minus248203
10minus248201
10minus248199
CRLB
of a
ngle
estim
atio
n (d
eg)
SNR (dB)
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
Figure 2 CRLB for estimating angle versus SNR when the range isknown
5 Simulation and Verification
In this section we consider several numerical examples thatcompare the CRLBs in different signal and noise modelsConsider an X-band FDA radar with the carrier frequency1198910
= 10GHz We assume a ULA of 119872 elements usedfor transmitting The array elements are spaced half of thewavelength apart from each other namely 119889 = 1205822 Onetarget of interest is supposed to reflect a plane wave thatimpinges on the array from direction of angle 120579 = 30
∘Under the signal model after matched filtering Figures
2 and 3 compare the CRLBs according to (9) and (13)respectively It can be noticed that the CRLBs are improvedwhen a larger number of elements are employed Howeverit has no significant difference when different frequencyincrements are used This is because sum
119872
119898=1(119898 minus 1)
2
1205822
≫
(Δ119891)2
sum119872
119898=1(119898 minus 1)
4
1198882
+ 2Δ119891sum119872
119898=1(119898 minus 1)
3
120582119888 thus thefrequency increment has a small impact on the CRLBsIn [33] a frequency offset selection strategy is derived toprecisely steer the beam toward a fixed range with a desiredangle
Figure 4 shows the CRLBs of angle and range whenboth the angle and range are unknown The CRLBs aresignificantly degraded due to the range-angle coupling Con-sequently the range and angle of targets cannot be estimateddirectly by the FDA radar However the CRLBs decreaseas the increase of the number of elements and frequencyincrement still holds Moreover generally more elementsmean that better CRLBs performance can be achieved for theFDA radar
To overcome the problem that the range and angle oftargets cannot be estimated directly by the FDA radar weuse the transmit subaperturing strategy on the transmitfrequency increments Figure 5 shows the corresponding
CRLB
of r
ange
estim
atio
n (m
)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
102
101
10020151050
SNR (dB)
Figure 3 CRLB for estimating range versus SNR when the angle isknown
CRLBs where 119872 = 32 is assumed It can be noticed that toobtain a lower CRLB the Δ119891
1and Δ119891
2should have inverse
signs that is one is passive and the other is negative Onereason is that in this case the FDA radar has a wider systembandwidth Figure 6 shows the CRLBs of angle and rangewhen 119872 = 20 is employed It can be noticed that theCRLBs performance improves with the increase of the sensornumber
Under the data model prior to matched filtering wesuppose the following signal parameters bandwidth 119861 =
10MHz repetition period 119879119901
= 1ms and pulse duration1198790
= 250 120583s In this case the approximate expressionsgiven in (31) are valid because the transmitted has a largetime-bandwidth product (119879
0sdot 119861 = 2500 ≫ 1) Figure 7
shows the CRLBs for direction range and Doppler shift asa function of SNR Note that when SNR = minus10 dB and119872 = 32 are employed we can get CRLB
119903119903FDA= 653m
and CRLBΩ119863Ω119863FDA
= 136937 rads that corresponds to theCRLB for velocity is 327 cms (since V = Ω
119863119888(2Ω
119888) and
Ω119888= 2120587119891
119888) Since (Δ119891)2 sum119872
119898=1(119898 minus 1)
4
1198882
≪ 1 the frequencyincrement has a small impact on the CRLBs In additionobserve that the CRLB for 120579 and 120578 is block-diagonal (see(32)) and therefore decoupled that is CRLB
120578120578FDAremains the
same whether or not 120579 is known and similarly CRLB120579120579FDA
is the same whether or not 120578 is known The decouplingis a consequence of the assumed space-time separability ofsignal and noise models and the assumption of the complexamplitude 120573 as an unknown deterministic constant
Figure 8 shows that the CRLBs versus SNR for differentcombinations of Δ119891
1and Δ119891
2 Comparing Figures 8 and
7 the CRLBs have been significantly improved Likewisecomparing Figures 8 and 9 theCRLBs performance improvesas the number of elements increases
8 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
104
103
102
101
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
108
107
106
105
10420151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
Figure 4 Both angle and range are unknown
6 Conclusion
In this paper we derive the CRLB to jointly estimate theattributes of a moving target using FDA radar and computethe corresponding CRLB expressions First we briefly intro-duce the FDA concept and make a summary on the FDAcharacteristics Then we consider two different data modelsnamely pre- and postmatched filtering Under differentsignal and noise models we compute the CRLB expressionsfor estimating the range direction and Doppler shift Wedemonstrate that the FDA radar beamforming is coupledin range and angle and that the targetrsquos range and anglecannot be estimated directly by the FDA radar To overcome
20151050
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
CRLB
of r
ange
estim
atio
n (m
)102
101
100
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
Figure 5 CRLBs of TS-FDA radar with119872 = 32
this problem this paper proposes a transmit subaperturingstrategy for the FDA radar In doing so the range and angle oftargets are estimated from the transmit-receive beamformingoutput Moreover we also specialize the CRLB results tothe case of temporally white noise and a chirp pulse signalExtensive simulation results verify the correctness of thederived CRLBs It is shown that the CRLBs decrease with theincrease of the number of elements and frequency incrementThe CRLBs can be further improved through three aspectsincreasing the number of elements enhancing the systembandwidth by employing a larger frequency increment andusing transmit subaperturing strategy with more subarrays
International Journal of Antennas and Propagation 9
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
CRLB
of r
ange
estim
atio
n (m
)
103
102
101
10020151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
Figure 6 CRLBs of TS-FDA radar with119872 = 20
Appendices
A Derive the CRLB for Angle WhenRange Is Known
To derive the CRLB we start with a well-known expressionfor the FIM under the data model in Section 3 We define thespatial noise covariance matrix as Rn = 120590
2
119899I119872
and signal-to-noise ratio (SNR) as SNR = |120573
0|2
1205902
119899 Suppose the target range
is known the FIM of 120579 is
119868120579120579
= 2Re 119863H120579(120595) (Rminus1
n )119863120579(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
(A1)
For a phased-array radar there is (120595) = 1205730a(120579) We then
have
120597a (120579)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579) (A2)
whereD = diag[0 1 119872 minus 1] and
120597aH (120579)
120597120579
120597a (120579)120597120579
=4120587
2
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A3)
The FIM of the phased-array radar is
119868120579120579phased-array
= 2SNR41205872
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A4)
Similarly for the FDA radar there is (120595) = 1205730a(120579 119903)The
derivation of a(120579 119903) with respect to 120579 is
120597a (120579 119903)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579 119903)
minus 1198952120587119889Δ119891 cos (120579)
119888
times diag [0 1 (119872 minus 1)2
] a (120579 119903)
120597aH (120579 119903)
120597120579
120597a (120579 119903)120597120579
= 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822
+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A5)
Accordingly the FIM of 120579 for the FDA can be expressedas
119868120579120579FDA
= 2SNR sdot 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A6)
10 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
(c) CRLB for Doppler shift versus SNR
Figure 7 General CRLB results of FDA radar
B Derive the CRLB for Range WhenAngle Is Known
Under the data model in Section 3 when the direction 120579 isknown the parameter to be estimated is 119903 The FIM of 119903 is
119868119903119903FDA
= 2Re 119863H119903(120595) (Rminus1
n )119863119903(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597aH (120579 119903)
120597119903Rminus1
n120597a (120579 119903)
120597119903
(B1)
The derivation of a(120579 119903) with respect to 119903 for FDA is
120597a (120579 119903)120597119903
= 1198952120587Δ119891
119888Da (120579 119903)
120597aH (120579 119903)
120597119903
120597a (120579 119903)120597119903
=4120587
2
Δ1198912
1198882
119872
sum
119898=1
(119898 minus 1)2
(B2)
The FIM of 119903 is thus given by
119868119903119903FDA
= 2SNR4120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(B3)
International Journal of Antennas and Propagation 11
C Derive the CRLB for Range and Angle
Under the data model in Section 3 both the angle and rangeare unknown The range and angle of targets are estimatedjointly The FIM for parameters 120579 and 119903 can be expressed as
IFDA = 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) = [
119868120579120579
119868120579119903
119868119903120579
119868119903119903
] (C1)
We then have
IFDA = 210038161003816100381610038161205730
1003816100381610038161003816
2
times
[[[[
[
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
]]]]
]
(C2)
Since (120597H(120595)120597119903)Rminus1
n (120597(120595)120597120579) = (120597H(120595)
120597120579)Rminus1
n (120597(120595)120597119903) then 119868120579119903= 119868
119903120579 We can get
119868120579120579
= 210038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
= 81205872
1198892cos2 (120579)
times SNRsum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
119868120579119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
= minus81205872
119889Δ119891 cos (120579)
times SNR[sum
119872
119898=1(119898 minus 1)
2
120582119888+Δ119891sum
119872
119898=1(119898 minus 1)
3
1198882]
119868119903119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
= SNR8120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(C3)
Since CRLB120579120579FDA
= [Iminus1
FDA]11 CRLB119903119903FDA
= [Iminus1
FDA]22 Theexpressions for the CRLB of angle and range estimationsgiven in (14) can be obtained by substituting (C3) into (C2)
D General CRLB Results
Under the data model in Section 4 we derive the continuousCRLB expressions for temporally and spatially white noiseand denote SNR = |120573|
2
s1205902
119899with s being the power We
also start with FIM
I120581119894120581119895
= 2Re 119863H120581119894
(120581) (Cminus1
n otimes Rminus1
n )119863120581119895(120581) (D1)
where 120581119894is the 119894th element of 120581 and 119863
120581119894(120595) = 120597(120581)120597120581
119894
Consider
IFDA =
[[[[[
[
119868120573120573
119868119879
120579120573119868119879
120578120573
119868120579120573
119868120579120579
119868119879
120578120579
119868120578120573
119868120578120579
119868120578120578
]]]]]
]
(D2)
For clarity we rewrite Fisherrsquos information matrix I as
IFDA = [A UV B] (D3)
where
V = U119879
(D4a)
V = [119868120579120573
119868120578120573
] (D4b)
B = [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] (D4c)
According to the matrix inversion lemma the inversematrix of IFDA is
Iminus1
FDA = [
[
(A minus UBminus1V)minus1
minusAminus1U(B minus VAminus1U)minus1
minusBminus1V(A minus UBminus1V)minus1
(B minus VAminus1U)minus1
]
]
(D5)
where the submatrix (A minus UBminus1V)minus1 corresponds to theCRLBof120573which is nuisance and (B minus VAminus1U)minus1 correspondsto the CRLB of 120579 and 120578 = [119903 Ω
119863]119879 which are of interest
CRLB120579120578120579120578FDA
= (B minus VAminus1U)minus1
= [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] minus [119868120579120573
119868120578120573
] 119868minus1
120573120573[119868
119879
120579120573119868119879
120578120573]
minus1
(D6)
where
119868120573120573
= 2 sdot[[[
[
s sdot 1198721205902
119899
0
0s sdot 1198721205902
119899
]]]
]
(D7a)
119868120579120573
= 2 sdot Re[1 119895] otimess1205902
119899
sdot 120573lowast
1198601 (D7b)
119868120578120573
= 2 sdot Re[1 119895] otimes120573
lowast
1205902
119908
sdot [119872 sdot 1198603+ s sdot 119860
4] (D7c)
119868120578120578
= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
[119872 sdot 1198606+ 119860
H3119860
4+ 119860
3119860
H4+ s sdot 119860
7]
(D7d)
12 International Journal of Antennas and Propagation
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHz
70001 70001 70001 70002100286514
100286515
100286515
kHz
(c) CRLB for Doppler shift versus SNR
Figure 8 General CRLB results of TS-FDA radar with119872 = 32
119868120579120579minus (119868
120579120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198602minus119860
1119860
H1
119872) (D8a)
119868120578120579minus (119868
120578120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198605minus119860
4119860
H1
119872) (D8b)
119868120578120578
minus (119868120578120573119868119879
120578120573)
1205902
119899
s119872
= 2
10038161003816100381610038161205731003816100381610038161003816
2
s1205902
119899
(119872119860
6
s+ 119860
7minus119872119860
3119860
H3
s2minus119860
4119860
H4
119872)
(D8c)
1198601= minus119895(
2120587119889 cos (120579)sum119872
119898=1(119898 minus 1)
120582
+2120587119889Δ119891 cos (120579)sum119872
119898=1(119898 minus 1)
2
119888)
(D9a)
1198602= 4120587
2
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(D9b)
International Journal of Antennas and Propagation 13
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(c) CRLB for Doppler shift versus SNR
Figure 9 General CRLB results of TS-FDA radar with119872 = 20
1198603= [int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905 int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
119879
(D9c)
1198604= [
2120587Δ119891sum119872
119898=1(119898 minus 1)
1198880]
119879
(D9d)
1198605= [minus4120587
2119889Δ119891 cos(120579) [
sum119872
119898=1(119898 minus 1)
2
12120582119888
+
Δ119891sum119872
119898=1(119898 minus 1)
3
121198882
] 0]
119879
(D9e)
1198606=[[
[
int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905 int
infin
minusinfin
119905119904H(119905 minus 120591)
119889119904 (119905 minus 120591)
119889119905
119889119905
int
infin
minusinfin
119905119904 (119905 minus 120591)
119889119904(119905 minus 120591)H
119889119905
119889119905 int
infin
minusinfin
1199052|119904 (119905 minus 120591)|
2119889119905
]]
]
(D9f)
1198607=[[
[
41205872
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
0
0 0
]]
]
(D9g)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework described in this paper was supported in part by theNational Natural Science Foundation of China under Grant
14 International Journal of Antennas and Propagation
41101317 the Program for New Century Excellent Talentsin University under Grant NCET-12-0095 Sichuan ProvinceScience Fund for Distinguished Young Scholars under Grant2013JQ0003 and Fundamental Research Fund for theCentralUniversities
References
[1] F Bandiera M Mancino and G Ricci ldquoLocalization strategiesfor multiple point-like radar targetsrdquo IEEE Transactions onSignal Processing vol 60 no 12 pp 6708ndash6712 2012
[2] D R Fuhrmann J P Browning and M Rangaswamy ldquoSignal-ing strategies for the hybrid MIMO phased-array radarrdquo IEEEJournal on Selected Topics in Signal Processing vol 4 no 1 pp66ndash78 2010
[3] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011
[4] M Compagnoni P Bestagini F Antonacci A Sarti and STubaro ldquoLocalization of acoustic sources through the fittingof propagation cones using multiple independent arraysrdquo IEEETransactions on Audio Speech and Language Processing vol 20no 7 pp 1964ndash1975 2012
[5] P Antonik M C Wicks H D Griffiths and C J BakerldquoFrequency diverse array radarsrdquo in Proceedings of the IEEERadar Conference (RADAR rsquo06) pp 215ndash217 Verona NY USAApril 2006
[6] P Antonik M CWicks H D Griffiths and C J Baker ldquoMulti-mission multi-mode waveform diversityrdquo in Proceedings of theIEEE Radar Conference (RADAR rsquo06) pp 580ndash582 Verona NYUSA April 2006
[7] P Antonik H D Griffiths and C J Baker ldquoRange depen-dent beamforming using element level waveform diversityrdquo inProceedings of the International Waveform Diversity and DesignConference pp 1ndash4 Las Vegas Nev USA January 2006
[8] P Baizert T B HaleM A Temple andM CWicks ldquoForward-looking radar GMTI benefits using a linear frequency diversearrayrdquo Electronics Letters vol 42 no 22 pp 1311ndash1312 2006
[9] B W Jung R S Adve and J Chun ldquoFrequency diversity inmultistatic radarsrdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo08) pp 1ndash6 Rome Italy May 2008
[10] M Secmen S Demir A Hizal and T Eker ldquoFrequencydiverse array antenna with periodic time modulated pattern inrange and anglerdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo07) pp 427ndash430 Boston Mass USA April 2007
[11] J Huang K F Tong and C J Baker ldquoFrequency diversearray with beam scanning featurerdquo in Proceedings of the IEEEAntennas and Propagation Conference (AP-S rsquo08) pp 1ndash4 SanDiego Calif USA July 2008
[12] T Higgins and S D Blunt ldquoAnalysis of range-angle coupledbeamforming with frequency-diverse chirpsrdquo in Proceedings ofthe International Waveform Diversity and Design Conference(WDD rsquo09) pp 140ndash144 Orlando Fla USA February 2009
[13] J Farooq M A Temple and M A Saville ldquoApplication of fre-quency diverse arrays to synthetic aperture radar imagingrdquo inProceedings of the International Conference on Electromagneticsin Advanced Applications (ICEAA rsquo07) pp 447ndash449 TorinoItaly September 2007
[14] J Farooq M A Temple and M A Saville ldquoExploiting fre-quency diverse array processing to improve SAR image reso-lutionrdquo in Proceedings of the IEEE Radar Conference (RADARrsquo08) pp 1ndash5 Rome Italy May 2008
[15] W QWang ldquoPhased-MIMO radar with frequency diversity forrangedependent beamformingrdquo IEEE Sensors Journal vol 13no 8 pp 1320ndash1328 2013
[16] A L Swindlehurst and P Stoica ldquoMaximum likelihood meth-ods in radar array signal processingrdquo Proceedings of the IEEEvol 86 no 2 pp 421ndash441 1998
[17] J Ward ldquoCramer-Rao bounds for target angle and Dopplerestimation with space-time adaptive processing radarrdquo in Pro-ceedings of the 29th Asilomar Conference on Signals Systems andComputers pp 1198ndash1202 Pacific Grove Calif USA November1995
[18] A Dogandzic and A Nehorai ldquoCramer-Rao bounds for esti-mating range velocity and direction with an active arrayrdquo IEEETransactions on Signal Processing vol 49 no 6 pp 1122ndash11372001
[19] A Dogandzic and A Nehorai ldquoEstimating range velocity anddirection with a radar arrayrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo99) pp 2773ndash2776 Phoenix Ariz USA March 1999
[20] J J Zhang G Maalouli A P Suppappola and D MorrellldquoCramer-Rao lower bounds for the joint estimation of targetattributes using MIMO radarrdquo in Proceedings of the Interna-tional Waveform Diversity and Design Conference (WDD rsquo09)pp 103ndash107 Orlando Fla USA February 2009
[21] A Hassanien S A Vorobyov and A B Gershman ldquoMovingtarget parameters estimation in noncoherent MIMO radarsystemsrdquo IEEE Transactions on Signal Processing vol 60 no 5pp 2354ndash2361 2012
[22] Q He R S Blum and A M Haimovich ldquoNoncoherent MIMOradar for location and velocity estimation more antennasmeans better performancerdquo IEEE Transactions on Signal Pro-cessing vol 58 no 7 pp 3661ndash3680 2010
[23] S M Kay Fundamentals of Statistical Signal Processing Estima-tion Theory vol 1 Pearson London UK 2nd edition 2011
[24] H L van TreesOptimum Array Processing JohnWiley amp SonsNew York NY USA 2002
[25] D Wilcox and M Sellathurai ldquoOn MIMO radar subarrayedtransmit beamformingrdquo IEEE Transactions on Signal Processingvol 60 no 4 pp 2076ndash2081 2012
[26] WQWang andH Z Shao ldquoRange-angle localization of targetsby a double-pulse frequency diverse array radarrdquo IEEE Journalon Selected Topics in Signal Processing vol 8 no 1 pp 106ndash1142014
[27] S Gogineni and A Nehorai ldquoTarget estimation using sparsemodeling for distributed MIMO radarrdquo IEEE Transactions onSignal Processing vol 59 no 11 pp 5315ndash5325 2011
[28] T Li andA Nehorai ldquoMaximum likelihood direction finding inspatially colored noise fields using sparse sensor arraysrdquo IEEETransactions on Signal Processing vol 59 no 3 pp 1048ndash10622011
[29] S Sen ldquoOFDMradar space-time adaptive processing by exploit-ing spatio-temporal sparsityrdquo IEEE Transactions on SignalProcessing vol 61 no 1 pp 118ndash130 2013
[30] J J Blanz A Papathanassiou M Haardt I Furio and P WBaier ldquoSmart antennas for combined DOA and joint channelestimation in time-slotted CDMA mobile radio systems withjoint detectionrdquo IEEE Transactions onVehicular Technology vol49 no 2 pp 293ndash306 2000
[31] P N Pathirana S C K Herath and A V Savkin ldquoMultitargettracking via space transformations using a single frequencycontinuous wave radarrdquo IEEE Transactions on Signal Processingvol 60 no 10 pp 5217ndash5229 2012
International Journal of Antennas and Propagation 15
[32] P H Leong T D Abhayapala and T A Lamahewa ldquoMultipletarget localization using wideband echo chirp signalsrdquo IEEETransactions on Signal Processing vol 61 no 16 pp 4077ndash40892013
[33] L Zhuang X Liu and W Yu ldquoPrecisely beam steering forfrequency diverse arrays based on frequency offset selectionrdquoin Proceedings of the International Radar Conference (RADARrsquo09) pp 1ndash4 Bordeaux France December 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 International Journal of Antennas and Propagation
d
1 2 N
middot middot middot
1 2 N
middot middot middot
Subarray 1 Subarray 2
Figure 1 Illustration of transmit subaperturing FDA radar
of targets cannot be estimated directly by a standard FDAradar To overcome this problem we present a transmitsubaperturing method for the FDA radar
324 CRLBs of Transmit Subaperturing FDA Radar Todecouple the range and angle peaks and estimate both therange and angle of target we divide the whole array into twoequal subarrays [25 26] Suppose that the number of elementsis 119872 each subarray has 119873 elements namely 119872 = 2119873The first subarray uses the frequency increment of Δ119891
1 and
the second subarray uses the frequency increment of Δ1198912
as shown in Figure 1 The resulting system is referred to astransmit subaperturing FDA (TS-FDA) radar
Taking the first element of the first subarray as thereference the new steering vector can be represented as thefollowing equation
b (120579 119903) = [1 119890minus1198951205931 sdot sdot sdot 119890
minus119895(119873minus1)1205931 119890minus119895120593119897 119890
minus119895(1205932+120593119873) sdot sdot sdot 119890minus119895(119873minus1)1205932+120593119873]
119879
(15)
where
1205931= (
2120587119889 sin (120579)120582
) minus (2120587119903 sdot Δ119891
1
119888) (16a)
120593119873= (
2120587119873119889 sin (120579)120582
) (16b)
1205932= (
2120587119889 sin (120579)120582
) minus (2120587119903 sdot Δ119891
2
119888) (16c)
The corresponding FIM can be derived as
ITS-FDA = 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) = [
119868120579120579
119868120579119903
119868119903120579
119868119903119903
] (17)
where
119868120579120579
= 2
10038161003816100381610038161205730
1003816100381610038161003816
2
1205902
119899
41205872
1198892cos2 (120579)1205822
2119873
sum
119898=1
(119898 minus 1)2
(18a)
119868119903119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2
1205902
119899
41205872
(Δ1198912
1+ Δ119891
2
2)
1198882
119873
sum
119898=1
(119898 minus 1)2
(18b)
119868120579119903= 119868
119903120579= minus
210038161003816100381610038161205730
1003816100381610038161003816
2
1205902
119899
41205872
119889 cos (120579)120582119888
times [(Δ1198911+ Δ119891
2)
119873
sum
119898=1
(119898 minus 1)2
+ Δ1198912119873
119873
sum
119898=1
(119898 minus 1)]
(19)
The angle and range CRLBs are the first two diagonalelements of the inverse of the FIM
CRLB120579120579TS-FDA
= [Iminus1
TS-FDA]11
(20a)
CRLB119903119903TS-FDA
= [Iminus1
TS-FDA]22
(20b)
where [sdot]119894119895is the element at the 119894th row and 119895th column of the
matrix
4 CRLB with Signal Model prior toMatched Filtering
41 Data Models Unlike the first data model the data modelprior to matched filtering allows to estimate the Doppler shift(velocity) Suppose an 119872-element antenna array receives atime delayed and Doppler-shifted echo of the transmittedsignal 119904(119905) exp(119895Ω
119888119905) where Ω
119888is the center frequency
Knowing the time delay and Doppler shift Ω119863(assuming
a target with constant radial velocity) the range and radialcomponent of velocity can be determined by 119903 = 1198881205912
and V = Ω119863119888(2Ω
119888) We denote the continuous-time signal
119904(119905) as 119904[119897] = 119904(119897 sdot Δ119905) where Δ119905 is the sampling intervalCorrespondingly the time delay and Doppler shift in thesampled signal domain are 119897
120591= 120591Δ119905 and 119908
119863= Ω
119863sdot Δ119905
respectively [27]After converting to baseband and sampling the received
signal at time 119897 sdot Δ119905 becomes
y [119899] = 120573 sdot a (120579 119903) sdot 119904 [119897 minus 119897120591]exp (119895119908
119863119897) + n [119897] 119897 = 1 119871
(21)
where 120573 is complex amplitude of the signal and n[119897] is theadditive noise [28]
We assume that the snapshots taken at 119897 = 1 119871 coverthe whole of a coherent processing interval (CPI) Thereforethe time duration of the CPI is 119879CPI = 119871 sdot Δ119905 Under thedata model of (21) the complex amplitude 120573 is assumed tobe an unknown deterministic constant during the CPI Tomodel the Doppler effect with a frequency shift the radialcomponent of the target velocity needs to be much smallerthan the propagation speed (ie V119888 ≪ 1) Then the time-bandwidth product of the complex envelope should be largerthan 1 In addition it is assumed that the propagation time ofthe signal across the array is much smaller than the reciprocal
International Journal of Antennas and Propagation 5
of the signal bandwidth which is the narrowband arrayassumption in array processing
Define the vector of unknown target parameters as120581 = [Re120573 Imag120573 120579 120578119879
] Stacking all samples into a singlevector (21) can be rewritten as
z = 120573 sdot 120601 (120578) otimes a (120579 119903) + n = (120581) + n (22)
where otimes denotes the Kronecker product (120581) =
120573 sdot 120601(120578) otimes a(120579 119903)
120601 (120578) = [119904 [1 minus2119903
119888 sdot Δ119905] exp (119895119908
1198631)
119904 [2 minus2119903
119888 sdot Δ119905] exp (119895119908
1198632)
119904 [119871 minus2119903
119888 sdot Δ119905] exp (119895119908
119863119871)]
119879
(23)
The noise n is assumed to be zero-mean Gaussian spatiallyand temporally correlated with spatiotemporal covariance[29]
119864 [nnH] = Cn otimes Rn (24)
where Cn is the temporal noise covariance matrixUnder the above data model the signal and noise param-
eters are disjoint and satisfy the space-time separability [30]
42 CRLBs of Angle Range and Doppler Shift Suppose thereceived signal is completely covered by the observations119897 = 1 119871 and the sampling is dense (ie Δ119905 rarr 0) TheRn is assumed to be constant in the frequency band 119891 isin
(minus1(2Δ119905) 1(2Δ119905)) where 119891 denotes the frequency in thecontinuous-time domain The vector of Doppler shift andtime delay in the continuous-time domain is defined as
120578 = [119903 Ω119863]119879
(25)
For Cn = I119871and Rn = 120590
2
119899I119872
with I119871being an 119871 times 119871
identity matrix We define the signal power s = 120601H 120601 The
CRLBs expression for 120579 and 120578 follows fromAppendix D as thefollowing
CRLB120579120578120579120578FDA
=
1205902
119899
210038161003816100381610038161205731003816100381610038161003816
2s
times
[[[[[[[[[[[
[
12058721198892cos2 (120579)119872(119872
2minus 1)
[
[
1
31205822
+
(Δ119891)2[(2119872minus 1) (8119872
2minus 3119872minus 11)]
451198882(119872+ 1)
+
2Δ119891 (119872minus 1)
3119888120582
]
]
minus1205872119889Δ119891 cos (120579)119872(119872
2minus 1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 0
minus1205872119889Δ119891 cos (120579)119872(119872
2minus 1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 1205872(Δ119891)2119872(119872
2minus 1)
31198882
+
12058511
sImag 12058512
s
0
Imag 12058512
s12058522
s
]]]]]]]]]]]
]
minus1
(26)
where
12058511
= int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905
minus1
s[int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905]
2
(27a)
12058522
= int
infin
minusinfin
1199052
|119904 (119905 minus 120591)|2
119889119905
minus1
s[int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
2
(27b)
12058512
= int
infin
minusinfin
119905119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905
minus [1
sint
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905
sdot int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905]
(28)
The terms 12058511
and 12058522
are proportional to the rootmean squared (RMS) bandwidth and RMS duration of 119904(119905)respectively Note that the decoupling is a consequence ofthe assumed space-time separability of signal and noisemodels and the assumption of the complex amplitude 120573 as anunknown deterministic constant In addition since the signaland additive noise parameters are disjoint the above CRLBexpressions hold regardless of whether the spatiotemporalnoise covariance Cn otimes Rn is known or unknown
43 CRLBs When Chirp Pulse Signal Is Employed In thissection we derive the CRLB expressions for the rangeDoppler shift and direction estimation when 119904(119905) is a chirppulse signal with a large time-bandwidth product [31 32] Itcan be expressed as
119904 (119905) =
119875minus1
sum
119901=0
1199040(119905 minus 119901119879
119901) (29)
6 International Journal of Antennas and Propagation
where 119879119901is the pulse repetition interval 119875 is the number of
chirp pulses and 1199040(119905) is expressed as [12]
1199040(119905) = exp [119895120587 119861
1198790
(119905 minus1
21198790)
2
] sdot [ℎ (119905) minus ℎ (119905 minus 1198790)]
(30)
where1198790is the chirp pulse duration119861 is the chirp bandwidth
and ℎ(119905) is the Heaviside step function
Assume the time-bandwidth product of the pulse is1198790sdot 119861 ≫ 1 Using the signal given in (29) and (30) in
continuous-time domain we obtain the signal power s =
1198751198790 120585
11= 1198754120587
2
1198612
11987903119888
2 Imag12058512 = minus(119875120587119861119879
2
03119888) and
12058522
= (119875MT3
012)[1 + (119879
0119879
119877)2
(1198752
minus 1)] Thus the CRLBexpressions of 120579 and 120578 for FDA are derived as the followingequation
CRLB120579120578120579120578FDA
=
1
2SNR
sdot
[[[[[[[[[[[[
[
12058721198892cos2(120579)119872(119872
2minus 1)
[
[
1
31205822+
(Δ119891)2[(2119872minus 1) (8119872
2minus3119872minus11)]
451198882119872(119872+ 1)
+
2Δ119891 (119872minus 1)
3119888120582
]
]
minus1205872119889Δ119891 cos(120579)119872(119872
2minus1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 0
minus1205872119889Δ119891 cos(120579)119872(119872
2minus 1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 1205872(Δ119891)2119872(119872
2minus1)
31198882
+
412058721198612119872
31198882
minus
1198721205871198611198790
3119888
0 minus
1198721205871198611198790
3119888
1198721198792
0
12
[1+(
119879119877
1198790
)
2
(1198752minus1)]
]]]]]]]]]]]]
]
minus1
(31)
Specially for the phased array (ie Δ119891 = 0) the CRLB is
CRLB120579120578120579120578phased-array
=1
2SNR
times
[[[[[[[[
[
1205872
1198892cos2 (120579)119872(119872
2
minus 1)
312058220 0
04120587
2
1198612
119872
31198882minus119872120587119861119879
0
3119888
0 minus119872120587119861119879
0
3119888
1198721198792
0
12[1 + (
119879119877
1198790
)
2
(1198752
minus 1)]
]]]]]]]]
]
minus1
(32)
When only one pulse (119875 = 1) is used the CRLBin (32) will be infinite because the model is not identifi-able and the range and Doppler shift cannot be uniquelyestimated [18] since phase-array radar has no rangeidentity capability In contrast (31) does not have sucha problem
According to the previous discussion the angle and rangeof targets cannot be estimated jointly due to the range-anglecoupling Using the similar transmit subaperturing approachpresented in Section 3 and steering vector b(120579 119903) (15) theCRLBs of angle range and Doppler shift are derived as inthe following equation
CRLB120579120578120579120578TS-FDA =
1
2SNR
times
[[[[[[[[[[[[[[[[[[[
[
412058721198892cos2 (120579)1205822
2119873
sum
119898=1
(119898 minus 1)2
minus
41205872119889 cos (120579)120582119888
[[[[
[
(Δ1198911 + Δ1198912)
119873
sum
119898=1
(119898 minus 1)2
+119873Δ1198912
119873
sum
119898=1
(119898 minus 1)
]]]]
]
0
minus
41205872119889 cos (120579)120582119888
[[[[
[
(Δ1198911 + Δ1198912)
119873
sum
119898=1
(119898 minus 1)2
+119873Δ1198912
119873
sum
119898=1
(119898 minus 1)
]]]]
]
(Δ1198912
1+ Δ1198912
2)
1198882
119873
sum
119898=1
(119898 minus 1)2+
412058721198612119872
31198882
minus
1198721205871198611198790
3119888
0 minus
1198721205871198611198790
3119888
1198721198792
0
12
[1 + (
119879119877
1198790
)
2
(1198752minus 1)]
]]]]]]]]]]]]]]]]]]]
]
minus1
(33)
International Journal of Antennas and Propagation 7
20151050
10minus2
10minus3
10minus47001 7002 7003
10minus248203
10minus248201
10minus248199
CRLB
of a
ngle
estim
atio
n (d
eg)
SNR (dB)
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
Figure 2 CRLB for estimating angle versus SNR when the range isknown
5 Simulation and Verification
In this section we consider several numerical examples thatcompare the CRLBs in different signal and noise modelsConsider an X-band FDA radar with the carrier frequency1198910
= 10GHz We assume a ULA of 119872 elements usedfor transmitting The array elements are spaced half of thewavelength apart from each other namely 119889 = 1205822 Onetarget of interest is supposed to reflect a plane wave thatimpinges on the array from direction of angle 120579 = 30
∘Under the signal model after matched filtering Figures
2 and 3 compare the CRLBs according to (9) and (13)respectively It can be noticed that the CRLBs are improvedwhen a larger number of elements are employed Howeverit has no significant difference when different frequencyincrements are used This is because sum
119872
119898=1(119898 minus 1)
2
1205822
≫
(Δ119891)2
sum119872
119898=1(119898 minus 1)
4
1198882
+ 2Δ119891sum119872
119898=1(119898 minus 1)
3
120582119888 thus thefrequency increment has a small impact on the CRLBsIn [33] a frequency offset selection strategy is derived toprecisely steer the beam toward a fixed range with a desiredangle
Figure 4 shows the CRLBs of angle and range whenboth the angle and range are unknown The CRLBs aresignificantly degraded due to the range-angle coupling Con-sequently the range and angle of targets cannot be estimateddirectly by the FDA radar However the CRLBs decreaseas the increase of the number of elements and frequencyincrement still holds Moreover generally more elementsmean that better CRLBs performance can be achieved for theFDA radar
To overcome the problem that the range and angle oftargets cannot be estimated directly by the FDA radar weuse the transmit subaperturing strategy on the transmitfrequency increments Figure 5 shows the corresponding
CRLB
of r
ange
estim
atio
n (m
)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
102
101
10020151050
SNR (dB)
Figure 3 CRLB for estimating range versus SNR when the angle isknown
CRLBs where 119872 = 32 is assumed It can be noticed that toobtain a lower CRLB the Δ119891
1and Δ119891
2should have inverse
signs that is one is passive and the other is negative Onereason is that in this case the FDA radar has a wider systembandwidth Figure 6 shows the CRLBs of angle and rangewhen 119872 = 20 is employed It can be noticed that theCRLBs performance improves with the increase of the sensornumber
Under the data model prior to matched filtering wesuppose the following signal parameters bandwidth 119861 =
10MHz repetition period 119879119901
= 1ms and pulse duration1198790
= 250 120583s In this case the approximate expressionsgiven in (31) are valid because the transmitted has a largetime-bandwidth product (119879
0sdot 119861 = 2500 ≫ 1) Figure 7
shows the CRLBs for direction range and Doppler shift asa function of SNR Note that when SNR = minus10 dB and119872 = 32 are employed we can get CRLB
119903119903FDA= 653m
and CRLBΩ119863Ω119863FDA
= 136937 rads that corresponds to theCRLB for velocity is 327 cms (since V = Ω
119863119888(2Ω
119888) and
Ω119888= 2120587119891
119888) Since (Δ119891)2 sum119872
119898=1(119898 minus 1)
4
1198882
≪ 1 the frequencyincrement has a small impact on the CRLBs In additionobserve that the CRLB for 120579 and 120578 is block-diagonal (see(32)) and therefore decoupled that is CRLB
120578120578FDAremains the
same whether or not 120579 is known and similarly CRLB120579120579FDA
is the same whether or not 120578 is known The decouplingis a consequence of the assumed space-time separability ofsignal and noise models and the assumption of the complexamplitude 120573 as an unknown deterministic constant
Figure 8 shows that the CRLBs versus SNR for differentcombinations of Δ119891
1and Δ119891
2 Comparing Figures 8 and
7 the CRLBs have been significantly improved Likewisecomparing Figures 8 and 9 theCRLBs performance improvesas the number of elements increases
8 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
104
103
102
101
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
108
107
106
105
10420151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
Figure 4 Both angle and range are unknown
6 Conclusion
In this paper we derive the CRLB to jointly estimate theattributes of a moving target using FDA radar and computethe corresponding CRLB expressions First we briefly intro-duce the FDA concept and make a summary on the FDAcharacteristics Then we consider two different data modelsnamely pre- and postmatched filtering Under differentsignal and noise models we compute the CRLB expressionsfor estimating the range direction and Doppler shift Wedemonstrate that the FDA radar beamforming is coupledin range and angle and that the targetrsquos range and anglecannot be estimated directly by the FDA radar To overcome
20151050
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
CRLB
of r
ange
estim
atio
n (m
)102
101
100
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
Figure 5 CRLBs of TS-FDA radar with119872 = 32
this problem this paper proposes a transmit subaperturingstrategy for the FDA radar In doing so the range and angle oftargets are estimated from the transmit-receive beamformingoutput Moreover we also specialize the CRLB results tothe case of temporally white noise and a chirp pulse signalExtensive simulation results verify the correctness of thederived CRLBs It is shown that the CRLBs decrease with theincrease of the number of elements and frequency incrementThe CRLBs can be further improved through three aspectsincreasing the number of elements enhancing the systembandwidth by employing a larger frequency increment andusing transmit subaperturing strategy with more subarrays
International Journal of Antennas and Propagation 9
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
CRLB
of r
ange
estim
atio
n (m
)
103
102
101
10020151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
Figure 6 CRLBs of TS-FDA radar with119872 = 20
Appendices
A Derive the CRLB for Angle WhenRange Is Known
To derive the CRLB we start with a well-known expressionfor the FIM under the data model in Section 3 We define thespatial noise covariance matrix as Rn = 120590
2
119899I119872
and signal-to-noise ratio (SNR) as SNR = |120573
0|2
1205902
119899 Suppose the target range
is known the FIM of 120579 is
119868120579120579
= 2Re 119863H120579(120595) (Rminus1
n )119863120579(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
(A1)
For a phased-array radar there is (120595) = 1205730a(120579) We then
have
120597a (120579)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579) (A2)
whereD = diag[0 1 119872 minus 1] and
120597aH (120579)
120597120579
120597a (120579)120597120579
=4120587
2
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A3)
The FIM of the phased-array radar is
119868120579120579phased-array
= 2SNR41205872
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A4)
Similarly for the FDA radar there is (120595) = 1205730a(120579 119903)The
derivation of a(120579 119903) with respect to 120579 is
120597a (120579 119903)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579 119903)
minus 1198952120587119889Δ119891 cos (120579)
119888
times diag [0 1 (119872 minus 1)2
] a (120579 119903)
120597aH (120579 119903)
120597120579
120597a (120579 119903)120597120579
= 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822
+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A5)
Accordingly the FIM of 120579 for the FDA can be expressedas
119868120579120579FDA
= 2SNR sdot 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A6)
10 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
(c) CRLB for Doppler shift versus SNR
Figure 7 General CRLB results of FDA radar
B Derive the CRLB for Range WhenAngle Is Known
Under the data model in Section 3 when the direction 120579 isknown the parameter to be estimated is 119903 The FIM of 119903 is
119868119903119903FDA
= 2Re 119863H119903(120595) (Rminus1
n )119863119903(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597aH (120579 119903)
120597119903Rminus1
n120597a (120579 119903)
120597119903
(B1)
The derivation of a(120579 119903) with respect to 119903 for FDA is
120597a (120579 119903)120597119903
= 1198952120587Δ119891
119888Da (120579 119903)
120597aH (120579 119903)
120597119903
120597a (120579 119903)120597119903
=4120587
2
Δ1198912
1198882
119872
sum
119898=1
(119898 minus 1)2
(B2)
The FIM of 119903 is thus given by
119868119903119903FDA
= 2SNR4120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(B3)
International Journal of Antennas and Propagation 11
C Derive the CRLB for Range and Angle
Under the data model in Section 3 both the angle and rangeare unknown The range and angle of targets are estimatedjointly The FIM for parameters 120579 and 119903 can be expressed as
IFDA = 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) = [
119868120579120579
119868120579119903
119868119903120579
119868119903119903
] (C1)
We then have
IFDA = 210038161003816100381610038161205730
1003816100381610038161003816
2
times
[[[[
[
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
]]]]
]
(C2)
Since (120597H(120595)120597119903)Rminus1
n (120597(120595)120597120579) = (120597H(120595)
120597120579)Rminus1
n (120597(120595)120597119903) then 119868120579119903= 119868
119903120579 We can get
119868120579120579
= 210038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
= 81205872
1198892cos2 (120579)
times SNRsum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
119868120579119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
= minus81205872
119889Δ119891 cos (120579)
times SNR[sum
119872
119898=1(119898 minus 1)
2
120582119888+Δ119891sum
119872
119898=1(119898 minus 1)
3
1198882]
119868119903119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
= SNR8120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(C3)
Since CRLB120579120579FDA
= [Iminus1
FDA]11 CRLB119903119903FDA
= [Iminus1
FDA]22 Theexpressions for the CRLB of angle and range estimationsgiven in (14) can be obtained by substituting (C3) into (C2)
D General CRLB Results
Under the data model in Section 4 we derive the continuousCRLB expressions for temporally and spatially white noiseand denote SNR = |120573|
2
s1205902
119899with s being the power We
also start with FIM
I120581119894120581119895
= 2Re 119863H120581119894
(120581) (Cminus1
n otimes Rminus1
n )119863120581119895(120581) (D1)
where 120581119894is the 119894th element of 120581 and 119863
120581119894(120595) = 120597(120581)120597120581
119894
Consider
IFDA =
[[[[[
[
119868120573120573
119868119879
120579120573119868119879
120578120573
119868120579120573
119868120579120579
119868119879
120578120579
119868120578120573
119868120578120579
119868120578120578
]]]]]
]
(D2)
For clarity we rewrite Fisherrsquos information matrix I as
IFDA = [A UV B] (D3)
where
V = U119879
(D4a)
V = [119868120579120573
119868120578120573
] (D4b)
B = [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] (D4c)
According to the matrix inversion lemma the inversematrix of IFDA is
Iminus1
FDA = [
[
(A minus UBminus1V)minus1
minusAminus1U(B minus VAminus1U)minus1
minusBminus1V(A minus UBminus1V)minus1
(B minus VAminus1U)minus1
]
]
(D5)
where the submatrix (A minus UBminus1V)minus1 corresponds to theCRLBof120573which is nuisance and (B minus VAminus1U)minus1 correspondsto the CRLB of 120579 and 120578 = [119903 Ω
119863]119879 which are of interest
CRLB120579120578120579120578FDA
= (B minus VAminus1U)minus1
= [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] minus [119868120579120573
119868120578120573
] 119868minus1
120573120573[119868
119879
120579120573119868119879
120578120573]
minus1
(D6)
where
119868120573120573
= 2 sdot[[[
[
s sdot 1198721205902
119899
0
0s sdot 1198721205902
119899
]]]
]
(D7a)
119868120579120573
= 2 sdot Re[1 119895] otimess1205902
119899
sdot 120573lowast
1198601 (D7b)
119868120578120573
= 2 sdot Re[1 119895] otimes120573
lowast
1205902
119908
sdot [119872 sdot 1198603+ s sdot 119860
4] (D7c)
119868120578120578
= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
[119872 sdot 1198606+ 119860
H3119860
4+ 119860
3119860
H4+ s sdot 119860
7]
(D7d)
12 International Journal of Antennas and Propagation
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHz
70001 70001 70001 70002100286514
100286515
100286515
kHz
(c) CRLB for Doppler shift versus SNR
Figure 8 General CRLB results of TS-FDA radar with119872 = 32
119868120579120579minus (119868
120579120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198602minus119860
1119860
H1
119872) (D8a)
119868120578120579minus (119868
120578120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198605minus119860
4119860
H1
119872) (D8b)
119868120578120578
minus (119868120578120573119868119879
120578120573)
1205902
119899
s119872
= 2
10038161003816100381610038161205731003816100381610038161003816
2
s1205902
119899
(119872119860
6
s+ 119860
7minus119872119860
3119860
H3
s2minus119860
4119860
H4
119872)
(D8c)
1198601= minus119895(
2120587119889 cos (120579)sum119872
119898=1(119898 minus 1)
120582
+2120587119889Δ119891 cos (120579)sum119872
119898=1(119898 minus 1)
2
119888)
(D9a)
1198602= 4120587
2
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(D9b)
International Journal of Antennas and Propagation 13
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(c) CRLB for Doppler shift versus SNR
Figure 9 General CRLB results of TS-FDA radar with119872 = 20
1198603= [int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905 int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
119879
(D9c)
1198604= [
2120587Δ119891sum119872
119898=1(119898 minus 1)
1198880]
119879
(D9d)
1198605= [minus4120587
2119889Δ119891 cos(120579) [
sum119872
119898=1(119898 minus 1)
2
12120582119888
+
Δ119891sum119872
119898=1(119898 minus 1)
3
121198882
] 0]
119879
(D9e)
1198606=[[
[
int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905 int
infin
minusinfin
119905119904H(119905 minus 120591)
119889119904 (119905 minus 120591)
119889119905
119889119905
int
infin
minusinfin
119905119904 (119905 minus 120591)
119889119904(119905 minus 120591)H
119889119905
119889119905 int
infin
minusinfin
1199052|119904 (119905 minus 120591)|
2119889119905
]]
]
(D9f)
1198607=[[
[
41205872
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
0
0 0
]]
]
(D9g)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework described in this paper was supported in part by theNational Natural Science Foundation of China under Grant
14 International Journal of Antennas and Propagation
41101317 the Program for New Century Excellent Talentsin University under Grant NCET-12-0095 Sichuan ProvinceScience Fund for Distinguished Young Scholars under Grant2013JQ0003 and Fundamental Research Fund for theCentralUniversities
References
[1] F Bandiera M Mancino and G Ricci ldquoLocalization strategiesfor multiple point-like radar targetsrdquo IEEE Transactions onSignal Processing vol 60 no 12 pp 6708ndash6712 2012
[2] D R Fuhrmann J P Browning and M Rangaswamy ldquoSignal-ing strategies for the hybrid MIMO phased-array radarrdquo IEEEJournal on Selected Topics in Signal Processing vol 4 no 1 pp66ndash78 2010
[3] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011
[4] M Compagnoni P Bestagini F Antonacci A Sarti and STubaro ldquoLocalization of acoustic sources through the fittingof propagation cones using multiple independent arraysrdquo IEEETransactions on Audio Speech and Language Processing vol 20no 7 pp 1964ndash1975 2012
[5] P Antonik M C Wicks H D Griffiths and C J BakerldquoFrequency diverse array radarsrdquo in Proceedings of the IEEERadar Conference (RADAR rsquo06) pp 215ndash217 Verona NY USAApril 2006
[6] P Antonik M CWicks H D Griffiths and C J Baker ldquoMulti-mission multi-mode waveform diversityrdquo in Proceedings of theIEEE Radar Conference (RADAR rsquo06) pp 580ndash582 Verona NYUSA April 2006
[7] P Antonik H D Griffiths and C J Baker ldquoRange depen-dent beamforming using element level waveform diversityrdquo inProceedings of the International Waveform Diversity and DesignConference pp 1ndash4 Las Vegas Nev USA January 2006
[8] P Baizert T B HaleM A Temple andM CWicks ldquoForward-looking radar GMTI benefits using a linear frequency diversearrayrdquo Electronics Letters vol 42 no 22 pp 1311ndash1312 2006
[9] B W Jung R S Adve and J Chun ldquoFrequency diversity inmultistatic radarsrdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo08) pp 1ndash6 Rome Italy May 2008
[10] M Secmen S Demir A Hizal and T Eker ldquoFrequencydiverse array antenna with periodic time modulated pattern inrange and anglerdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo07) pp 427ndash430 Boston Mass USA April 2007
[11] J Huang K F Tong and C J Baker ldquoFrequency diversearray with beam scanning featurerdquo in Proceedings of the IEEEAntennas and Propagation Conference (AP-S rsquo08) pp 1ndash4 SanDiego Calif USA July 2008
[12] T Higgins and S D Blunt ldquoAnalysis of range-angle coupledbeamforming with frequency-diverse chirpsrdquo in Proceedings ofthe International Waveform Diversity and Design Conference(WDD rsquo09) pp 140ndash144 Orlando Fla USA February 2009
[13] J Farooq M A Temple and M A Saville ldquoApplication of fre-quency diverse arrays to synthetic aperture radar imagingrdquo inProceedings of the International Conference on Electromagneticsin Advanced Applications (ICEAA rsquo07) pp 447ndash449 TorinoItaly September 2007
[14] J Farooq M A Temple and M A Saville ldquoExploiting fre-quency diverse array processing to improve SAR image reso-lutionrdquo in Proceedings of the IEEE Radar Conference (RADARrsquo08) pp 1ndash5 Rome Italy May 2008
[15] W QWang ldquoPhased-MIMO radar with frequency diversity forrangedependent beamformingrdquo IEEE Sensors Journal vol 13no 8 pp 1320ndash1328 2013
[16] A L Swindlehurst and P Stoica ldquoMaximum likelihood meth-ods in radar array signal processingrdquo Proceedings of the IEEEvol 86 no 2 pp 421ndash441 1998
[17] J Ward ldquoCramer-Rao bounds for target angle and Dopplerestimation with space-time adaptive processing radarrdquo in Pro-ceedings of the 29th Asilomar Conference on Signals Systems andComputers pp 1198ndash1202 Pacific Grove Calif USA November1995
[18] A Dogandzic and A Nehorai ldquoCramer-Rao bounds for esti-mating range velocity and direction with an active arrayrdquo IEEETransactions on Signal Processing vol 49 no 6 pp 1122ndash11372001
[19] A Dogandzic and A Nehorai ldquoEstimating range velocity anddirection with a radar arrayrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo99) pp 2773ndash2776 Phoenix Ariz USA March 1999
[20] J J Zhang G Maalouli A P Suppappola and D MorrellldquoCramer-Rao lower bounds for the joint estimation of targetattributes using MIMO radarrdquo in Proceedings of the Interna-tional Waveform Diversity and Design Conference (WDD rsquo09)pp 103ndash107 Orlando Fla USA February 2009
[21] A Hassanien S A Vorobyov and A B Gershman ldquoMovingtarget parameters estimation in noncoherent MIMO radarsystemsrdquo IEEE Transactions on Signal Processing vol 60 no 5pp 2354ndash2361 2012
[22] Q He R S Blum and A M Haimovich ldquoNoncoherent MIMOradar for location and velocity estimation more antennasmeans better performancerdquo IEEE Transactions on Signal Pro-cessing vol 58 no 7 pp 3661ndash3680 2010
[23] S M Kay Fundamentals of Statistical Signal Processing Estima-tion Theory vol 1 Pearson London UK 2nd edition 2011
[24] H L van TreesOptimum Array Processing JohnWiley amp SonsNew York NY USA 2002
[25] D Wilcox and M Sellathurai ldquoOn MIMO radar subarrayedtransmit beamformingrdquo IEEE Transactions on Signal Processingvol 60 no 4 pp 2076ndash2081 2012
[26] WQWang andH Z Shao ldquoRange-angle localization of targetsby a double-pulse frequency diverse array radarrdquo IEEE Journalon Selected Topics in Signal Processing vol 8 no 1 pp 106ndash1142014
[27] S Gogineni and A Nehorai ldquoTarget estimation using sparsemodeling for distributed MIMO radarrdquo IEEE Transactions onSignal Processing vol 59 no 11 pp 5315ndash5325 2011
[28] T Li andA Nehorai ldquoMaximum likelihood direction finding inspatially colored noise fields using sparse sensor arraysrdquo IEEETransactions on Signal Processing vol 59 no 3 pp 1048ndash10622011
[29] S Sen ldquoOFDMradar space-time adaptive processing by exploit-ing spatio-temporal sparsityrdquo IEEE Transactions on SignalProcessing vol 61 no 1 pp 118ndash130 2013
[30] J J Blanz A Papathanassiou M Haardt I Furio and P WBaier ldquoSmart antennas for combined DOA and joint channelestimation in time-slotted CDMA mobile radio systems withjoint detectionrdquo IEEE Transactions onVehicular Technology vol49 no 2 pp 293ndash306 2000
[31] P N Pathirana S C K Herath and A V Savkin ldquoMultitargettracking via space transformations using a single frequencycontinuous wave radarrdquo IEEE Transactions on Signal Processingvol 60 no 10 pp 5217ndash5229 2012
International Journal of Antennas and Propagation 15
[32] P H Leong T D Abhayapala and T A Lamahewa ldquoMultipletarget localization using wideband echo chirp signalsrdquo IEEETransactions on Signal Processing vol 61 no 16 pp 4077ndash40892013
[33] L Zhuang X Liu and W Yu ldquoPrecisely beam steering forfrequency diverse arrays based on frequency offset selectionrdquoin Proceedings of the International Radar Conference (RADARrsquo09) pp 1ndash4 Bordeaux France December 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 5
of the signal bandwidth which is the narrowband arrayassumption in array processing
Define the vector of unknown target parameters as120581 = [Re120573 Imag120573 120579 120578119879
] Stacking all samples into a singlevector (21) can be rewritten as
z = 120573 sdot 120601 (120578) otimes a (120579 119903) + n = (120581) + n (22)
where otimes denotes the Kronecker product (120581) =
120573 sdot 120601(120578) otimes a(120579 119903)
120601 (120578) = [119904 [1 minus2119903
119888 sdot Δ119905] exp (119895119908
1198631)
119904 [2 minus2119903
119888 sdot Δ119905] exp (119895119908
1198632)
119904 [119871 minus2119903
119888 sdot Δ119905] exp (119895119908
119863119871)]
119879
(23)
The noise n is assumed to be zero-mean Gaussian spatiallyand temporally correlated with spatiotemporal covariance[29]
119864 [nnH] = Cn otimes Rn (24)
where Cn is the temporal noise covariance matrixUnder the above data model the signal and noise param-
eters are disjoint and satisfy the space-time separability [30]
42 CRLBs of Angle Range and Doppler Shift Suppose thereceived signal is completely covered by the observations119897 = 1 119871 and the sampling is dense (ie Δ119905 rarr 0) TheRn is assumed to be constant in the frequency band 119891 isin
(minus1(2Δ119905) 1(2Δ119905)) where 119891 denotes the frequency in thecontinuous-time domain The vector of Doppler shift andtime delay in the continuous-time domain is defined as
120578 = [119903 Ω119863]119879
(25)
For Cn = I119871and Rn = 120590
2
119899I119872
with I119871being an 119871 times 119871
identity matrix We define the signal power s = 120601H 120601 The
CRLBs expression for 120579 and 120578 follows fromAppendix D as thefollowing
CRLB120579120578120579120578FDA
=
1205902
119899
210038161003816100381610038161205731003816100381610038161003816
2s
times
[[[[[[[[[[[
[
12058721198892cos2 (120579)119872(119872
2minus 1)
[
[
1
31205822
+
(Δ119891)2[(2119872minus 1) (8119872
2minus 3119872minus 11)]
451198882(119872+ 1)
+
2Δ119891 (119872minus 1)
3119888120582
]
]
minus1205872119889Δ119891 cos (120579)119872(119872
2minus 1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 0
minus1205872119889Δ119891 cos (120579)119872(119872
2minus 1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 1205872(Δ119891)2119872(119872
2minus 1)
31198882
+
12058511
sImag 12058512
s
0
Imag 12058512
s12058522
s
]]]]]]]]]]]
]
minus1
(26)
where
12058511
= int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905
minus1
s[int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905]
2
(27a)
12058522
= int
infin
minusinfin
1199052
|119904 (119905 minus 120591)|2
119889119905
minus1
s[int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
2
(27b)
12058512
= int
infin
minusinfin
119905119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905
minus [1
sint
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905
sdot int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905]
(28)
The terms 12058511
and 12058522
are proportional to the rootmean squared (RMS) bandwidth and RMS duration of 119904(119905)respectively Note that the decoupling is a consequence ofthe assumed space-time separability of signal and noisemodels and the assumption of the complex amplitude 120573 as anunknown deterministic constant In addition since the signaland additive noise parameters are disjoint the above CRLBexpressions hold regardless of whether the spatiotemporalnoise covariance Cn otimes Rn is known or unknown
43 CRLBs When Chirp Pulse Signal Is Employed In thissection we derive the CRLB expressions for the rangeDoppler shift and direction estimation when 119904(119905) is a chirppulse signal with a large time-bandwidth product [31 32] Itcan be expressed as
119904 (119905) =
119875minus1
sum
119901=0
1199040(119905 minus 119901119879
119901) (29)
6 International Journal of Antennas and Propagation
where 119879119901is the pulse repetition interval 119875 is the number of
chirp pulses and 1199040(119905) is expressed as [12]
1199040(119905) = exp [119895120587 119861
1198790
(119905 minus1
21198790)
2
] sdot [ℎ (119905) minus ℎ (119905 minus 1198790)]
(30)
where1198790is the chirp pulse duration119861 is the chirp bandwidth
and ℎ(119905) is the Heaviside step function
Assume the time-bandwidth product of the pulse is1198790sdot 119861 ≫ 1 Using the signal given in (29) and (30) in
continuous-time domain we obtain the signal power s =
1198751198790 120585
11= 1198754120587
2
1198612
11987903119888
2 Imag12058512 = minus(119875120587119861119879
2
03119888) and
12058522
= (119875MT3
012)[1 + (119879
0119879
119877)2
(1198752
minus 1)] Thus the CRLBexpressions of 120579 and 120578 for FDA are derived as the followingequation
CRLB120579120578120579120578FDA
=
1
2SNR
sdot
[[[[[[[[[[[[
[
12058721198892cos2(120579)119872(119872
2minus 1)
[
[
1
31205822+
(Δ119891)2[(2119872minus 1) (8119872
2minus3119872minus11)]
451198882119872(119872+ 1)
+
2Δ119891 (119872minus 1)
3119888120582
]
]
minus1205872119889Δ119891 cos(120579)119872(119872
2minus1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 0
minus1205872119889Δ119891 cos(120579)119872(119872
2minus 1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 1205872(Δ119891)2119872(119872
2minus1)
31198882
+
412058721198612119872
31198882
minus
1198721205871198611198790
3119888
0 minus
1198721205871198611198790
3119888
1198721198792
0
12
[1+(
119879119877
1198790
)
2
(1198752minus1)]
]]]]]]]]]]]]
]
minus1
(31)
Specially for the phased array (ie Δ119891 = 0) the CRLB is
CRLB120579120578120579120578phased-array
=1
2SNR
times
[[[[[[[[
[
1205872
1198892cos2 (120579)119872(119872
2
minus 1)
312058220 0
04120587
2
1198612
119872
31198882minus119872120587119861119879
0
3119888
0 minus119872120587119861119879
0
3119888
1198721198792
0
12[1 + (
119879119877
1198790
)
2
(1198752
minus 1)]
]]]]]]]]
]
minus1
(32)
When only one pulse (119875 = 1) is used the CRLBin (32) will be infinite because the model is not identifi-able and the range and Doppler shift cannot be uniquelyestimated [18] since phase-array radar has no rangeidentity capability In contrast (31) does not have sucha problem
According to the previous discussion the angle and rangeof targets cannot be estimated jointly due to the range-anglecoupling Using the similar transmit subaperturing approachpresented in Section 3 and steering vector b(120579 119903) (15) theCRLBs of angle range and Doppler shift are derived as inthe following equation
CRLB120579120578120579120578TS-FDA =
1
2SNR
times
[[[[[[[[[[[[[[[[[[[
[
412058721198892cos2 (120579)1205822
2119873
sum
119898=1
(119898 minus 1)2
minus
41205872119889 cos (120579)120582119888
[[[[
[
(Δ1198911 + Δ1198912)
119873
sum
119898=1
(119898 minus 1)2
+119873Δ1198912
119873
sum
119898=1
(119898 minus 1)
]]]]
]
0
minus
41205872119889 cos (120579)120582119888
[[[[
[
(Δ1198911 + Δ1198912)
119873
sum
119898=1
(119898 minus 1)2
+119873Δ1198912
119873
sum
119898=1
(119898 minus 1)
]]]]
]
(Δ1198912
1+ Δ1198912
2)
1198882
119873
sum
119898=1
(119898 minus 1)2+
412058721198612119872
31198882
minus
1198721205871198611198790
3119888
0 minus
1198721205871198611198790
3119888
1198721198792
0
12
[1 + (
119879119877
1198790
)
2
(1198752minus 1)]
]]]]]]]]]]]]]]]]]]]
]
minus1
(33)
International Journal of Antennas and Propagation 7
20151050
10minus2
10minus3
10minus47001 7002 7003
10minus248203
10minus248201
10minus248199
CRLB
of a
ngle
estim
atio
n (d
eg)
SNR (dB)
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
Figure 2 CRLB for estimating angle versus SNR when the range isknown
5 Simulation and Verification
In this section we consider several numerical examples thatcompare the CRLBs in different signal and noise modelsConsider an X-band FDA radar with the carrier frequency1198910
= 10GHz We assume a ULA of 119872 elements usedfor transmitting The array elements are spaced half of thewavelength apart from each other namely 119889 = 1205822 Onetarget of interest is supposed to reflect a plane wave thatimpinges on the array from direction of angle 120579 = 30
∘Under the signal model after matched filtering Figures
2 and 3 compare the CRLBs according to (9) and (13)respectively It can be noticed that the CRLBs are improvedwhen a larger number of elements are employed Howeverit has no significant difference when different frequencyincrements are used This is because sum
119872
119898=1(119898 minus 1)
2
1205822
≫
(Δ119891)2
sum119872
119898=1(119898 minus 1)
4
1198882
+ 2Δ119891sum119872
119898=1(119898 minus 1)
3
120582119888 thus thefrequency increment has a small impact on the CRLBsIn [33] a frequency offset selection strategy is derived toprecisely steer the beam toward a fixed range with a desiredangle
Figure 4 shows the CRLBs of angle and range whenboth the angle and range are unknown The CRLBs aresignificantly degraded due to the range-angle coupling Con-sequently the range and angle of targets cannot be estimateddirectly by the FDA radar However the CRLBs decreaseas the increase of the number of elements and frequencyincrement still holds Moreover generally more elementsmean that better CRLBs performance can be achieved for theFDA radar
To overcome the problem that the range and angle oftargets cannot be estimated directly by the FDA radar weuse the transmit subaperturing strategy on the transmitfrequency increments Figure 5 shows the corresponding
CRLB
of r
ange
estim
atio
n (m
)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
102
101
10020151050
SNR (dB)
Figure 3 CRLB for estimating range versus SNR when the angle isknown
CRLBs where 119872 = 32 is assumed It can be noticed that toobtain a lower CRLB the Δ119891
1and Δ119891
2should have inverse
signs that is one is passive and the other is negative Onereason is that in this case the FDA radar has a wider systembandwidth Figure 6 shows the CRLBs of angle and rangewhen 119872 = 20 is employed It can be noticed that theCRLBs performance improves with the increase of the sensornumber
Under the data model prior to matched filtering wesuppose the following signal parameters bandwidth 119861 =
10MHz repetition period 119879119901
= 1ms and pulse duration1198790
= 250 120583s In this case the approximate expressionsgiven in (31) are valid because the transmitted has a largetime-bandwidth product (119879
0sdot 119861 = 2500 ≫ 1) Figure 7
shows the CRLBs for direction range and Doppler shift asa function of SNR Note that when SNR = minus10 dB and119872 = 32 are employed we can get CRLB
119903119903FDA= 653m
and CRLBΩ119863Ω119863FDA
= 136937 rads that corresponds to theCRLB for velocity is 327 cms (since V = Ω
119863119888(2Ω
119888) and
Ω119888= 2120587119891
119888) Since (Δ119891)2 sum119872
119898=1(119898 minus 1)
4
1198882
≪ 1 the frequencyincrement has a small impact on the CRLBs In additionobserve that the CRLB for 120579 and 120578 is block-diagonal (see(32)) and therefore decoupled that is CRLB
120578120578FDAremains the
same whether or not 120579 is known and similarly CRLB120579120579FDA
is the same whether or not 120578 is known The decouplingis a consequence of the assumed space-time separability ofsignal and noise models and the assumption of the complexamplitude 120573 as an unknown deterministic constant
Figure 8 shows that the CRLBs versus SNR for differentcombinations of Δ119891
1and Δ119891
2 Comparing Figures 8 and
7 the CRLBs have been significantly improved Likewisecomparing Figures 8 and 9 theCRLBs performance improvesas the number of elements increases
8 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
104
103
102
101
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
108
107
106
105
10420151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
Figure 4 Both angle and range are unknown
6 Conclusion
In this paper we derive the CRLB to jointly estimate theattributes of a moving target using FDA radar and computethe corresponding CRLB expressions First we briefly intro-duce the FDA concept and make a summary on the FDAcharacteristics Then we consider two different data modelsnamely pre- and postmatched filtering Under differentsignal and noise models we compute the CRLB expressionsfor estimating the range direction and Doppler shift Wedemonstrate that the FDA radar beamforming is coupledin range and angle and that the targetrsquos range and anglecannot be estimated directly by the FDA radar To overcome
20151050
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
CRLB
of r
ange
estim
atio
n (m
)102
101
100
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
Figure 5 CRLBs of TS-FDA radar with119872 = 32
this problem this paper proposes a transmit subaperturingstrategy for the FDA radar In doing so the range and angle oftargets are estimated from the transmit-receive beamformingoutput Moreover we also specialize the CRLB results tothe case of temporally white noise and a chirp pulse signalExtensive simulation results verify the correctness of thederived CRLBs It is shown that the CRLBs decrease with theincrease of the number of elements and frequency incrementThe CRLBs can be further improved through three aspectsincreasing the number of elements enhancing the systembandwidth by employing a larger frequency increment andusing transmit subaperturing strategy with more subarrays
International Journal of Antennas and Propagation 9
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
CRLB
of r
ange
estim
atio
n (m
)
103
102
101
10020151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
Figure 6 CRLBs of TS-FDA radar with119872 = 20
Appendices
A Derive the CRLB for Angle WhenRange Is Known
To derive the CRLB we start with a well-known expressionfor the FIM under the data model in Section 3 We define thespatial noise covariance matrix as Rn = 120590
2
119899I119872
and signal-to-noise ratio (SNR) as SNR = |120573
0|2
1205902
119899 Suppose the target range
is known the FIM of 120579 is
119868120579120579
= 2Re 119863H120579(120595) (Rminus1
n )119863120579(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
(A1)
For a phased-array radar there is (120595) = 1205730a(120579) We then
have
120597a (120579)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579) (A2)
whereD = diag[0 1 119872 minus 1] and
120597aH (120579)
120597120579
120597a (120579)120597120579
=4120587
2
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A3)
The FIM of the phased-array radar is
119868120579120579phased-array
= 2SNR41205872
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A4)
Similarly for the FDA radar there is (120595) = 1205730a(120579 119903)The
derivation of a(120579 119903) with respect to 120579 is
120597a (120579 119903)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579 119903)
minus 1198952120587119889Δ119891 cos (120579)
119888
times diag [0 1 (119872 minus 1)2
] a (120579 119903)
120597aH (120579 119903)
120597120579
120597a (120579 119903)120597120579
= 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822
+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A5)
Accordingly the FIM of 120579 for the FDA can be expressedas
119868120579120579FDA
= 2SNR sdot 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A6)
10 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
(c) CRLB for Doppler shift versus SNR
Figure 7 General CRLB results of FDA radar
B Derive the CRLB for Range WhenAngle Is Known
Under the data model in Section 3 when the direction 120579 isknown the parameter to be estimated is 119903 The FIM of 119903 is
119868119903119903FDA
= 2Re 119863H119903(120595) (Rminus1
n )119863119903(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597aH (120579 119903)
120597119903Rminus1
n120597a (120579 119903)
120597119903
(B1)
The derivation of a(120579 119903) with respect to 119903 for FDA is
120597a (120579 119903)120597119903
= 1198952120587Δ119891
119888Da (120579 119903)
120597aH (120579 119903)
120597119903
120597a (120579 119903)120597119903
=4120587
2
Δ1198912
1198882
119872
sum
119898=1
(119898 minus 1)2
(B2)
The FIM of 119903 is thus given by
119868119903119903FDA
= 2SNR4120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(B3)
International Journal of Antennas and Propagation 11
C Derive the CRLB for Range and Angle
Under the data model in Section 3 both the angle and rangeare unknown The range and angle of targets are estimatedjointly The FIM for parameters 120579 and 119903 can be expressed as
IFDA = 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) = [
119868120579120579
119868120579119903
119868119903120579
119868119903119903
] (C1)
We then have
IFDA = 210038161003816100381610038161205730
1003816100381610038161003816
2
times
[[[[
[
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
]]]]
]
(C2)
Since (120597H(120595)120597119903)Rminus1
n (120597(120595)120597120579) = (120597H(120595)
120597120579)Rminus1
n (120597(120595)120597119903) then 119868120579119903= 119868
119903120579 We can get
119868120579120579
= 210038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
= 81205872
1198892cos2 (120579)
times SNRsum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
119868120579119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
= minus81205872
119889Δ119891 cos (120579)
times SNR[sum
119872
119898=1(119898 minus 1)
2
120582119888+Δ119891sum
119872
119898=1(119898 minus 1)
3
1198882]
119868119903119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
= SNR8120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(C3)
Since CRLB120579120579FDA
= [Iminus1
FDA]11 CRLB119903119903FDA
= [Iminus1
FDA]22 Theexpressions for the CRLB of angle and range estimationsgiven in (14) can be obtained by substituting (C3) into (C2)
D General CRLB Results
Under the data model in Section 4 we derive the continuousCRLB expressions for temporally and spatially white noiseand denote SNR = |120573|
2
s1205902
119899with s being the power We
also start with FIM
I120581119894120581119895
= 2Re 119863H120581119894
(120581) (Cminus1
n otimes Rminus1
n )119863120581119895(120581) (D1)
where 120581119894is the 119894th element of 120581 and 119863
120581119894(120595) = 120597(120581)120597120581
119894
Consider
IFDA =
[[[[[
[
119868120573120573
119868119879
120579120573119868119879
120578120573
119868120579120573
119868120579120579
119868119879
120578120579
119868120578120573
119868120578120579
119868120578120578
]]]]]
]
(D2)
For clarity we rewrite Fisherrsquos information matrix I as
IFDA = [A UV B] (D3)
where
V = U119879
(D4a)
V = [119868120579120573
119868120578120573
] (D4b)
B = [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] (D4c)
According to the matrix inversion lemma the inversematrix of IFDA is
Iminus1
FDA = [
[
(A minus UBminus1V)minus1
minusAminus1U(B minus VAminus1U)minus1
minusBminus1V(A minus UBminus1V)minus1
(B minus VAminus1U)minus1
]
]
(D5)
where the submatrix (A minus UBminus1V)minus1 corresponds to theCRLBof120573which is nuisance and (B minus VAminus1U)minus1 correspondsto the CRLB of 120579 and 120578 = [119903 Ω
119863]119879 which are of interest
CRLB120579120578120579120578FDA
= (B minus VAminus1U)minus1
= [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] minus [119868120579120573
119868120578120573
] 119868minus1
120573120573[119868
119879
120579120573119868119879
120578120573]
minus1
(D6)
where
119868120573120573
= 2 sdot[[[
[
s sdot 1198721205902
119899
0
0s sdot 1198721205902
119899
]]]
]
(D7a)
119868120579120573
= 2 sdot Re[1 119895] otimess1205902
119899
sdot 120573lowast
1198601 (D7b)
119868120578120573
= 2 sdot Re[1 119895] otimes120573
lowast
1205902
119908
sdot [119872 sdot 1198603+ s sdot 119860
4] (D7c)
119868120578120578
= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
[119872 sdot 1198606+ 119860
H3119860
4+ 119860
3119860
H4+ s sdot 119860
7]
(D7d)
12 International Journal of Antennas and Propagation
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHz
70001 70001 70001 70002100286514
100286515
100286515
kHz
(c) CRLB for Doppler shift versus SNR
Figure 8 General CRLB results of TS-FDA radar with119872 = 32
119868120579120579minus (119868
120579120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198602minus119860
1119860
H1
119872) (D8a)
119868120578120579minus (119868
120578120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198605minus119860
4119860
H1
119872) (D8b)
119868120578120578
minus (119868120578120573119868119879
120578120573)
1205902
119899
s119872
= 2
10038161003816100381610038161205731003816100381610038161003816
2
s1205902
119899
(119872119860
6
s+ 119860
7minus119872119860
3119860
H3
s2minus119860
4119860
H4
119872)
(D8c)
1198601= minus119895(
2120587119889 cos (120579)sum119872
119898=1(119898 minus 1)
120582
+2120587119889Δ119891 cos (120579)sum119872
119898=1(119898 minus 1)
2
119888)
(D9a)
1198602= 4120587
2
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(D9b)
International Journal of Antennas and Propagation 13
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(c) CRLB for Doppler shift versus SNR
Figure 9 General CRLB results of TS-FDA radar with119872 = 20
1198603= [int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905 int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
119879
(D9c)
1198604= [
2120587Δ119891sum119872
119898=1(119898 minus 1)
1198880]
119879
(D9d)
1198605= [minus4120587
2119889Δ119891 cos(120579) [
sum119872
119898=1(119898 minus 1)
2
12120582119888
+
Δ119891sum119872
119898=1(119898 minus 1)
3
121198882
] 0]
119879
(D9e)
1198606=[[
[
int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905 int
infin
minusinfin
119905119904H(119905 minus 120591)
119889119904 (119905 minus 120591)
119889119905
119889119905
int
infin
minusinfin
119905119904 (119905 minus 120591)
119889119904(119905 minus 120591)H
119889119905
119889119905 int
infin
minusinfin
1199052|119904 (119905 minus 120591)|
2119889119905
]]
]
(D9f)
1198607=[[
[
41205872
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
0
0 0
]]
]
(D9g)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework described in this paper was supported in part by theNational Natural Science Foundation of China under Grant
14 International Journal of Antennas and Propagation
41101317 the Program for New Century Excellent Talentsin University under Grant NCET-12-0095 Sichuan ProvinceScience Fund for Distinguished Young Scholars under Grant2013JQ0003 and Fundamental Research Fund for theCentralUniversities
References
[1] F Bandiera M Mancino and G Ricci ldquoLocalization strategiesfor multiple point-like radar targetsrdquo IEEE Transactions onSignal Processing vol 60 no 12 pp 6708ndash6712 2012
[2] D R Fuhrmann J P Browning and M Rangaswamy ldquoSignal-ing strategies for the hybrid MIMO phased-array radarrdquo IEEEJournal on Selected Topics in Signal Processing vol 4 no 1 pp66ndash78 2010
[3] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011
[4] M Compagnoni P Bestagini F Antonacci A Sarti and STubaro ldquoLocalization of acoustic sources through the fittingof propagation cones using multiple independent arraysrdquo IEEETransactions on Audio Speech and Language Processing vol 20no 7 pp 1964ndash1975 2012
[5] P Antonik M C Wicks H D Griffiths and C J BakerldquoFrequency diverse array radarsrdquo in Proceedings of the IEEERadar Conference (RADAR rsquo06) pp 215ndash217 Verona NY USAApril 2006
[6] P Antonik M CWicks H D Griffiths and C J Baker ldquoMulti-mission multi-mode waveform diversityrdquo in Proceedings of theIEEE Radar Conference (RADAR rsquo06) pp 580ndash582 Verona NYUSA April 2006
[7] P Antonik H D Griffiths and C J Baker ldquoRange depen-dent beamforming using element level waveform diversityrdquo inProceedings of the International Waveform Diversity and DesignConference pp 1ndash4 Las Vegas Nev USA January 2006
[8] P Baizert T B HaleM A Temple andM CWicks ldquoForward-looking radar GMTI benefits using a linear frequency diversearrayrdquo Electronics Letters vol 42 no 22 pp 1311ndash1312 2006
[9] B W Jung R S Adve and J Chun ldquoFrequency diversity inmultistatic radarsrdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo08) pp 1ndash6 Rome Italy May 2008
[10] M Secmen S Demir A Hizal and T Eker ldquoFrequencydiverse array antenna with periodic time modulated pattern inrange and anglerdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo07) pp 427ndash430 Boston Mass USA April 2007
[11] J Huang K F Tong and C J Baker ldquoFrequency diversearray with beam scanning featurerdquo in Proceedings of the IEEEAntennas and Propagation Conference (AP-S rsquo08) pp 1ndash4 SanDiego Calif USA July 2008
[12] T Higgins and S D Blunt ldquoAnalysis of range-angle coupledbeamforming with frequency-diverse chirpsrdquo in Proceedings ofthe International Waveform Diversity and Design Conference(WDD rsquo09) pp 140ndash144 Orlando Fla USA February 2009
[13] J Farooq M A Temple and M A Saville ldquoApplication of fre-quency diverse arrays to synthetic aperture radar imagingrdquo inProceedings of the International Conference on Electromagneticsin Advanced Applications (ICEAA rsquo07) pp 447ndash449 TorinoItaly September 2007
[14] J Farooq M A Temple and M A Saville ldquoExploiting fre-quency diverse array processing to improve SAR image reso-lutionrdquo in Proceedings of the IEEE Radar Conference (RADARrsquo08) pp 1ndash5 Rome Italy May 2008
[15] W QWang ldquoPhased-MIMO radar with frequency diversity forrangedependent beamformingrdquo IEEE Sensors Journal vol 13no 8 pp 1320ndash1328 2013
[16] A L Swindlehurst and P Stoica ldquoMaximum likelihood meth-ods in radar array signal processingrdquo Proceedings of the IEEEvol 86 no 2 pp 421ndash441 1998
[17] J Ward ldquoCramer-Rao bounds for target angle and Dopplerestimation with space-time adaptive processing radarrdquo in Pro-ceedings of the 29th Asilomar Conference on Signals Systems andComputers pp 1198ndash1202 Pacific Grove Calif USA November1995
[18] A Dogandzic and A Nehorai ldquoCramer-Rao bounds for esti-mating range velocity and direction with an active arrayrdquo IEEETransactions on Signal Processing vol 49 no 6 pp 1122ndash11372001
[19] A Dogandzic and A Nehorai ldquoEstimating range velocity anddirection with a radar arrayrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo99) pp 2773ndash2776 Phoenix Ariz USA March 1999
[20] J J Zhang G Maalouli A P Suppappola and D MorrellldquoCramer-Rao lower bounds for the joint estimation of targetattributes using MIMO radarrdquo in Proceedings of the Interna-tional Waveform Diversity and Design Conference (WDD rsquo09)pp 103ndash107 Orlando Fla USA February 2009
[21] A Hassanien S A Vorobyov and A B Gershman ldquoMovingtarget parameters estimation in noncoherent MIMO radarsystemsrdquo IEEE Transactions on Signal Processing vol 60 no 5pp 2354ndash2361 2012
[22] Q He R S Blum and A M Haimovich ldquoNoncoherent MIMOradar for location and velocity estimation more antennasmeans better performancerdquo IEEE Transactions on Signal Pro-cessing vol 58 no 7 pp 3661ndash3680 2010
[23] S M Kay Fundamentals of Statistical Signal Processing Estima-tion Theory vol 1 Pearson London UK 2nd edition 2011
[24] H L van TreesOptimum Array Processing JohnWiley amp SonsNew York NY USA 2002
[25] D Wilcox and M Sellathurai ldquoOn MIMO radar subarrayedtransmit beamformingrdquo IEEE Transactions on Signal Processingvol 60 no 4 pp 2076ndash2081 2012
[26] WQWang andH Z Shao ldquoRange-angle localization of targetsby a double-pulse frequency diverse array radarrdquo IEEE Journalon Selected Topics in Signal Processing vol 8 no 1 pp 106ndash1142014
[27] S Gogineni and A Nehorai ldquoTarget estimation using sparsemodeling for distributed MIMO radarrdquo IEEE Transactions onSignal Processing vol 59 no 11 pp 5315ndash5325 2011
[28] T Li andA Nehorai ldquoMaximum likelihood direction finding inspatially colored noise fields using sparse sensor arraysrdquo IEEETransactions on Signal Processing vol 59 no 3 pp 1048ndash10622011
[29] S Sen ldquoOFDMradar space-time adaptive processing by exploit-ing spatio-temporal sparsityrdquo IEEE Transactions on SignalProcessing vol 61 no 1 pp 118ndash130 2013
[30] J J Blanz A Papathanassiou M Haardt I Furio and P WBaier ldquoSmart antennas for combined DOA and joint channelestimation in time-slotted CDMA mobile radio systems withjoint detectionrdquo IEEE Transactions onVehicular Technology vol49 no 2 pp 293ndash306 2000
[31] P N Pathirana S C K Herath and A V Savkin ldquoMultitargettracking via space transformations using a single frequencycontinuous wave radarrdquo IEEE Transactions on Signal Processingvol 60 no 10 pp 5217ndash5229 2012
International Journal of Antennas and Propagation 15
[32] P H Leong T D Abhayapala and T A Lamahewa ldquoMultipletarget localization using wideband echo chirp signalsrdquo IEEETransactions on Signal Processing vol 61 no 16 pp 4077ndash40892013
[33] L Zhuang X Liu and W Yu ldquoPrecisely beam steering forfrequency diverse arrays based on frequency offset selectionrdquoin Proceedings of the International Radar Conference (RADARrsquo09) pp 1ndash4 Bordeaux France December 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 International Journal of Antennas and Propagation
where 119879119901is the pulse repetition interval 119875 is the number of
chirp pulses and 1199040(119905) is expressed as [12]
1199040(119905) = exp [119895120587 119861
1198790
(119905 minus1
21198790)
2
] sdot [ℎ (119905) minus ℎ (119905 minus 1198790)]
(30)
where1198790is the chirp pulse duration119861 is the chirp bandwidth
and ℎ(119905) is the Heaviside step function
Assume the time-bandwidth product of the pulse is1198790sdot 119861 ≫ 1 Using the signal given in (29) and (30) in
continuous-time domain we obtain the signal power s =
1198751198790 120585
11= 1198754120587
2
1198612
11987903119888
2 Imag12058512 = minus(119875120587119861119879
2
03119888) and
12058522
= (119875MT3
012)[1 + (119879
0119879
119877)2
(1198752
minus 1)] Thus the CRLBexpressions of 120579 and 120578 for FDA are derived as the followingequation
CRLB120579120578120579120578FDA
=
1
2SNR
sdot
[[[[[[[[[[[[
[
12058721198892cos2(120579)119872(119872
2minus 1)
[
[
1
31205822+
(Δ119891)2[(2119872minus 1) (8119872
2minus3119872minus11)]
451198882119872(119872+ 1)
+
2Δ119891 (119872minus 1)
3119888120582
]
]
minus1205872119889Δ119891 cos(120579)119872(119872
2minus1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 0
minus1205872119889Δ119891 cos(120579)119872(119872
2minus 1) [
1
3120582119888
+
Δ119891 (119872minus 1)
31198882
] 1205872(Δ119891)2119872(119872
2minus1)
31198882
+
412058721198612119872
31198882
minus
1198721205871198611198790
3119888
0 minus
1198721205871198611198790
3119888
1198721198792
0
12
[1+(
119879119877
1198790
)
2
(1198752minus1)]
]]]]]]]]]]]]
]
minus1
(31)
Specially for the phased array (ie Δ119891 = 0) the CRLB is
CRLB120579120578120579120578phased-array
=1
2SNR
times
[[[[[[[[
[
1205872
1198892cos2 (120579)119872(119872
2
minus 1)
312058220 0
04120587
2
1198612
119872
31198882minus119872120587119861119879
0
3119888
0 minus119872120587119861119879
0
3119888
1198721198792
0
12[1 + (
119879119877
1198790
)
2
(1198752
minus 1)]
]]]]]]]]
]
minus1
(32)
When only one pulse (119875 = 1) is used the CRLBin (32) will be infinite because the model is not identifi-able and the range and Doppler shift cannot be uniquelyestimated [18] since phase-array radar has no rangeidentity capability In contrast (31) does not have sucha problem
According to the previous discussion the angle and rangeof targets cannot be estimated jointly due to the range-anglecoupling Using the similar transmit subaperturing approachpresented in Section 3 and steering vector b(120579 119903) (15) theCRLBs of angle range and Doppler shift are derived as inthe following equation
CRLB120579120578120579120578TS-FDA =
1
2SNR
times
[[[[[[[[[[[[[[[[[[[
[
412058721198892cos2 (120579)1205822
2119873
sum
119898=1
(119898 minus 1)2
minus
41205872119889 cos (120579)120582119888
[[[[
[
(Δ1198911 + Δ1198912)
119873
sum
119898=1
(119898 minus 1)2
+119873Δ1198912
119873
sum
119898=1
(119898 minus 1)
]]]]
]
0
minus
41205872119889 cos (120579)120582119888
[[[[
[
(Δ1198911 + Δ1198912)
119873
sum
119898=1
(119898 minus 1)2
+119873Δ1198912
119873
sum
119898=1
(119898 minus 1)
]]]]
]
(Δ1198912
1+ Δ1198912
2)
1198882
119873
sum
119898=1
(119898 minus 1)2+
412058721198612119872
31198882
minus
1198721205871198611198790
3119888
0 minus
1198721205871198611198790
3119888
1198721198792
0
12
[1 + (
119879119877
1198790
)
2
(1198752minus 1)]
]]]]]]]]]]]]]]]]]]]
]
minus1
(33)
International Journal of Antennas and Propagation 7
20151050
10minus2
10minus3
10minus47001 7002 7003
10minus248203
10minus248201
10minus248199
CRLB
of a
ngle
estim
atio
n (d
eg)
SNR (dB)
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
Figure 2 CRLB for estimating angle versus SNR when the range isknown
5 Simulation and Verification
In this section we consider several numerical examples thatcompare the CRLBs in different signal and noise modelsConsider an X-band FDA radar with the carrier frequency1198910
= 10GHz We assume a ULA of 119872 elements usedfor transmitting The array elements are spaced half of thewavelength apart from each other namely 119889 = 1205822 Onetarget of interest is supposed to reflect a plane wave thatimpinges on the array from direction of angle 120579 = 30
∘Under the signal model after matched filtering Figures
2 and 3 compare the CRLBs according to (9) and (13)respectively It can be noticed that the CRLBs are improvedwhen a larger number of elements are employed Howeverit has no significant difference when different frequencyincrements are used This is because sum
119872
119898=1(119898 minus 1)
2
1205822
≫
(Δ119891)2
sum119872
119898=1(119898 minus 1)
4
1198882
+ 2Δ119891sum119872
119898=1(119898 minus 1)
3
120582119888 thus thefrequency increment has a small impact on the CRLBsIn [33] a frequency offset selection strategy is derived toprecisely steer the beam toward a fixed range with a desiredangle
Figure 4 shows the CRLBs of angle and range whenboth the angle and range are unknown The CRLBs aresignificantly degraded due to the range-angle coupling Con-sequently the range and angle of targets cannot be estimateddirectly by the FDA radar However the CRLBs decreaseas the increase of the number of elements and frequencyincrement still holds Moreover generally more elementsmean that better CRLBs performance can be achieved for theFDA radar
To overcome the problem that the range and angle oftargets cannot be estimated directly by the FDA radar weuse the transmit subaperturing strategy on the transmitfrequency increments Figure 5 shows the corresponding
CRLB
of r
ange
estim
atio
n (m
)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
102
101
10020151050
SNR (dB)
Figure 3 CRLB for estimating range versus SNR when the angle isknown
CRLBs where 119872 = 32 is assumed It can be noticed that toobtain a lower CRLB the Δ119891
1and Δ119891
2should have inverse
signs that is one is passive and the other is negative Onereason is that in this case the FDA radar has a wider systembandwidth Figure 6 shows the CRLBs of angle and rangewhen 119872 = 20 is employed It can be noticed that theCRLBs performance improves with the increase of the sensornumber
Under the data model prior to matched filtering wesuppose the following signal parameters bandwidth 119861 =
10MHz repetition period 119879119901
= 1ms and pulse duration1198790
= 250 120583s In this case the approximate expressionsgiven in (31) are valid because the transmitted has a largetime-bandwidth product (119879
0sdot 119861 = 2500 ≫ 1) Figure 7
shows the CRLBs for direction range and Doppler shift asa function of SNR Note that when SNR = minus10 dB and119872 = 32 are employed we can get CRLB
119903119903FDA= 653m
and CRLBΩ119863Ω119863FDA
= 136937 rads that corresponds to theCRLB for velocity is 327 cms (since V = Ω
119863119888(2Ω
119888) and
Ω119888= 2120587119891
119888) Since (Δ119891)2 sum119872
119898=1(119898 minus 1)
4
1198882
≪ 1 the frequencyincrement has a small impact on the CRLBs In additionobserve that the CRLB for 120579 and 120578 is block-diagonal (see(32)) and therefore decoupled that is CRLB
120578120578FDAremains the
same whether or not 120579 is known and similarly CRLB120579120579FDA
is the same whether or not 120578 is known The decouplingis a consequence of the assumed space-time separability ofsignal and noise models and the assumption of the complexamplitude 120573 as an unknown deterministic constant
Figure 8 shows that the CRLBs versus SNR for differentcombinations of Δ119891
1and Δ119891
2 Comparing Figures 8 and
7 the CRLBs have been significantly improved Likewisecomparing Figures 8 and 9 theCRLBs performance improvesas the number of elements increases
8 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
104
103
102
101
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
108
107
106
105
10420151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
Figure 4 Both angle and range are unknown
6 Conclusion
In this paper we derive the CRLB to jointly estimate theattributes of a moving target using FDA radar and computethe corresponding CRLB expressions First we briefly intro-duce the FDA concept and make a summary on the FDAcharacteristics Then we consider two different data modelsnamely pre- and postmatched filtering Under differentsignal and noise models we compute the CRLB expressionsfor estimating the range direction and Doppler shift Wedemonstrate that the FDA radar beamforming is coupledin range and angle and that the targetrsquos range and anglecannot be estimated directly by the FDA radar To overcome
20151050
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
CRLB
of r
ange
estim
atio
n (m
)102
101
100
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
Figure 5 CRLBs of TS-FDA radar with119872 = 32
this problem this paper proposes a transmit subaperturingstrategy for the FDA radar In doing so the range and angle oftargets are estimated from the transmit-receive beamformingoutput Moreover we also specialize the CRLB results tothe case of temporally white noise and a chirp pulse signalExtensive simulation results verify the correctness of thederived CRLBs It is shown that the CRLBs decrease with theincrease of the number of elements and frequency incrementThe CRLBs can be further improved through three aspectsincreasing the number of elements enhancing the systembandwidth by employing a larger frequency increment andusing transmit subaperturing strategy with more subarrays
International Journal of Antennas and Propagation 9
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
CRLB
of r
ange
estim
atio
n (m
)
103
102
101
10020151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
Figure 6 CRLBs of TS-FDA radar with119872 = 20
Appendices
A Derive the CRLB for Angle WhenRange Is Known
To derive the CRLB we start with a well-known expressionfor the FIM under the data model in Section 3 We define thespatial noise covariance matrix as Rn = 120590
2
119899I119872
and signal-to-noise ratio (SNR) as SNR = |120573
0|2
1205902
119899 Suppose the target range
is known the FIM of 120579 is
119868120579120579
= 2Re 119863H120579(120595) (Rminus1
n )119863120579(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
(A1)
For a phased-array radar there is (120595) = 1205730a(120579) We then
have
120597a (120579)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579) (A2)
whereD = diag[0 1 119872 minus 1] and
120597aH (120579)
120597120579
120597a (120579)120597120579
=4120587
2
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A3)
The FIM of the phased-array radar is
119868120579120579phased-array
= 2SNR41205872
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A4)
Similarly for the FDA radar there is (120595) = 1205730a(120579 119903)The
derivation of a(120579 119903) with respect to 120579 is
120597a (120579 119903)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579 119903)
minus 1198952120587119889Δ119891 cos (120579)
119888
times diag [0 1 (119872 minus 1)2
] a (120579 119903)
120597aH (120579 119903)
120597120579
120597a (120579 119903)120597120579
= 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822
+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A5)
Accordingly the FIM of 120579 for the FDA can be expressedas
119868120579120579FDA
= 2SNR sdot 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A6)
10 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
(c) CRLB for Doppler shift versus SNR
Figure 7 General CRLB results of FDA radar
B Derive the CRLB for Range WhenAngle Is Known
Under the data model in Section 3 when the direction 120579 isknown the parameter to be estimated is 119903 The FIM of 119903 is
119868119903119903FDA
= 2Re 119863H119903(120595) (Rminus1
n )119863119903(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597aH (120579 119903)
120597119903Rminus1
n120597a (120579 119903)
120597119903
(B1)
The derivation of a(120579 119903) with respect to 119903 for FDA is
120597a (120579 119903)120597119903
= 1198952120587Δ119891
119888Da (120579 119903)
120597aH (120579 119903)
120597119903
120597a (120579 119903)120597119903
=4120587
2
Δ1198912
1198882
119872
sum
119898=1
(119898 minus 1)2
(B2)
The FIM of 119903 is thus given by
119868119903119903FDA
= 2SNR4120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(B3)
International Journal of Antennas and Propagation 11
C Derive the CRLB for Range and Angle
Under the data model in Section 3 both the angle and rangeare unknown The range and angle of targets are estimatedjointly The FIM for parameters 120579 and 119903 can be expressed as
IFDA = 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) = [
119868120579120579
119868120579119903
119868119903120579
119868119903119903
] (C1)
We then have
IFDA = 210038161003816100381610038161205730
1003816100381610038161003816
2
times
[[[[
[
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
]]]]
]
(C2)
Since (120597H(120595)120597119903)Rminus1
n (120597(120595)120597120579) = (120597H(120595)
120597120579)Rminus1
n (120597(120595)120597119903) then 119868120579119903= 119868
119903120579 We can get
119868120579120579
= 210038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
= 81205872
1198892cos2 (120579)
times SNRsum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
119868120579119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
= minus81205872
119889Δ119891 cos (120579)
times SNR[sum
119872
119898=1(119898 minus 1)
2
120582119888+Δ119891sum
119872
119898=1(119898 minus 1)
3
1198882]
119868119903119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
= SNR8120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(C3)
Since CRLB120579120579FDA
= [Iminus1
FDA]11 CRLB119903119903FDA
= [Iminus1
FDA]22 Theexpressions for the CRLB of angle and range estimationsgiven in (14) can be obtained by substituting (C3) into (C2)
D General CRLB Results
Under the data model in Section 4 we derive the continuousCRLB expressions for temporally and spatially white noiseand denote SNR = |120573|
2
s1205902
119899with s being the power We
also start with FIM
I120581119894120581119895
= 2Re 119863H120581119894
(120581) (Cminus1
n otimes Rminus1
n )119863120581119895(120581) (D1)
where 120581119894is the 119894th element of 120581 and 119863
120581119894(120595) = 120597(120581)120597120581
119894
Consider
IFDA =
[[[[[
[
119868120573120573
119868119879
120579120573119868119879
120578120573
119868120579120573
119868120579120579
119868119879
120578120579
119868120578120573
119868120578120579
119868120578120578
]]]]]
]
(D2)
For clarity we rewrite Fisherrsquos information matrix I as
IFDA = [A UV B] (D3)
where
V = U119879
(D4a)
V = [119868120579120573
119868120578120573
] (D4b)
B = [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] (D4c)
According to the matrix inversion lemma the inversematrix of IFDA is
Iminus1
FDA = [
[
(A minus UBminus1V)minus1
minusAminus1U(B minus VAminus1U)minus1
minusBminus1V(A minus UBminus1V)minus1
(B minus VAminus1U)minus1
]
]
(D5)
where the submatrix (A minus UBminus1V)minus1 corresponds to theCRLBof120573which is nuisance and (B minus VAminus1U)minus1 correspondsto the CRLB of 120579 and 120578 = [119903 Ω
119863]119879 which are of interest
CRLB120579120578120579120578FDA
= (B minus VAminus1U)minus1
= [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] minus [119868120579120573
119868120578120573
] 119868minus1
120573120573[119868
119879
120579120573119868119879
120578120573]
minus1
(D6)
where
119868120573120573
= 2 sdot[[[
[
s sdot 1198721205902
119899
0
0s sdot 1198721205902
119899
]]]
]
(D7a)
119868120579120573
= 2 sdot Re[1 119895] otimess1205902
119899
sdot 120573lowast
1198601 (D7b)
119868120578120573
= 2 sdot Re[1 119895] otimes120573
lowast
1205902
119908
sdot [119872 sdot 1198603+ s sdot 119860
4] (D7c)
119868120578120578
= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
[119872 sdot 1198606+ 119860
H3119860
4+ 119860
3119860
H4+ s sdot 119860
7]
(D7d)
12 International Journal of Antennas and Propagation
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHz
70001 70001 70001 70002100286514
100286515
100286515
kHz
(c) CRLB for Doppler shift versus SNR
Figure 8 General CRLB results of TS-FDA radar with119872 = 32
119868120579120579minus (119868
120579120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198602minus119860
1119860
H1
119872) (D8a)
119868120578120579minus (119868
120578120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198605minus119860
4119860
H1
119872) (D8b)
119868120578120578
minus (119868120578120573119868119879
120578120573)
1205902
119899
s119872
= 2
10038161003816100381610038161205731003816100381610038161003816
2
s1205902
119899
(119872119860
6
s+ 119860
7minus119872119860
3119860
H3
s2minus119860
4119860
H4
119872)
(D8c)
1198601= minus119895(
2120587119889 cos (120579)sum119872
119898=1(119898 minus 1)
120582
+2120587119889Δ119891 cos (120579)sum119872
119898=1(119898 minus 1)
2
119888)
(D9a)
1198602= 4120587
2
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(D9b)
International Journal of Antennas and Propagation 13
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(c) CRLB for Doppler shift versus SNR
Figure 9 General CRLB results of TS-FDA radar with119872 = 20
1198603= [int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905 int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
119879
(D9c)
1198604= [
2120587Δ119891sum119872
119898=1(119898 minus 1)
1198880]
119879
(D9d)
1198605= [minus4120587
2119889Δ119891 cos(120579) [
sum119872
119898=1(119898 minus 1)
2
12120582119888
+
Δ119891sum119872
119898=1(119898 minus 1)
3
121198882
] 0]
119879
(D9e)
1198606=[[
[
int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905 int
infin
minusinfin
119905119904H(119905 minus 120591)
119889119904 (119905 minus 120591)
119889119905
119889119905
int
infin
minusinfin
119905119904 (119905 minus 120591)
119889119904(119905 minus 120591)H
119889119905
119889119905 int
infin
minusinfin
1199052|119904 (119905 minus 120591)|
2119889119905
]]
]
(D9f)
1198607=[[
[
41205872
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
0
0 0
]]
]
(D9g)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework described in this paper was supported in part by theNational Natural Science Foundation of China under Grant
14 International Journal of Antennas and Propagation
41101317 the Program for New Century Excellent Talentsin University under Grant NCET-12-0095 Sichuan ProvinceScience Fund for Distinguished Young Scholars under Grant2013JQ0003 and Fundamental Research Fund for theCentralUniversities
References
[1] F Bandiera M Mancino and G Ricci ldquoLocalization strategiesfor multiple point-like radar targetsrdquo IEEE Transactions onSignal Processing vol 60 no 12 pp 6708ndash6712 2012
[2] D R Fuhrmann J P Browning and M Rangaswamy ldquoSignal-ing strategies for the hybrid MIMO phased-array radarrdquo IEEEJournal on Selected Topics in Signal Processing vol 4 no 1 pp66ndash78 2010
[3] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011
[4] M Compagnoni P Bestagini F Antonacci A Sarti and STubaro ldquoLocalization of acoustic sources through the fittingof propagation cones using multiple independent arraysrdquo IEEETransactions on Audio Speech and Language Processing vol 20no 7 pp 1964ndash1975 2012
[5] P Antonik M C Wicks H D Griffiths and C J BakerldquoFrequency diverse array radarsrdquo in Proceedings of the IEEERadar Conference (RADAR rsquo06) pp 215ndash217 Verona NY USAApril 2006
[6] P Antonik M CWicks H D Griffiths and C J Baker ldquoMulti-mission multi-mode waveform diversityrdquo in Proceedings of theIEEE Radar Conference (RADAR rsquo06) pp 580ndash582 Verona NYUSA April 2006
[7] P Antonik H D Griffiths and C J Baker ldquoRange depen-dent beamforming using element level waveform diversityrdquo inProceedings of the International Waveform Diversity and DesignConference pp 1ndash4 Las Vegas Nev USA January 2006
[8] P Baizert T B HaleM A Temple andM CWicks ldquoForward-looking radar GMTI benefits using a linear frequency diversearrayrdquo Electronics Letters vol 42 no 22 pp 1311ndash1312 2006
[9] B W Jung R S Adve and J Chun ldquoFrequency diversity inmultistatic radarsrdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo08) pp 1ndash6 Rome Italy May 2008
[10] M Secmen S Demir A Hizal and T Eker ldquoFrequencydiverse array antenna with periodic time modulated pattern inrange and anglerdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo07) pp 427ndash430 Boston Mass USA April 2007
[11] J Huang K F Tong and C J Baker ldquoFrequency diversearray with beam scanning featurerdquo in Proceedings of the IEEEAntennas and Propagation Conference (AP-S rsquo08) pp 1ndash4 SanDiego Calif USA July 2008
[12] T Higgins and S D Blunt ldquoAnalysis of range-angle coupledbeamforming with frequency-diverse chirpsrdquo in Proceedings ofthe International Waveform Diversity and Design Conference(WDD rsquo09) pp 140ndash144 Orlando Fla USA February 2009
[13] J Farooq M A Temple and M A Saville ldquoApplication of fre-quency diverse arrays to synthetic aperture radar imagingrdquo inProceedings of the International Conference on Electromagneticsin Advanced Applications (ICEAA rsquo07) pp 447ndash449 TorinoItaly September 2007
[14] J Farooq M A Temple and M A Saville ldquoExploiting fre-quency diverse array processing to improve SAR image reso-lutionrdquo in Proceedings of the IEEE Radar Conference (RADARrsquo08) pp 1ndash5 Rome Italy May 2008
[15] W QWang ldquoPhased-MIMO radar with frequency diversity forrangedependent beamformingrdquo IEEE Sensors Journal vol 13no 8 pp 1320ndash1328 2013
[16] A L Swindlehurst and P Stoica ldquoMaximum likelihood meth-ods in radar array signal processingrdquo Proceedings of the IEEEvol 86 no 2 pp 421ndash441 1998
[17] J Ward ldquoCramer-Rao bounds for target angle and Dopplerestimation with space-time adaptive processing radarrdquo in Pro-ceedings of the 29th Asilomar Conference on Signals Systems andComputers pp 1198ndash1202 Pacific Grove Calif USA November1995
[18] A Dogandzic and A Nehorai ldquoCramer-Rao bounds for esti-mating range velocity and direction with an active arrayrdquo IEEETransactions on Signal Processing vol 49 no 6 pp 1122ndash11372001
[19] A Dogandzic and A Nehorai ldquoEstimating range velocity anddirection with a radar arrayrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo99) pp 2773ndash2776 Phoenix Ariz USA March 1999
[20] J J Zhang G Maalouli A P Suppappola and D MorrellldquoCramer-Rao lower bounds for the joint estimation of targetattributes using MIMO radarrdquo in Proceedings of the Interna-tional Waveform Diversity and Design Conference (WDD rsquo09)pp 103ndash107 Orlando Fla USA February 2009
[21] A Hassanien S A Vorobyov and A B Gershman ldquoMovingtarget parameters estimation in noncoherent MIMO radarsystemsrdquo IEEE Transactions on Signal Processing vol 60 no 5pp 2354ndash2361 2012
[22] Q He R S Blum and A M Haimovich ldquoNoncoherent MIMOradar for location and velocity estimation more antennasmeans better performancerdquo IEEE Transactions on Signal Pro-cessing vol 58 no 7 pp 3661ndash3680 2010
[23] S M Kay Fundamentals of Statistical Signal Processing Estima-tion Theory vol 1 Pearson London UK 2nd edition 2011
[24] H L van TreesOptimum Array Processing JohnWiley amp SonsNew York NY USA 2002
[25] D Wilcox and M Sellathurai ldquoOn MIMO radar subarrayedtransmit beamformingrdquo IEEE Transactions on Signal Processingvol 60 no 4 pp 2076ndash2081 2012
[26] WQWang andH Z Shao ldquoRange-angle localization of targetsby a double-pulse frequency diverse array radarrdquo IEEE Journalon Selected Topics in Signal Processing vol 8 no 1 pp 106ndash1142014
[27] S Gogineni and A Nehorai ldquoTarget estimation using sparsemodeling for distributed MIMO radarrdquo IEEE Transactions onSignal Processing vol 59 no 11 pp 5315ndash5325 2011
[28] T Li andA Nehorai ldquoMaximum likelihood direction finding inspatially colored noise fields using sparse sensor arraysrdquo IEEETransactions on Signal Processing vol 59 no 3 pp 1048ndash10622011
[29] S Sen ldquoOFDMradar space-time adaptive processing by exploit-ing spatio-temporal sparsityrdquo IEEE Transactions on SignalProcessing vol 61 no 1 pp 118ndash130 2013
[30] J J Blanz A Papathanassiou M Haardt I Furio and P WBaier ldquoSmart antennas for combined DOA and joint channelestimation in time-slotted CDMA mobile radio systems withjoint detectionrdquo IEEE Transactions onVehicular Technology vol49 no 2 pp 293ndash306 2000
[31] P N Pathirana S C K Herath and A V Savkin ldquoMultitargettracking via space transformations using a single frequencycontinuous wave radarrdquo IEEE Transactions on Signal Processingvol 60 no 10 pp 5217ndash5229 2012
International Journal of Antennas and Propagation 15
[32] P H Leong T D Abhayapala and T A Lamahewa ldquoMultipletarget localization using wideband echo chirp signalsrdquo IEEETransactions on Signal Processing vol 61 no 16 pp 4077ndash40892013
[33] L Zhuang X Liu and W Yu ldquoPrecisely beam steering forfrequency diverse arrays based on frequency offset selectionrdquoin Proceedings of the International Radar Conference (RADARrsquo09) pp 1ndash4 Bordeaux France December 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 7
20151050
10minus2
10minus3
10minus47001 7002 7003
10minus248203
10minus248201
10minus248199
CRLB
of a
ngle
estim
atio
n (d
eg)
SNR (dB)
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
Figure 2 CRLB for estimating angle versus SNR when the range isknown
5 Simulation and Verification
In this section we consider several numerical examples thatcompare the CRLBs in different signal and noise modelsConsider an X-band FDA radar with the carrier frequency1198910
= 10GHz We assume a ULA of 119872 elements usedfor transmitting The array elements are spaced half of thewavelength apart from each other namely 119889 = 1205822 Onetarget of interest is supposed to reflect a plane wave thatimpinges on the array from direction of angle 120579 = 30
∘Under the signal model after matched filtering Figures
2 and 3 compare the CRLBs according to (9) and (13)respectively It can be noticed that the CRLBs are improvedwhen a larger number of elements are employed Howeverit has no significant difference when different frequencyincrements are used This is because sum
119872
119898=1(119898 minus 1)
2
1205822
≫
(Δ119891)2
sum119872
119898=1(119898 minus 1)
4
1198882
+ 2Δ119891sum119872
119898=1(119898 minus 1)
3
120582119888 thus thefrequency increment has a small impact on the CRLBsIn [33] a frequency offset selection strategy is derived toprecisely steer the beam toward a fixed range with a desiredangle
Figure 4 shows the CRLBs of angle and range whenboth the angle and range are unknown The CRLBs aresignificantly degraded due to the range-angle coupling Con-sequently the range and angle of targets cannot be estimateddirectly by the FDA radar However the CRLBs decreaseas the increase of the number of elements and frequencyincrement still holds Moreover generally more elementsmean that better CRLBs performance can be achieved for theFDA radar
To overcome the problem that the range and angle oftargets cannot be estimated directly by the FDA radar weuse the transmit subaperturing strategy on the transmitfrequency increments Figure 5 shows the corresponding
CRLB
of r
ange
estim
atio
n (m
)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
102
101
10020151050
SNR (dB)
Figure 3 CRLB for estimating range versus SNR when the angle isknown
CRLBs where 119872 = 32 is assumed It can be noticed that toobtain a lower CRLB the Δ119891
1and Δ119891
2should have inverse
signs that is one is passive and the other is negative Onereason is that in this case the FDA radar has a wider systembandwidth Figure 6 shows the CRLBs of angle and rangewhen 119872 = 20 is employed It can be noticed that theCRLBs performance improves with the increase of the sensornumber
Under the data model prior to matched filtering wesuppose the following signal parameters bandwidth 119861 =
10MHz repetition period 119879119901
= 1ms and pulse duration1198790
= 250 120583s In this case the approximate expressionsgiven in (31) are valid because the transmitted has a largetime-bandwidth product (119879
0sdot 119861 = 2500 ≫ 1) Figure 7
shows the CRLBs for direction range and Doppler shift asa function of SNR Note that when SNR = minus10 dB and119872 = 32 are employed we can get CRLB
119903119903FDA= 653m
and CRLBΩ119863Ω119863FDA
= 136937 rads that corresponds to theCRLB for velocity is 327 cms (since V = Ω
119863119888(2Ω
119888) and
Ω119888= 2120587119891
119888) Since (Δ119891)2 sum119872
119898=1(119898 minus 1)
4
1198882
≪ 1 the frequencyincrement has a small impact on the CRLBs In additionobserve that the CRLB for 120579 and 120578 is block-diagonal (see(32)) and therefore decoupled that is CRLB
120578120578FDAremains the
same whether or not 120579 is known and similarly CRLB120579120579FDA
is the same whether or not 120578 is known The decouplingis a consequence of the assumed space-time separability ofsignal and noise models and the assumption of the complexamplitude 120573 as an unknown deterministic constant
Figure 8 shows that the CRLBs versus SNR for differentcombinations of Δ119891
1and Δ119891
2 Comparing Figures 8 and
7 the CRLBs have been significantly improved Likewisecomparing Figures 8 and 9 theCRLBs performance improvesas the number of elements increases
8 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
104
103
102
101
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
108
107
106
105
10420151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
Figure 4 Both angle and range are unknown
6 Conclusion
In this paper we derive the CRLB to jointly estimate theattributes of a moving target using FDA radar and computethe corresponding CRLB expressions First we briefly intro-duce the FDA concept and make a summary on the FDAcharacteristics Then we consider two different data modelsnamely pre- and postmatched filtering Under differentsignal and noise models we compute the CRLB expressionsfor estimating the range direction and Doppler shift Wedemonstrate that the FDA radar beamforming is coupledin range and angle and that the targetrsquos range and anglecannot be estimated directly by the FDA radar To overcome
20151050
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
CRLB
of r
ange
estim
atio
n (m
)102
101
100
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
Figure 5 CRLBs of TS-FDA radar with119872 = 32
this problem this paper proposes a transmit subaperturingstrategy for the FDA radar In doing so the range and angle oftargets are estimated from the transmit-receive beamformingoutput Moreover we also specialize the CRLB results tothe case of temporally white noise and a chirp pulse signalExtensive simulation results verify the correctness of thederived CRLBs It is shown that the CRLBs decrease with theincrease of the number of elements and frequency incrementThe CRLBs can be further improved through three aspectsincreasing the number of elements enhancing the systembandwidth by employing a larger frequency increment andusing transmit subaperturing strategy with more subarrays
International Journal of Antennas and Propagation 9
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
CRLB
of r
ange
estim
atio
n (m
)
103
102
101
10020151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
Figure 6 CRLBs of TS-FDA radar with119872 = 20
Appendices
A Derive the CRLB for Angle WhenRange Is Known
To derive the CRLB we start with a well-known expressionfor the FIM under the data model in Section 3 We define thespatial noise covariance matrix as Rn = 120590
2
119899I119872
and signal-to-noise ratio (SNR) as SNR = |120573
0|2
1205902
119899 Suppose the target range
is known the FIM of 120579 is
119868120579120579
= 2Re 119863H120579(120595) (Rminus1
n )119863120579(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
(A1)
For a phased-array radar there is (120595) = 1205730a(120579) We then
have
120597a (120579)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579) (A2)
whereD = diag[0 1 119872 minus 1] and
120597aH (120579)
120597120579
120597a (120579)120597120579
=4120587
2
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A3)
The FIM of the phased-array radar is
119868120579120579phased-array
= 2SNR41205872
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A4)
Similarly for the FDA radar there is (120595) = 1205730a(120579 119903)The
derivation of a(120579 119903) with respect to 120579 is
120597a (120579 119903)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579 119903)
minus 1198952120587119889Δ119891 cos (120579)
119888
times diag [0 1 (119872 minus 1)2
] a (120579 119903)
120597aH (120579 119903)
120597120579
120597a (120579 119903)120597120579
= 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822
+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A5)
Accordingly the FIM of 120579 for the FDA can be expressedas
119868120579120579FDA
= 2SNR sdot 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A6)
10 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
(c) CRLB for Doppler shift versus SNR
Figure 7 General CRLB results of FDA radar
B Derive the CRLB for Range WhenAngle Is Known
Under the data model in Section 3 when the direction 120579 isknown the parameter to be estimated is 119903 The FIM of 119903 is
119868119903119903FDA
= 2Re 119863H119903(120595) (Rminus1
n )119863119903(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597aH (120579 119903)
120597119903Rminus1
n120597a (120579 119903)
120597119903
(B1)
The derivation of a(120579 119903) with respect to 119903 for FDA is
120597a (120579 119903)120597119903
= 1198952120587Δ119891
119888Da (120579 119903)
120597aH (120579 119903)
120597119903
120597a (120579 119903)120597119903
=4120587
2
Δ1198912
1198882
119872
sum
119898=1
(119898 minus 1)2
(B2)
The FIM of 119903 is thus given by
119868119903119903FDA
= 2SNR4120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(B3)
International Journal of Antennas and Propagation 11
C Derive the CRLB for Range and Angle
Under the data model in Section 3 both the angle and rangeare unknown The range and angle of targets are estimatedjointly The FIM for parameters 120579 and 119903 can be expressed as
IFDA = 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) = [
119868120579120579
119868120579119903
119868119903120579
119868119903119903
] (C1)
We then have
IFDA = 210038161003816100381610038161205730
1003816100381610038161003816
2
times
[[[[
[
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
]]]]
]
(C2)
Since (120597H(120595)120597119903)Rminus1
n (120597(120595)120597120579) = (120597H(120595)
120597120579)Rminus1
n (120597(120595)120597119903) then 119868120579119903= 119868
119903120579 We can get
119868120579120579
= 210038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
= 81205872
1198892cos2 (120579)
times SNRsum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
119868120579119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
= minus81205872
119889Δ119891 cos (120579)
times SNR[sum
119872
119898=1(119898 minus 1)
2
120582119888+Δ119891sum
119872
119898=1(119898 minus 1)
3
1198882]
119868119903119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
= SNR8120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(C3)
Since CRLB120579120579FDA
= [Iminus1
FDA]11 CRLB119903119903FDA
= [Iminus1
FDA]22 Theexpressions for the CRLB of angle and range estimationsgiven in (14) can be obtained by substituting (C3) into (C2)
D General CRLB Results
Under the data model in Section 4 we derive the continuousCRLB expressions for temporally and spatially white noiseand denote SNR = |120573|
2
s1205902
119899with s being the power We
also start with FIM
I120581119894120581119895
= 2Re 119863H120581119894
(120581) (Cminus1
n otimes Rminus1
n )119863120581119895(120581) (D1)
where 120581119894is the 119894th element of 120581 and 119863
120581119894(120595) = 120597(120581)120597120581
119894
Consider
IFDA =
[[[[[
[
119868120573120573
119868119879
120579120573119868119879
120578120573
119868120579120573
119868120579120579
119868119879
120578120579
119868120578120573
119868120578120579
119868120578120578
]]]]]
]
(D2)
For clarity we rewrite Fisherrsquos information matrix I as
IFDA = [A UV B] (D3)
where
V = U119879
(D4a)
V = [119868120579120573
119868120578120573
] (D4b)
B = [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] (D4c)
According to the matrix inversion lemma the inversematrix of IFDA is
Iminus1
FDA = [
[
(A minus UBminus1V)minus1
minusAminus1U(B minus VAminus1U)minus1
minusBminus1V(A minus UBminus1V)minus1
(B minus VAminus1U)minus1
]
]
(D5)
where the submatrix (A minus UBminus1V)minus1 corresponds to theCRLBof120573which is nuisance and (B minus VAminus1U)minus1 correspondsto the CRLB of 120579 and 120578 = [119903 Ω
119863]119879 which are of interest
CRLB120579120578120579120578FDA
= (B minus VAminus1U)minus1
= [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] minus [119868120579120573
119868120578120573
] 119868minus1
120573120573[119868
119879
120579120573119868119879
120578120573]
minus1
(D6)
where
119868120573120573
= 2 sdot[[[
[
s sdot 1198721205902
119899
0
0s sdot 1198721205902
119899
]]]
]
(D7a)
119868120579120573
= 2 sdot Re[1 119895] otimess1205902
119899
sdot 120573lowast
1198601 (D7b)
119868120578120573
= 2 sdot Re[1 119895] otimes120573
lowast
1205902
119908
sdot [119872 sdot 1198603+ s sdot 119860
4] (D7c)
119868120578120578
= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
[119872 sdot 1198606+ 119860
H3119860
4+ 119860
3119860
H4+ s sdot 119860
7]
(D7d)
12 International Journal of Antennas and Propagation
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHz
70001 70001 70001 70002100286514
100286515
100286515
kHz
(c) CRLB for Doppler shift versus SNR
Figure 8 General CRLB results of TS-FDA radar with119872 = 32
119868120579120579minus (119868
120579120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198602minus119860
1119860
H1
119872) (D8a)
119868120578120579minus (119868
120578120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198605minus119860
4119860
H1
119872) (D8b)
119868120578120578
minus (119868120578120573119868119879
120578120573)
1205902
119899
s119872
= 2
10038161003816100381610038161205731003816100381610038161003816
2
s1205902
119899
(119872119860
6
s+ 119860
7minus119872119860
3119860
H3
s2minus119860
4119860
H4
119872)
(D8c)
1198601= minus119895(
2120587119889 cos (120579)sum119872
119898=1(119898 minus 1)
120582
+2120587119889Δ119891 cos (120579)sum119872
119898=1(119898 minus 1)
2
119888)
(D9a)
1198602= 4120587
2
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(D9b)
International Journal of Antennas and Propagation 13
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(c) CRLB for Doppler shift versus SNR
Figure 9 General CRLB results of TS-FDA radar with119872 = 20
1198603= [int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905 int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
119879
(D9c)
1198604= [
2120587Δ119891sum119872
119898=1(119898 minus 1)
1198880]
119879
(D9d)
1198605= [minus4120587
2119889Δ119891 cos(120579) [
sum119872
119898=1(119898 minus 1)
2
12120582119888
+
Δ119891sum119872
119898=1(119898 minus 1)
3
121198882
] 0]
119879
(D9e)
1198606=[[
[
int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905 int
infin
minusinfin
119905119904H(119905 minus 120591)
119889119904 (119905 minus 120591)
119889119905
119889119905
int
infin
minusinfin
119905119904 (119905 minus 120591)
119889119904(119905 minus 120591)H
119889119905
119889119905 int
infin
minusinfin
1199052|119904 (119905 minus 120591)|
2119889119905
]]
]
(D9f)
1198607=[[
[
41205872
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
0
0 0
]]
]
(D9g)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework described in this paper was supported in part by theNational Natural Science Foundation of China under Grant
14 International Journal of Antennas and Propagation
41101317 the Program for New Century Excellent Talentsin University under Grant NCET-12-0095 Sichuan ProvinceScience Fund for Distinguished Young Scholars under Grant2013JQ0003 and Fundamental Research Fund for theCentralUniversities
References
[1] F Bandiera M Mancino and G Ricci ldquoLocalization strategiesfor multiple point-like radar targetsrdquo IEEE Transactions onSignal Processing vol 60 no 12 pp 6708ndash6712 2012
[2] D R Fuhrmann J P Browning and M Rangaswamy ldquoSignal-ing strategies for the hybrid MIMO phased-array radarrdquo IEEEJournal on Selected Topics in Signal Processing vol 4 no 1 pp66ndash78 2010
[3] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011
[4] M Compagnoni P Bestagini F Antonacci A Sarti and STubaro ldquoLocalization of acoustic sources through the fittingof propagation cones using multiple independent arraysrdquo IEEETransactions on Audio Speech and Language Processing vol 20no 7 pp 1964ndash1975 2012
[5] P Antonik M C Wicks H D Griffiths and C J BakerldquoFrequency diverse array radarsrdquo in Proceedings of the IEEERadar Conference (RADAR rsquo06) pp 215ndash217 Verona NY USAApril 2006
[6] P Antonik M CWicks H D Griffiths and C J Baker ldquoMulti-mission multi-mode waveform diversityrdquo in Proceedings of theIEEE Radar Conference (RADAR rsquo06) pp 580ndash582 Verona NYUSA April 2006
[7] P Antonik H D Griffiths and C J Baker ldquoRange depen-dent beamforming using element level waveform diversityrdquo inProceedings of the International Waveform Diversity and DesignConference pp 1ndash4 Las Vegas Nev USA January 2006
[8] P Baizert T B HaleM A Temple andM CWicks ldquoForward-looking radar GMTI benefits using a linear frequency diversearrayrdquo Electronics Letters vol 42 no 22 pp 1311ndash1312 2006
[9] B W Jung R S Adve and J Chun ldquoFrequency diversity inmultistatic radarsrdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo08) pp 1ndash6 Rome Italy May 2008
[10] M Secmen S Demir A Hizal and T Eker ldquoFrequencydiverse array antenna with periodic time modulated pattern inrange and anglerdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo07) pp 427ndash430 Boston Mass USA April 2007
[11] J Huang K F Tong and C J Baker ldquoFrequency diversearray with beam scanning featurerdquo in Proceedings of the IEEEAntennas and Propagation Conference (AP-S rsquo08) pp 1ndash4 SanDiego Calif USA July 2008
[12] T Higgins and S D Blunt ldquoAnalysis of range-angle coupledbeamforming with frequency-diverse chirpsrdquo in Proceedings ofthe International Waveform Diversity and Design Conference(WDD rsquo09) pp 140ndash144 Orlando Fla USA February 2009
[13] J Farooq M A Temple and M A Saville ldquoApplication of fre-quency diverse arrays to synthetic aperture radar imagingrdquo inProceedings of the International Conference on Electromagneticsin Advanced Applications (ICEAA rsquo07) pp 447ndash449 TorinoItaly September 2007
[14] J Farooq M A Temple and M A Saville ldquoExploiting fre-quency diverse array processing to improve SAR image reso-lutionrdquo in Proceedings of the IEEE Radar Conference (RADARrsquo08) pp 1ndash5 Rome Italy May 2008
[15] W QWang ldquoPhased-MIMO radar with frequency diversity forrangedependent beamformingrdquo IEEE Sensors Journal vol 13no 8 pp 1320ndash1328 2013
[16] A L Swindlehurst and P Stoica ldquoMaximum likelihood meth-ods in radar array signal processingrdquo Proceedings of the IEEEvol 86 no 2 pp 421ndash441 1998
[17] J Ward ldquoCramer-Rao bounds for target angle and Dopplerestimation with space-time adaptive processing radarrdquo in Pro-ceedings of the 29th Asilomar Conference on Signals Systems andComputers pp 1198ndash1202 Pacific Grove Calif USA November1995
[18] A Dogandzic and A Nehorai ldquoCramer-Rao bounds for esti-mating range velocity and direction with an active arrayrdquo IEEETransactions on Signal Processing vol 49 no 6 pp 1122ndash11372001
[19] A Dogandzic and A Nehorai ldquoEstimating range velocity anddirection with a radar arrayrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo99) pp 2773ndash2776 Phoenix Ariz USA March 1999
[20] J J Zhang G Maalouli A P Suppappola and D MorrellldquoCramer-Rao lower bounds for the joint estimation of targetattributes using MIMO radarrdquo in Proceedings of the Interna-tional Waveform Diversity and Design Conference (WDD rsquo09)pp 103ndash107 Orlando Fla USA February 2009
[21] A Hassanien S A Vorobyov and A B Gershman ldquoMovingtarget parameters estimation in noncoherent MIMO radarsystemsrdquo IEEE Transactions on Signal Processing vol 60 no 5pp 2354ndash2361 2012
[22] Q He R S Blum and A M Haimovich ldquoNoncoherent MIMOradar for location and velocity estimation more antennasmeans better performancerdquo IEEE Transactions on Signal Pro-cessing vol 58 no 7 pp 3661ndash3680 2010
[23] S M Kay Fundamentals of Statistical Signal Processing Estima-tion Theory vol 1 Pearson London UK 2nd edition 2011
[24] H L van TreesOptimum Array Processing JohnWiley amp SonsNew York NY USA 2002
[25] D Wilcox and M Sellathurai ldquoOn MIMO radar subarrayedtransmit beamformingrdquo IEEE Transactions on Signal Processingvol 60 no 4 pp 2076ndash2081 2012
[26] WQWang andH Z Shao ldquoRange-angle localization of targetsby a double-pulse frequency diverse array radarrdquo IEEE Journalon Selected Topics in Signal Processing vol 8 no 1 pp 106ndash1142014
[27] S Gogineni and A Nehorai ldquoTarget estimation using sparsemodeling for distributed MIMO radarrdquo IEEE Transactions onSignal Processing vol 59 no 11 pp 5315ndash5325 2011
[28] T Li andA Nehorai ldquoMaximum likelihood direction finding inspatially colored noise fields using sparse sensor arraysrdquo IEEETransactions on Signal Processing vol 59 no 3 pp 1048ndash10622011
[29] S Sen ldquoOFDMradar space-time adaptive processing by exploit-ing spatio-temporal sparsityrdquo IEEE Transactions on SignalProcessing vol 61 no 1 pp 118ndash130 2013
[30] J J Blanz A Papathanassiou M Haardt I Furio and P WBaier ldquoSmart antennas for combined DOA and joint channelestimation in time-slotted CDMA mobile radio systems withjoint detectionrdquo IEEE Transactions onVehicular Technology vol49 no 2 pp 293ndash306 2000
[31] P N Pathirana S C K Herath and A V Savkin ldquoMultitargettracking via space transformations using a single frequencycontinuous wave radarrdquo IEEE Transactions on Signal Processingvol 60 no 10 pp 5217ndash5229 2012
International Journal of Antennas and Propagation 15
[32] P H Leong T D Abhayapala and T A Lamahewa ldquoMultipletarget localization using wideband echo chirp signalsrdquo IEEETransactions on Signal Processing vol 61 no 16 pp 4077ndash40892013
[33] L Zhuang X Liu and W Yu ldquoPrecisely beam steering forfrequency diverse arrays based on frequency offset selectionrdquoin Proceedings of the International Radar Conference (RADARrsquo09) pp 1ndash4 Bordeaux France December 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
104
103
102
101
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
108
107
106
105
10420151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
Figure 4 Both angle and range are unknown
6 Conclusion
In this paper we derive the CRLB to jointly estimate theattributes of a moving target using FDA radar and computethe corresponding CRLB expressions First we briefly intro-duce the FDA concept and make a summary on the FDAcharacteristics Then we consider two different data modelsnamely pre- and postmatched filtering Under differentsignal and noise models we compute the CRLB expressionsfor estimating the range direction and Doppler shift Wedemonstrate that the FDA radar beamforming is coupledin range and angle and that the targetrsquos range and anglecannot be estimated directly by the FDA radar To overcome
20151050
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
CRLB
of r
ange
estim
atio
n (m
)102
101
100
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
Figure 5 CRLBs of TS-FDA radar with119872 = 32
this problem this paper proposes a transmit subaperturingstrategy for the FDA radar In doing so the range and angle oftargets are estimated from the transmit-receive beamformingoutput Moreover we also specialize the CRLB results tothe case of temporally white noise and a chirp pulse signalExtensive simulation results verify the correctness of thederived CRLBs It is shown that the CRLBs decrease with theincrease of the number of elements and frequency incrementThe CRLBs can be further improved through three aspectsincreasing the number of elements enhancing the systembandwidth by employing a larger frequency increment andusing transmit subaperturing strategy with more subarrays
International Journal of Antennas and Propagation 9
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
CRLB
of r
ange
estim
atio
n (m
)
103
102
101
10020151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
Figure 6 CRLBs of TS-FDA radar with119872 = 20
Appendices
A Derive the CRLB for Angle WhenRange Is Known
To derive the CRLB we start with a well-known expressionfor the FIM under the data model in Section 3 We define thespatial noise covariance matrix as Rn = 120590
2
119899I119872
and signal-to-noise ratio (SNR) as SNR = |120573
0|2
1205902
119899 Suppose the target range
is known the FIM of 120579 is
119868120579120579
= 2Re 119863H120579(120595) (Rminus1
n )119863120579(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
(A1)
For a phased-array radar there is (120595) = 1205730a(120579) We then
have
120597a (120579)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579) (A2)
whereD = diag[0 1 119872 minus 1] and
120597aH (120579)
120597120579
120597a (120579)120597120579
=4120587
2
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A3)
The FIM of the phased-array radar is
119868120579120579phased-array
= 2SNR41205872
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A4)
Similarly for the FDA radar there is (120595) = 1205730a(120579 119903)The
derivation of a(120579 119903) with respect to 120579 is
120597a (120579 119903)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579 119903)
minus 1198952120587119889Δ119891 cos (120579)
119888
times diag [0 1 (119872 minus 1)2
] a (120579 119903)
120597aH (120579 119903)
120597120579
120597a (120579 119903)120597120579
= 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822
+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A5)
Accordingly the FIM of 120579 for the FDA can be expressedas
119868120579120579FDA
= 2SNR sdot 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A6)
10 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
(c) CRLB for Doppler shift versus SNR
Figure 7 General CRLB results of FDA radar
B Derive the CRLB for Range WhenAngle Is Known
Under the data model in Section 3 when the direction 120579 isknown the parameter to be estimated is 119903 The FIM of 119903 is
119868119903119903FDA
= 2Re 119863H119903(120595) (Rminus1
n )119863119903(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597aH (120579 119903)
120597119903Rminus1
n120597a (120579 119903)
120597119903
(B1)
The derivation of a(120579 119903) with respect to 119903 for FDA is
120597a (120579 119903)120597119903
= 1198952120587Δ119891
119888Da (120579 119903)
120597aH (120579 119903)
120597119903
120597a (120579 119903)120597119903
=4120587
2
Δ1198912
1198882
119872
sum
119898=1
(119898 minus 1)2
(B2)
The FIM of 119903 is thus given by
119868119903119903FDA
= 2SNR4120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(B3)
International Journal of Antennas and Propagation 11
C Derive the CRLB for Range and Angle
Under the data model in Section 3 both the angle and rangeare unknown The range and angle of targets are estimatedjointly The FIM for parameters 120579 and 119903 can be expressed as
IFDA = 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) = [
119868120579120579
119868120579119903
119868119903120579
119868119903119903
] (C1)
We then have
IFDA = 210038161003816100381610038161205730
1003816100381610038161003816
2
times
[[[[
[
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
]]]]
]
(C2)
Since (120597H(120595)120597119903)Rminus1
n (120597(120595)120597120579) = (120597H(120595)
120597120579)Rminus1
n (120597(120595)120597119903) then 119868120579119903= 119868
119903120579 We can get
119868120579120579
= 210038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
= 81205872
1198892cos2 (120579)
times SNRsum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
119868120579119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
= minus81205872
119889Δ119891 cos (120579)
times SNR[sum
119872
119898=1(119898 minus 1)
2
120582119888+Δ119891sum
119872
119898=1(119898 minus 1)
3
1198882]
119868119903119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
= SNR8120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(C3)
Since CRLB120579120579FDA
= [Iminus1
FDA]11 CRLB119903119903FDA
= [Iminus1
FDA]22 Theexpressions for the CRLB of angle and range estimationsgiven in (14) can be obtained by substituting (C3) into (C2)
D General CRLB Results
Under the data model in Section 4 we derive the continuousCRLB expressions for temporally and spatially white noiseand denote SNR = |120573|
2
s1205902
119899with s being the power We
also start with FIM
I120581119894120581119895
= 2Re 119863H120581119894
(120581) (Cminus1
n otimes Rminus1
n )119863120581119895(120581) (D1)
where 120581119894is the 119894th element of 120581 and 119863
120581119894(120595) = 120597(120581)120597120581
119894
Consider
IFDA =
[[[[[
[
119868120573120573
119868119879
120579120573119868119879
120578120573
119868120579120573
119868120579120579
119868119879
120578120579
119868120578120573
119868120578120579
119868120578120578
]]]]]
]
(D2)
For clarity we rewrite Fisherrsquos information matrix I as
IFDA = [A UV B] (D3)
where
V = U119879
(D4a)
V = [119868120579120573
119868120578120573
] (D4b)
B = [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] (D4c)
According to the matrix inversion lemma the inversematrix of IFDA is
Iminus1
FDA = [
[
(A minus UBminus1V)minus1
minusAminus1U(B minus VAminus1U)minus1
minusBminus1V(A minus UBminus1V)minus1
(B minus VAminus1U)minus1
]
]
(D5)
where the submatrix (A minus UBminus1V)minus1 corresponds to theCRLBof120573which is nuisance and (B minus VAminus1U)minus1 correspondsto the CRLB of 120579 and 120578 = [119903 Ω
119863]119879 which are of interest
CRLB120579120578120579120578FDA
= (B minus VAminus1U)minus1
= [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] minus [119868120579120573
119868120578120573
] 119868minus1
120573120573[119868
119879
120579120573119868119879
120578120573]
minus1
(D6)
where
119868120573120573
= 2 sdot[[[
[
s sdot 1198721205902
119899
0
0s sdot 1198721205902
119899
]]]
]
(D7a)
119868120579120573
= 2 sdot Re[1 119895] otimess1205902
119899
sdot 120573lowast
1198601 (D7b)
119868120578120573
= 2 sdot Re[1 119895] otimes120573
lowast
1205902
119908
sdot [119872 sdot 1198603+ s sdot 119860
4] (D7c)
119868120578120578
= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
[119872 sdot 1198606+ 119860
H3119860
4+ 119860
3119860
H4+ s sdot 119860
7]
(D7d)
12 International Journal of Antennas and Propagation
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHz
70001 70001 70001 70002100286514
100286515
100286515
kHz
(c) CRLB for Doppler shift versus SNR
Figure 8 General CRLB results of TS-FDA radar with119872 = 32
119868120579120579minus (119868
120579120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198602minus119860
1119860
H1
119872) (D8a)
119868120578120579minus (119868
120578120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198605minus119860
4119860
H1
119872) (D8b)
119868120578120578
minus (119868120578120573119868119879
120578120573)
1205902
119899
s119872
= 2
10038161003816100381610038161205731003816100381610038161003816
2
s1205902
119899
(119872119860
6
s+ 119860
7minus119872119860
3119860
H3
s2minus119860
4119860
H4
119872)
(D8c)
1198601= minus119895(
2120587119889 cos (120579)sum119872
119898=1(119898 minus 1)
120582
+2120587119889Δ119891 cos (120579)sum119872
119898=1(119898 minus 1)
2
119888)
(D9a)
1198602= 4120587
2
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(D9b)
International Journal of Antennas and Propagation 13
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(c) CRLB for Doppler shift versus SNR
Figure 9 General CRLB results of TS-FDA radar with119872 = 20
1198603= [int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905 int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
119879
(D9c)
1198604= [
2120587Δ119891sum119872
119898=1(119898 minus 1)
1198880]
119879
(D9d)
1198605= [minus4120587
2119889Δ119891 cos(120579) [
sum119872
119898=1(119898 minus 1)
2
12120582119888
+
Δ119891sum119872
119898=1(119898 minus 1)
3
121198882
] 0]
119879
(D9e)
1198606=[[
[
int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905 int
infin
minusinfin
119905119904H(119905 minus 120591)
119889119904 (119905 minus 120591)
119889119905
119889119905
int
infin
minusinfin
119905119904 (119905 minus 120591)
119889119904(119905 minus 120591)H
119889119905
119889119905 int
infin
minusinfin
1199052|119904 (119905 minus 120591)|
2119889119905
]]
]
(D9f)
1198607=[[
[
41205872
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
0
0 0
]]
]
(D9g)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework described in this paper was supported in part by theNational Natural Science Foundation of China under Grant
14 International Journal of Antennas and Propagation
41101317 the Program for New Century Excellent Talentsin University under Grant NCET-12-0095 Sichuan ProvinceScience Fund for Distinguished Young Scholars under Grant2013JQ0003 and Fundamental Research Fund for theCentralUniversities
References
[1] F Bandiera M Mancino and G Ricci ldquoLocalization strategiesfor multiple point-like radar targetsrdquo IEEE Transactions onSignal Processing vol 60 no 12 pp 6708ndash6712 2012
[2] D R Fuhrmann J P Browning and M Rangaswamy ldquoSignal-ing strategies for the hybrid MIMO phased-array radarrdquo IEEEJournal on Selected Topics in Signal Processing vol 4 no 1 pp66ndash78 2010
[3] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011
[4] M Compagnoni P Bestagini F Antonacci A Sarti and STubaro ldquoLocalization of acoustic sources through the fittingof propagation cones using multiple independent arraysrdquo IEEETransactions on Audio Speech and Language Processing vol 20no 7 pp 1964ndash1975 2012
[5] P Antonik M C Wicks H D Griffiths and C J BakerldquoFrequency diverse array radarsrdquo in Proceedings of the IEEERadar Conference (RADAR rsquo06) pp 215ndash217 Verona NY USAApril 2006
[6] P Antonik M CWicks H D Griffiths and C J Baker ldquoMulti-mission multi-mode waveform diversityrdquo in Proceedings of theIEEE Radar Conference (RADAR rsquo06) pp 580ndash582 Verona NYUSA April 2006
[7] P Antonik H D Griffiths and C J Baker ldquoRange depen-dent beamforming using element level waveform diversityrdquo inProceedings of the International Waveform Diversity and DesignConference pp 1ndash4 Las Vegas Nev USA January 2006
[8] P Baizert T B HaleM A Temple andM CWicks ldquoForward-looking radar GMTI benefits using a linear frequency diversearrayrdquo Electronics Letters vol 42 no 22 pp 1311ndash1312 2006
[9] B W Jung R S Adve and J Chun ldquoFrequency diversity inmultistatic radarsrdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo08) pp 1ndash6 Rome Italy May 2008
[10] M Secmen S Demir A Hizal and T Eker ldquoFrequencydiverse array antenna with periodic time modulated pattern inrange and anglerdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo07) pp 427ndash430 Boston Mass USA April 2007
[11] J Huang K F Tong and C J Baker ldquoFrequency diversearray with beam scanning featurerdquo in Proceedings of the IEEEAntennas and Propagation Conference (AP-S rsquo08) pp 1ndash4 SanDiego Calif USA July 2008
[12] T Higgins and S D Blunt ldquoAnalysis of range-angle coupledbeamforming with frequency-diverse chirpsrdquo in Proceedings ofthe International Waveform Diversity and Design Conference(WDD rsquo09) pp 140ndash144 Orlando Fla USA February 2009
[13] J Farooq M A Temple and M A Saville ldquoApplication of fre-quency diverse arrays to synthetic aperture radar imagingrdquo inProceedings of the International Conference on Electromagneticsin Advanced Applications (ICEAA rsquo07) pp 447ndash449 TorinoItaly September 2007
[14] J Farooq M A Temple and M A Saville ldquoExploiting fre-quency diverse array processing to improve SAR image reso-lutionrdquo in Proceedings of the IEEE Radar Conference (RADARrsquo08) pp 1ndash5 Rome Italy May 2008
[15] W QWang ldquoPhased-MIMO radar with frequency diversity forrangedependent beamformingrdquo IEEE Sensors Journal vol 13no 8 pp 1320ndash1328 2013
[16] A L Swindlehurst and P Stoica ldquoMaximum likelihood meth-ods in radar array signal processingrdquo Proceedings of the IEEEvol 86 no 2 pp 421ndash441 1998
[17] J Ward ldquoCramer-Rao bounds for target angle and Dopplerestimation with space-time adaptive processing radarrdquo in Pro-ceedings of the 29th Asilomar Conference on Signals Systems andComputers pp 1198ndash1202 Pacific Grove Calif USA November1995
[18] A Dogandzic and A Nehorai ldquoCramer-Rao bounds for esti-mating range velocity and direction with an active arrayrdquo IEEETransactions on Signal Processing vol 49 no 6 pp 1122ndash11372001
[19] A Dogandzic and A Nehorai ldquoEstimating range velocity anddirection with a radar arrayrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo99) pp 2773ndash2776 Phoenix Ariz USA March 1999
[20] J J Zhang G Maalouli A P Suppappola and D MorrellldquoCramer-Rao lower bounds for the joint estimation of targetattributes using MIMO radarrdquo in Proceedings of the Interna-tional Waveform Diversity and Design Conference (WDD rsquo09)pp 103ndash107 Orlando Fla USA February 2009
[21] A Hassanien S A Vorobyov and A B Gershman ldquoMovingtarget parameters estimation in noncoherent MIMO radarsystemsrdquo IEEE Transactions on Signal Processing vol 60 no 5pp 2354ndash2361 2012
[22] Q He R S Blum and A M Haimovich ldquoNoncoherent MIMOradar for location and velocity estimation more antennasmeans better performancerdquo IEEE Transactions on Signal Pro-cessing vol 58 no 7 pp 3661ndash3680 2010
[23] S M Kay Fundamentals of Statistical Signal Processing Estima-tion Theory vol 1 Pearson London UK 2nd edition 2011
[24] H L van TreesOptimum Array Processing JohnWiley amp SonsNew York NY USA 2002
[25] D Wilcox and M Sellathurai ldquoOn MIMO radar subarrayedtransmit beamformingrdquo IEEE Transactions on Signal Processingvol 60 no 4 pp 2076ndash2081 2012
[26] WQWang andH Z Shao ldquoRange-angle localization of targetsby a double-pulse frequency diverse array radarrdquo IEEE Journalon Selected Topics in Signal Processing vol 8 no 1 pp 106ndash1142014
[27] S Gogineni and A Nehorai ldquoTarget estimation using sparsemodeling for distributed MIMO radarrdquo IEEE Transactions onSignal Processing vol 59 no 11 pp 5315ndash5325 2011
[28] T Li andA Nehorai ldquoMaximum likelihood direction finding inspatially colored noise fields using sparse sensor arraysrdquo IEEETransactions on Signal Processing vol 59 no 3 pp 1048ndash10622011
[29] S Sen ldquoOFDMradar space-time adaptive processing by exploit-ing spatio-temporal sparsityrdquo IEEE Transactions on SignalProcessing vol 61 no 1 pp 118ndash130 2013
[30] J J Blanz A Papathanassiou M Haardt I Furio and P WBaier ldquoSmart antennas for combined DOA and joint channelestimation in time-slotted CDMA mobile radio systems withjoint detectionrdquo IEEE Transactions onVehicular Technology vol49 no 2 pp 293ndash306 2000
[31] P N Pathirana S C K Herath and A V Savkin ldquoMultitargettracking via space transformations using a single frequencycontinuous wave radarrdquo IEEE Transactions on Signal Processingvol 60 no 10 pp 5217ndash5229 2012
International Journal of Antennas and Propagation 15
[32] P H Leong T D Abhayapala and T A Lamahewa ldquoMultipletarget localization using wideband echo chirp signalsrdquo IEEETransactions on Signal Processing vol 61 no 16 pp 4077ndash40892013
[33] L Zhuang X Liu and W Yu ldquoPrecisely beam steering forfrequency diverse arrays based on frequency offset selectionrdquoin Proceedings of the International Radar Conference (RADARrsquo09) pp 1ndash4 Bordeaux France December 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 9
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
CRLB
of r
ange
estim
atio
n (m
)
103
102
101
10020151050
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
Figure 6 CRLBs of TS-FDA radar with119872 = 20
Appendices
A Derive the CRLB for Angle WhenRange Is Known
To derive the CRLB we start with a well-known expressionfor the FIM under the data model in Section 3 We define thespatial noise covariance matrix as Rn = 120590
2
119899I119872
and signal-to-noise ratio (SNR) as SNR = |120573
0|2
1205902
119899 Suppose the target range
is known the FIM of 120579 is
119868120579120579
= 2Re 119863H120579(120595) (Rminus1
n )119863120579(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
(A1)
For a phased-array radar there is (120595) = 1205730a(120579) We then
have
120597a (120579)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579) (A2)
whereD = diag[0 1 119872 minus 1] and
120597aH (120579)
120597120579
120597a (120579)120597120579
=4120587
2
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A3)
The FIM of the phased-array radar is
119868120579120579phased-array
= 2SNR41205872
1198892cos2 (120579)1205822
119872
sum
119898=1
(119898 minus 1)2
(A4)
Similarly for the FDA radar there is (120595) = 1205730a(120579 119903)The
derivation of a(120579 119903) with respect to 120579 is
120597a (120579 119903)120597120579
= minus1198952120587119889 cos (120579)
120582Da (120579 119903)
minus 1198952120587119889Δ119891 cos (120579)
119888
times diag [0 1 (119872 minus 1)2
] a (120579 119903)
120597aH (120579 119903)
120597120579
120597a (120579 119903)120597120579
= 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822
+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A5)
Accordingly the FIM of 120579 for the FDA can be expressedas
119868120579120579FDA
= 2SNR sdot 41205872
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(A6)
10 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
(c) CRLB for Doppler shift versus SNR
Figure 7 General CRLB results of FDA radar
B Derive the CRLB for Range WhenAngle Is Known
Under the data model in Section 3 when the direction 120579 isknown the parameter to be estimated is 119903 The FIM of 119903 is
119868119903119903FDA
= 2Re 119863H119903(120595) (Rminus1
n )119863119903(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597aH (120579 119903)
120597119903Rminus1
n120597a (120579 119903)
120597119903
(B1)
The derivation of a(120579 119903) with respect to 119903 for FDA is
120597a (120579 119903)120597119903
= 1198952120587Δ119891
119888Da (120579 119903)
120597aH (120579 119903)
120597119903
120597a (120579 119903)120597119903
=4120587
2
Δ1198912
1198882
119872
sum
119898=1
(119898 minus 1)2
(B2)
The FIM of 119903 is thus given by
119868119903119903FDA
= 2SNR4120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(B3)
International Journal of Antennas and Propagation 11
C Derive the CRLB for Range and Angle
Under the data model in Section 3 both the angle and rangeare unknown The range and angle of targets are estimatedjointly The FIM for parameters 120579 and 119903 can be expressed as
IFDA = 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) = [
119868120579120579
119868120579119903
119868119903120579
119868119903119903
] (C1)
We then have
IFDA = 210038161003816100381610038161205730
1003816100381610038161003816
2
times
[[[[
[
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
]]]]
]
(C2)
Since (120597H(120595)120597119903)Rminus1
n (120597(120595)120597120579) = (120597H(120595)
120597120579)Rminus1
n (120597(120595)120597119903) then 119868120579119903= 119868
119903120579 We can get
119868120579120579
= 210038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
= 81205872
1198892cos2 (120579)
times SNRsum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
119868120579119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
= minus81205872
119889Δ119891 cos (120579)
times SNR[sum
119872
119898=1(119898 minus 1)
2
120582119888+Δ119891sum
119872
119898=1(119898 minus 1)
3
1198882]
119868119903119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
= SNR8120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(C3)
Since CRLB120579120579FDA
= [Iminus1
FDA]11 CRLB119903119903FDA
= [Iminus1
FDA]22 Theexpressions for the CRLB of angle and range estimationsgiven in (14) can be obtained by substituting (C3) into (C2)
D General CRLB Results
Under the data model in Section 4 we derive the continuousCRLB expressions for temporally and spatially white noiseand denote SNR = |120573|
2
s1205902
119899with s being the power We
also start with FIM
I120581119894120581119895
= 2Re 119863H120581119894
(120581) (Cminus1
n otimes Rminus1
n )119863120581119895(120581) (D1)
where 120581119894is the 119894th element of 120581 and 119863
120581119894(120595) = 120597(120581)120597120581
119894
Consider
IFDA =
[[[[[
[
119868120573120573
119868119879
120579120573119868119879
120578120573
119868120579120573
119868120579120579
119868119879
120578120579
119868120578120573
119868120578120579
119868120578120578
]]]]]
]
(D2)
For clarity we rewrite Fisherrsquos information matrix I as
IFDA = [A UV B] (D3)
where
V = U119879
(D4a)
V = [119868120579120573
119868120578120573
] (D4b)
B = [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] (D4c)
According to the matrix inversion lemma the inversematrix of IFDA is
Iminus1
FDA = [
[
(A minus UBminus1V)minus1
minusAminus1U(B minus VAminus1U)minus1
minusBminus1V(A minus UBminus1V)minus1
(B minus VAminus1U)minus1
]
]
(D5)
where the submatrix (A minus UBminus1V)minus1 corresponds to theCRLBof120573which is nuisance and (B minus VAminus1U)minus1 correspondsto the CRLB of 120579 and 120578 = [119903 Ω
119863]119879 which are of interest
CRLB120579120578120579120578FDA
= (B minus VAminus1U)minus1
= [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] minus [119868120579120573
119868120578120573
] 119868minus1
120573120573[119868
119879
120579120573119868119879
120578120573]
minus1
(D6)
where
119868120573120573
= 2 sdot[[[
[
s sdot 1198721205902
119899
0
0s sdot 1198721205902
119899
]]]
]
(D7a)
119868120579120573
= 2 sdot Re[1 119895] otimess1205902
119899
sdot 120573lowast
1198601 (D7b)
119868120578120573
= 2 sdot Re[1 119895] otimes120573
lowast
1205902
119908
sdot [119872 sdot 1198603+ s sdot 119860
4] (D7c)
119868120578120578
= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
[119872 sdot 1198606+ 119860
H3119860
4+ 119860
3119860
H4+ s sdot 119860
7]
(D7d)
12 International Journal of Antennas and Propagation
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHz
70001 70001 70001 70002100286514
100286515
100286515
kHz
(c) CRLB for Doppler shift versus SNR
Figure 8 General CRLB results of TS-FDA radar with119872 = 32
119868120579120579minus (119868
120579120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198602minus119860
1119860
H1
119872) (D8a)
119868120578120579minus (119868
120578120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198605minus119860
4119860
H1
119872) (D8b)
119868120578120578
minus (119868120578120573119868119879
120578120573)
1205902
119899
s119872
= 2
10038161003816100381610038161205731003816100381610038161003816
2
s1205902
119899
(119872119860
6
s+ 119860
7minus119872119860
3119860
H3
s2minus119860
4119860
H4
119872)
(D8c)
1198601= minus119895(
2120587119889 cos (120579)sum119872
119898=1(119898 minus 1)
120582
+2120587119889Δ119891 cos (120579)sum119872
119898=1(119898 minus 1)
2
119888)
(D9a)
1198602= 4120587
2
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(D9b)
International Journal of Antennas and Propagation 13
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(c) CRLB for Doppler shift versus SNR
Figure 9 General CRLB results of TS-FDA radar with119872 = 20
1198603= [int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905 int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
119879
(D9c)
1198604= [
2120587Δ119891sum119872
119898=1(119898 minus 1)
1198880]
119879
(D9d)
1198605= [minus4120587
2119889Δ119891 cos(120579) [
sum119872
119898=1(119898 minus 1)
2
12120582119888
+
Δ119891sum119872
119898=1(119898 minus 1)
3
121198882
] 0]
119879
(D9e)
1198606=[[
[
int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905 int
infin
minusinfin
119905119904H(119905 minus 120591)
119889119904 (119905 minus 120591)
119889119905
119889119905
int
infin
minusinfin
119905119904 (119905 minus 120591)
119889119904(119905 minus 120591)H
119889119905
119889119905 int
infin
minusinfin
1199052|119904 (119905 minus 120591)|
2119889119905
]]
]
(D9f)
1198607=[[
[
41205872
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
0
0 0
]]
]
(D9g)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework described in this paper was supported in part by theNational Natural Science Foundation of China under Grant
14 International Journal of Antennas and Propagation
41101317 the Program for New Century Excellent Talentsin University under Grant NCET-12-0095 Sichuan ProvinceScience Fund for Distinguished Young Scholars under Grant2013JQ0003 and Fundamental Research Fund for theCentralUniversities
References
[1] F Bandiera M Mancino and G Ricci ldquoLocalization strategiesfor multiple point-like radar targetsrdquo IEEE Transactions onSignal Processing vol 60 no 12 pp 6708ndash6712 2012
[2] D R Fuhrmann J P Browning and M Rangaswamy ldquoSignal-ing strategies for the hybrid MIMO phased-array radarrdquo IEEEJournal on Selected Topics in Signal Processing vol 4 no 1 pp66ndash78 2010
[3] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011
[4] M Compagnoni P Bestagini F Antonacci A Sarti and STubaro ldquoLocalization of acoustic sources through the fittingof propagation cones using multiple independent arraysrdquo IEEETransactions on Audio Speech and Language Processing vol 20no 7 pp 1964ndash1975 2012
[5] P Antonik M C Wicks H D Griffiths and C J BakerldquoFrequency diverse array radarsrdquo in Proceedings of the IEEERadar Conference (RADAR rsquo06) pp 215ndash217 Verona NY USAApril 2006
[6] P Antonik M CWicks H D Griffiths and C J Baker ldquoMulti-mission multi-mode waveform diversityrdquo in Proceedings of theIEEE Radar Conference (RADAR rsquo06) pp 580ndash582 Verona NYUSA April 2006
[7] P Antonik H D Griffiths and C J Baker ldquoRange depen-dent beamforming using element level waveform diversityrdquo inProceedings of the International Waveform Diversity and DesignConference pp 1ndash4 Las Vegas Nev USA January 2006
[8] P Baizert T B HaleM A Temple andM CWicks ldquoForward-looking radar GMTI benefits using a linear frequency diversearrayrdquo Electronics Letters vol 42 no 22 pp 1311ndash1312 2006
[9] B W Jung R S Adve and J Chun ldquoFrequency diversity inmultistatic radarsrdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo08) pp 1ndash6 Rome Italy May 2008
[10] M Secmen S Demir A Hizal and T Eker ldquoFrequencydiverse array antenna with periodic time modulated pattern inrange and anglerdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo07) pp 427ndash430 Boston Mass USA April 2007
[11] J Huang K F Tong and C J Baker ldquoFrequency diversearray with beam scanning featurerdquo in Proceedings of the IEEEAntennas and Propagation Conference (AP-S rsquo08) pp 1ndash4 SanDiego Calif USA July 2008
[12] T Higgins and S D Blunt ldquoAnalysis of range-angle coupledbeamforming with frequency-diverse chirpsrdquo in Proceedings ofthe International Waveform Diversity and Design Conference(WDD rsquo09) pp 140ndash144 Orlando Fla USA February 2009
[13] J Farooq M A Temple and M A Saville ldquoApplication of fre-quency diverse arrays to synthetic aperture radar imagingrdquo inProceedings of the International Conference on Electromagneticsin Advanced Applications (ICEAA rsquo07) pp 447ndash449 TorinoItaly September 2007
[14] J Farooq M A Temple and M A Saville ldquoExploiting fre-quency diverse array processing to improve SAR image reso-lutionrdquo in Proceedings of the IEEE Radar Conference (RADARrsquo08) pp 1ndash5 Rome Italy May 2008
[15] W QWang ldquoPhased-MIMO radar with frequency diversity forrangedependent beamformingrdquo IEEE Sensors Journal vol 13no 8 pp 1320ndash1328 2013
[16] A L Swindlehurst and P Stoica ldquoMaximum likelihood meth-ods in radar array signal processingrdquo Proceedings of the IEEEvol 86 no 2 pp 421ndash441 1998
[17] J Ward ldquoCramer-Rao bounds for target angle and Dopplerestimation with space-time adaptive processing radarrdquo in Pro-ceedings of the 29th Asilomar Conference on Signals Systems andComputers pp 1198ndash1202 Pacific Grove Calif USA November1995
[18] A Dogandzic and A Nehorai ldquoCramer-Rao bounds for esti-mating range velocity and direction with an active arrayrdquo IEEETransactions on Signal Processing vol 49 no 6 pp 1122ndash11372001
[19] A Dogandzic and A Nehorai ldquoEstimating range velocity anddirection with a radar arrayrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo99) pp 2773ndash2776 Phoenix Ariz USA March 1999
[20] J J Zhang G Maalouli A P Suppappola and D MorrellldquoCramer-Rao lower bounds for the joint estimation of targetattributes using MIMO radarrdquo in Proceedings of the Interna-tional Waveform Diversity and Design Conference (WDD rsquo09)pp 103ndash107 Orlando Fla USA February 2009
[21] A Hassanien S A Vorobyov and A B Gershman ldquoMovingtarget parameters estimation in noncoherent MIMO radarsystemsrdquo IEEE Transactions on Signal Processing vol 60 no 5pp 2354ndash2361 2012
[22] Q He R S Blum and A M Haimovich ldquoNoncoherent MIMOradar for location and velocity estimation more antennasmeans better performancerdquo IEEE Transactions on Signal Pro-cessing vol 58 no 7 pp 3661ndash3680 2010
[23] S M Kay Fundamentals of Statistical Signal Processing Estima-tion Theory vol 1 Pearson London UK 2nd edition 2011
[24] H L van TreesOptimum Array Processing JohnWiley amp SonsNew York NY USA 2002
[25] D Wilcox and M Sellathurai ldquoOn MIMO radar subarrayedtransmit beamformingrdquo IEEE Transactions on Signal Processingvol 60 no 4 pp 2076ndash2081 2012
[26] WQWang andH Z Shao ldquoRange-angle localization of targetsby a double-pulse frequency diverse array radarrdquo IEEE Journalon Selected Topics in Signal Processing vol 8 no 1 pp 106ndash1142014
[27] S Gogineni and A Nehorai ldquoTarget estimation using sparsemodeling for distributed MIMO radarrdquo IEEE Transactions onSignal Processing vol 59 no 11 pp 5315ndash5325 2011
[28] T Li andA Nehorai ldquoMaximum likelihood direction finding inspatially colored noise fields using sparse sensor arraysrdquo IEEETransactions on Signal Processing vol 59 no 3 pp 1048ndash10622011
[29] S Sen ldquoOFDMradar space-time adaptive processing by exploit-ing spatio-temporal sparsityrdquo IEEE Transactions on SignalProcessing vol 61 no 1 pp 118ndash130 2013
[30] J J Blanz A Papathanassiou M Haardt I Furio and P WBaier ldquoSmart antennas for combined DOA and joint channelestimation in time-slotted CDMA mobile radio systems withjoint detectionrdquo IEEE Transactions onVehicular Technology vol49 no 2 pp 293ndash306 2000
[31] P N Pathirana S C K Herath and A V Savkin ldquoMultitargettracking via space transformations using a single frequencycontinuous wave radarrdquo IEEE Transactions on Signal Processingvol 60 no 10 pp 5217ndash5229 2012
International Journal of Antennas and Propagation 15
[32] P H Leong T D Abhayapala and T A Lamahewa ldquoMultipletarget localization using wideband echo chirp signalsrdquo IEEETransactions on Signal Processing vol 61 no 16 pp 4077ndash40892013
[33] L Zhuang X Liu and W Yu ldquoPrecisely beam steering forfrequency diverse arrays based on frequency offset selectionrdquoin Proceedings of the International Radar Conference (RADARrsquo09) pp 1ndash4 Bordeaux France December 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 International Journal of Antennas and Propagation
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
(a) CRLB for estimating angle versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
(b) CRLB for estimating range versus SNR
20151050
SNR (dB)
M = 16 Δf = 15kHzM = 16 Δf = 30kHz
M = 32 Δf = 15kHzM = 32 Δf = 30kHz
minus5minus10
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
(c) CRLB for Doppler shift versus SNR
Figure 7 General CRLB results of FDA radar
B Derive the CRLB for Range WhenAngle Is Known
Under the data model in Section 3 when the direction 120579 isknown the parameter to be estimated is 119903 The FIM of 119903 is
119868119903119903FDA
= 2Re 119863H119903(120595) (Rminus1
n )119863119903(120595)
= 210038161003816100381610038161205730
1003816100381610038161003816
2 Re120597aH (120579 119903)
120597119903Rminus1
n120597a (120579 119903)
120597119903
(B1)
The derivation of a(120579 119903) with respect to 119903 for FDA is
120597a (120579 119903)120597119903
= 1198952120587Δ119891
119888Da (120579 119903)
120597aH (120579 119903)
120597119903
120597a (120579 119903)120597119903
=4120587
2
Δ1198912
1198882
119872
sum
119898=1
(119898 minus 1)2
(B2)
The FIM of 119903 is thus given by
119868119903119903FDA
= 2SNR4120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(B3)
International Journal of Antennas and Propagation 11
C Derive the CRLB for Range and Angle
Under the data model in Section 3 both the angle and rangeare unknown The range and angle of targets are estimatedjointly The FIM for parameters 120579 and 119903 can be expressed as
IFDA = 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) = [
119868120579120579
119868120579119903
119868119903120579
119868119903119903
] (C1)
We then have
IFDA = 210038161003816100381610038161205730
1003816100381610038161003816
2
times
[[[[
[
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
]]]]
]
(C2)
Since (120597H(120595)120597119903)Rminus1
n (120597(120595)120597120579) = (120597H(120595)
120597120579)Rminus1
n (120597(120595)120597119903) then 119868120579119903= 119868
119903120579 We can get
119868120579120579
= 210038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
= 81205872
1198892cos2 (120579)
times SNRsum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
119868120579119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
= minus81205872
119889Δ119891 cos (120579)
times SNR[sum
119872
119898=1(119898 minus 1)
2
120582119888+Δ119891sum
119872
119898=1(119898 minus 1)
3
1198882]
119868119903119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
= SNR8120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(C3)
Since CRLB120579120579FDA
= [Iminus1
FDA]11 CRLB119903119903FDA
= [Iminus1
FDA]22 Theexpressions for the CRLB of angle and range estimationsgiven in (14) can be obtained by substituting (C3) into (C2)
D General CRLB Results
Under the data model in Section 4 we derive the continuousCRLB expressions for temporally and spatially white noiseand denote SNR = |120573|
2
s1205902
119899with s being the power We
also start with FIM
I120581119894120581119895
= 2Re 119863H120581119894
(120581) (Cminus1
n otimes Rminus1
n )119863120581119895(120581) (D1)
where 120581119894is the 119894th element of 120581 and 119863
120581119894(120595) = 120597(120581)120597120581
119894
Consider
IFDA =
[[[[[
[
119868120573120573
119868119879
120579120573119868119879
120578120573
119868120579120573
119868120579120579
119868119879
120578120579
119868120578120573
119868120578120579
119868120578120578
]]]]]
]
(D2)
For clarity we rewrite Fisherrsquos information matrix I as
IFDA = [A UV B] (D3)
where
V = U119879
(D4a)
V = [119868120579120573
119868120578120573
] (D4b)
B = [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] (D4c)
According to the matrix inversion lemma the inversematrix of IFDA is
Iminus1
FDA = [
[
(A minus UBminus1V)minus1
minusAminus1U(B minus VAminus1U)minus1
minusBminus1V(A minus UBminus1V)minus1
(B minus VAminus1U)minus1
]
]
(D5)
where the submatrix (A minus UBminus1V)minus1 corresponds to theCRLBof120573which is nuisance and (B minus VAminus1U)minus1 correspondsto the CRLB of 120579 and 120578 = [119903 Ω
119863]119879 which are of interest
CRLB120579120578120579120578FDA
= (B minus VAminus1U)minus1
= [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] minus [119868120579120573
119868120578120573
] 119868minus1
120573120573[119868
119879
120579120573119868119879
120578120573]
minus1
(D6)
where
119868120573120573
= 2 sdot[[[
[
s sdot 1198721205902
119899
0
0s sdot 1198721205902
119899
]]]
]
(D7a)
119868120579120573
= 2 sdot Re[1 119895] otimess1205902
119899
sdot 120573lowast
1198601 (D7b)
119868120578120573
= 2 sdot Re[1 119895] otimes120573
lowast
1205902
119908
sdot [119872 sdot 1198603+ s sdot 119860
4] (D7c)
119868120578120578
= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
[119872 sdot 1198606+ 119860
H3119860
4+ 119860
3119860
H4+ s sdot 119860
7]
(D7d)
12 International Journal of Antennas and Propagation
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHz
70001 70001 70001 70002100286514
100286515
100286515
kHz
(c) CRLB for Doppler shift versus SNR
Figure 8 General CRLB results of TS-FDA radar with119872 = 32
119868120579120579minus (119868
120579120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198602minus119860
1119860
H1
119872) (D8a)
119868120578120579minus (119868
120578120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198605minus119860
4119860
H1
119872) (D8b)
119868120578120578
minus (119868120578120573119868119879
120578120573)
1205902
119899
s119872
= 2
10038161003816100381610038161205731003816100381610038161003816
2
s1205902
119899
(119872119860
6
s+ 119860
7minus119872119860
3119860
H3
s2minus119860
4119860
H4
119872)
(D8c)
1198601= minus119895(
2120587119889 cos (120579)sum119872
119898=1(119898 minus 1)
120582
+2120587119889Δ119891 cos (120579)sum119872
119898=1(119898 minus 1)
2
119888)
(D9a)
1198602= 4120587
2
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(D9b)
International Journal of Antennas and Propagation 13
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(c) CRLB for Doppler shift versus SNR
Figure 9 General CRLB results of TS-FDA radar with119872 = 20
1198603= [int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905 int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
119879
(D9c)
1198604= [
2120587Δ119891sum119872
119898=1(119898 minus 1)
1198880]
119879
(D9d)
1198605= [minus4120587
2119889Δ119891 cos(120579) [
sum119872
119898=1(119898 minus 1)
2
12120582119888
+
Δ119891sum119872
119898=1(119898 minus 1)
3
121198882
] 0]
119879
(D9e)
1198606=[[
[
int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905 int
infin
minusinfin
119905119904H(119905 minus 120591)
119889119904 (119905 minus 120591)
119889119905
119889119905
int
infin
minusinfin
119905119904 (119905 minus 120591)
119889119904(119905 minus 120591)H
119889119905
119889119905 int
infin
minusinfin
1199052|119904 (119905 minus 120591)|
2119889119905
]]
]
(D9f)
1198607=[[
[
41205872
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
0
0 0
]]
]
(D9g)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework described in this paper was supported in part by theNational Natural Science Foundation of China under Grant
14 International Journal of Antennas and Propagation
41101317 the Program for New Century Excellent Talentsin University under Grant NCET-12-0095 Sichuan ProvinceScience Fund for Distinguished Young Scholars under Grant2013JQ0003 and Fundamental Research Fund for theCentralUniversities
References
[1] F Bandiera M Mancino and G Ricci ldquoLocalization strategiesfor multiple point-like radar targetsrdquo IEEE Transactions onSignal Processing vol 60 no 12 pp 6708ndash6712 2012
[2] D R Fuhrmann J P Browning and M Rangaswamy ldquoSignal-ing strategies for the hybrid MIMO phased-array radarrdquo IEEEJournal on Selected Topics in Signal Processing vol 4 no 1 pp66ndash78 2010
[3] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011
[4] M Compagnoni P Bestagini F Antonacci A Sarti and STubaro ldquoLocalization of acoustic sources through the fittingof propagation cones using multiple independent arraysrdquo IEEETransactions on Audio Speech and Language Processing vol 20no 7 pp 1964ndash1975 2012
[5] P Antonik M C Wicks H D Griffiths and C J BakerldquoFrequency diverse array radarsrdquo in Proceedings of the IEEERadar Conference (RADAR rsquo06) pp 215ndash217 Verona NY USAApril 2006
[6] P Antonik M CWicks H D Griffiths and C J Baker ldquoMulti-mission multi-mode waveform diversityrdquo in Proceedings of theIEEE Radar Conference (RADAR rsquo06) pp 580ndash582 Verona NYUSA April 2006
[7] P Antonik H D Griffiths and C J Baker ldquoRange depen-dent beamforming using element level waveform diversityrdquo inProceedings of the International Waveform Diversity and DesignConference pp 1ndash4 Las Vegas Nev USA January 2006
[8] P Baizert T B HaleM A Temple andM CWicks ldquoForward-looking radar GMTI benefits using a linear frequency diversearrayrdquo Electronics Letters vol 42 no 22 pp 1311ndash1312 2006
[9] B W Jung R S Adve and J Chun ldquoFrequency diversity inmultistatic radarsrdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo08) pp 1ndash6 Rome Italy May 2008
[10] M Secmen S Demir A Hizal and T Eker ldquoFrequencydiverse array antenna with periodic time modulated pattern inrange and anglerdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo07) pp 427ndash430 Boston Mass USA April 2007
[11] J Huang K F Tong and C J Baker ldquoFrequency diversearray with beam scanning featurerdquo in Proceedings of the IEEEAntennas and Propagation Conference (AP-S rsquo08) pp 1ndash4 SanDiego Calif USA July 2008
[12] T Higgins and S D Blunt ldquoAnalysis of range-angle coupledbeamforming with frequency-diverse chirpsrdquo in Proceedings ofthe International Waveform Diversity and Design Conference(WDD rsquo09) pp 140ndash144 Orlando Fla USA February 2009
[13] J Farooq M A Temple and M A Saville ldquoApplication of fre-quency diverse arrays to synthetic aperture radar imagingrdquo inProceedings of the International Conference on Electromagneticsin Advanced Applications (ICEAA rsquo07) pp 447ndash449 TorinoItaly September 2007
[14] J Farooq M A Temple and M A Saville ldquoExploiting fre-quency diverse array processing to improve SAR image reso-lutionrdquo in Proceedings of the IEEE Radar Conference (RADARrsquo08) pp 1ndash5 Rome Italy May 2008
[15] W QWang ldquoPhased-MIMO radar with frequency diversity forrangedependent beamformingrdquo IEEE Sensors Journal vol 13no 8 pp 1320ndash1328 2013
[16] A L Swindlehurst and P Stoica ldquoMaximum likelihood meth-ods in radar array signal processingrdquo Proceedings of the IEEEvol 86 no 2 pp 421ndash441 1998
[17] J Ward ldquoCramer-Rao bounds for target angle and Dopplerestimation with space-time adaptive processing radarrdquo in Pro-ceedings of the 29th Asilomar Conference on Signals Systems andComputers pp 1198ndash1202 Pacific Grove Calif USA November1995
[18] A Dogandzic and A Nehorai ldquoCramer-Rao bounds for esti-mating range velocity and direction with an active arrayrdquo IEEETransactions on Signal Processing vol 49 no 6 pp 1122ndash11372001
[19] A Dogandzic and A Nehorai ldquoEstimating range velocity anddirection with a radar arrayrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo99) pp 2773ndash2776 Phoenix Ariz USA March 1999
[20] J J Zhang G Maalouli A P Suppappola and D MorrellldquoCramer-Rao lower bounds for the joint estimation of targetattributes using MIMO radarrdquo in Proceedings of the Interna-tional Waveform Diversity and Design Conference (WDD rsquo09)pp 103ndash107 Orlando Fla USA February 2009
[21] A Hassanien S A Vorobyov and A B Gershman ldquoMovingtarget parameters estimation in noncoherent MIMO radarsystemsrdquo IEEE Transactions on Signal Processing vol 60 no 5pp 2354ndash2361 2012
[22] Q He R S Blum and A M Haimovich ldquoNoncoherent MIMOradar for location and velocity estimation more antennasmeans better performancerdquo IEEE Transactions on Signal Pro-cessing vol 58 no 7 pp 3661ndash3680 2010
[23] S M Kay Fundamentals of Statistical Signal Processing Estima-tion Theory vol 1 Pearson London UK 2nd edition 2011
[24] H L van TreesOptimum Array Processing JohnWiley amp SonsNew York NY USA 2002
[25] D Wilcox and M Sellathurai ldquoOn MIMO radar subarrayedtransmit beamformingrdquo IEEE Transactions on Signal Processingvol 60 no 4 pp 2076ndash2081 2012
[26] WQWang andH Z Shao ldquoRange-angle localization of targetsby a double-pulse frequency diverse array radarrdquo IEEE Journalon Selected Topics in Signal Processing vol 8 no 1 pp 106ndash1142014
[27] S Gogineni and A Nehorai ldquoTarget estimation using sparsemodeling for distributed MIMO radarrdquo IEEE Transactions onSignal Processing vol 59 no 11 pp 5315ndash5325 2011
[28] T Li andA Nehorai ldquoMaximum likelihood direction finding inspatially colored noise fields using sparse sensor arraysrdquo IEEETransactions on Signal Processing vol 59 no 3 pp 1048ndash10622011
[29] S Sen ldquoOFDMradar space-time adaptive processing by exploit-ing spatio-temporal sparsityrdquo IEEE Transactions on SignalProcessing vol 61 no 1 pp 118ndash130 2013
[30] J J Blanz A Papathanassiou M Haardt I Furio and P WBaier ldquoSmart antennas for combined DOA and joint channelestimation in time-slotted CDMA mobile radio systems withjoint detectionrdquo IEEE Transactions onVehicular Technology vol49 no 2 pp 293ndash306 2000
[31] P N Pathirana S C K Herath and A V Savkin ldquoMultitargettracking via space transformations using a single frequencycontinuous wave radarrdquo IEEE Transactions on Signal Processingvol 60 no 10 pp 5217ndash5229 2012
International Journal of Antennas and Propagation 15
[32] P H Leong T D Abhayapala and T A Lamahewa ldquoMultipletarget localization using wideband echo chirp signalsrdquo IEEETransactions on Signal Processing vol 61 no 16 pp 4077ndash40892013
[33] L Zhuang X Liu and W Yu ldquoPrecisely beam steering forfrequency diverse arrays based on frequency offset selectionrdquoin Proceedings of the International Radar Conference (RADARrsquo09) pp 1ndash4 Bordeaux France December 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 11
C Derive the CRLB for Range and Angle
Under the data model in Section 3 both the angle and rangeare unknown The range and angle of targets are estimatedjointly The FIM for parameters 120579 and 119903 can be expressed as
IFDA = 2Re 119863H120595119894
(120595) (Rminus1
n )119863120595119895(120595) = [
119868120579120579
119868120579119903
119868119903120579
119868119903119903
] (C1)
We then have
IFDA = 210038161003816100381610038161205730
1003816100381610038161003816
2
times
[[[[
[
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597120579
120597H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
120597H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
]]]]
]
(C2)
Since (120597H(120595)120597119903)Rminus1
n (120597(120595)120597120579) = (120597H(120595)
120597120579)Rminus1
n (120597(120595)120597119903) then 119868120579119903= 119868
119903120579 We can get
119868120579120579
= 210038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597120579
= 81205872
1198892cos2 (120579)
times SNRsum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
119868120579119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597120579Rminus1
n120597 (120595)
120597119903
= minus81205872
119889Δ119891 cos (120579)
times SNR[sum
119872
119898=1(119898 minus 1)
2
120582119888+Δ119891sum
119872
119898=1(119898 minus 1)
3
1198882]
119868119903119903= 2
10038161003816100381610038161205730
1003816100381610038161003816
2120597
H(120595)
120597119903Rminus1
n120597 (120595)
120597119903
= SNR8120587
2
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
(C3)
Since CRLB120579120579FDA
= [Iminus1
FDA]11 CRLB119903119903FDA
= [Iminus1
FDA]22 Theexpressions for the CRLB of angle and range estimationsgiven in (14) can be obtained by substituting (C3) into (C2)
D General CRLB Results
Under the data model in Section 4 we derive the continuousCRLB expressions for temporally and spatially white noiseand denote SNR = |120573|
2
s1205902
119899with s being the power We
also start with FIM
I120581119894120581119895
= 2Re 119863H120581119894
(120581) (Cminus1
n otimes Rminus1
n )119863120581119895(120581) (D1)
where 120581119894is the 119894th element of 120581 and 119863
120581119894(120595) = 120597(120581)120597120581
119894
Consider
IFDA =
[[[[[
[
119868120573120573
119868119879
120579120573119868119879
120578120573
119868120579120573
119868120579120579
119868119879
120578120579
119868120578120573
119868120578120579
119868120578120578
]]]]]
]
(D2)
For clarity we rewrite Fisherrsquos information matrix I as
IFDA = [A UV B] (D3)
where
V = U119879
(D4a)
V = [119868120579120573
119868120578120573
] (D4b)
B = [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] (D4c)
According to the matrix inversion lemma the inversematrix of IFDA is
Iminus1
FDA = [
[
(A minus UBminus1V)minus1
minusAminus1U(B minus VAminus1U)minus1
minusBminus1V(A minus UBminus1V)minus1
(B minus VAminus1U)minus1
]
]
(D5)
where the submatrix (A minus UBminus1V)minus1 corresponds to theCRLBof120573which is nuisance and (B minus VAminus1U)minus1 correspondsto the CRLB of 120579 and 120578 = [119903 Ω
119863]119879 which are of interest
CRLB120579120578120579120578FDA
= (B minus VAminus1U)minus1
= [119868120579120579
119868119879
120578120579
119868120578120579
119868120578120578
] minus [119868120579120573
119868120578120573
] 119868minus1
120573120573[119868
119879
120579120573119868119879
120578120573]
minus1
(D6)
where
119868120573120573
= 2 sdot[[[
[
s sdot 1198721205902
119899
0
0s sdot 1198721205902
119899
]]]
]
(D7a)
119868120579120573
= 2 sdot Re[1 119895] otimess1205902
119899
sdot 120573lowast
1198601 (D7b)
119868120578120573
= 2 sdot Re[1 119895] otimes120573
lowast
1205902
119908
sdot [119872 sdot 1198603+ s sdot 119860
4] (D7c)
119868120578120578
= 2 sdot Re10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
[119872 sdot 1198606+ 119860
H3119860
4+ 119860
3119860
H4+ s sdot 119860
7]
(D7d)
12 International Journal of Antennas and Propagation
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHz
70001 70001 70001 70002100286514
100286515
100286515
kHz
(c) CRLB for Doppler shift versus SNR
Figure 8 General CRLB results of TS-FDA radar with119872 = 32
119868120579120579minus (119868
120579120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198602minus119860
1119860
H1
119872) (D8a)
119868120578120579minus (119868
120578120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198605minus119860
4119860
H1
119872) (D8b)
119868120578120578
minus (119868120578120573119868119879
120578120573)
1205902
119899
s119872
= 2
10038161003816100381610038161205731003816100381610038161003816
2
s1205902
119899
(119872119860
6
s+ 119860
7minus119872119860
3119860
H3
s2minus119860
4119860
H4
119872)
(D8c)
1198601= minus119895(
2120587119889 cos (120579)sum119872
119898=1(119898 minus 1)
120582
+2120587119889Δ119891 cos (120579)sum119872
119898=1(119898 minus 1)
2
119888)
(D9a)
1198602= 4120587
2
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(D9b)
International Journal of Antennas and Propagation 13
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(c) CRLB for Doppler shift versus SNR
Figure 9 General CRLB results of TS-FDA radar with119872 = 20
1198603= [int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905 int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
119879
(D9c)
1198604= [
2120587Δ119891sum119872
119898=1(119898 minus 1)
1198880]
119879
(D9d)
1198605= [minus4120587
2119889Δ119891 cos(120579) [
sum119872
119898=1(119898 minus 1)
2
12120582119888
+
Δ119891sum119872
119898=1(119898 minus 1)
3
121198882
] 0]
119879
(D9e)
1198606=[[
[
int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905 int
infin
minusinfin
119905119904H(119905 minus 120591)
119889119904 (119905 minus 120591)
119889119905
119889119905
int
infin
minusinfin
119905119904 (119905 minus 120591)
119889119904(119905 minus 120591)H
119889119905
119889119905 int
infin
minusinfin
1199052|119904 (119905 minus 120591)|
2119889119905
]]
]
(D9f)
1198607=[[
[
41205872
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
0
0 0
]]
]
(D9g)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework described in this paper was supported in part by theNational Natural Science Foundation of China under Grant
14 International Journal of Antennas and Propagation
41101317 the Program for New Century Excellent Talentsin University under Grant NCET-12-0095 Sichuan ProvinceScience Fund for Distinguished Young Scholars under Grant2013JQ0003 and Fundamental Research Fund for theCentralUniversities
References
[1] F Bandiera M Mancino and G Ricci ldquoLocalization strategiesfor multiple point-like radar targetsrdquo IEEE Transactions onSignal Processing vol 60 no 12 pp 6708ndash6712 2012
[2] D R Fuhrmann J P Browning and M Rangaswamy ldquoSignal-ing strategies for the hybrid MIMO phased-array radarrdquo IEEEJournal on Selected Topics in Signal Processing vol 4 no 1 pp66ndash78 2010
[3] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011
[4] M Compagnoni P Bestagini F Antonacci A Sarti and STubaro ldquoLocalization of acoustic sources through the fittingof propagation cones using multiple independent arraysrdquo IEEETransactions on Audio Speech and Language Processing vol 20no 7 pp 1964ndash1975 2012
[5] P Antonik M C Wicks H D Griffiths and C J BakerldquoFrequency diverse array radarsrdquo in Proceedings of the IEEERadar Conference (RADAR rsquo06) pp 215ndash217 Verona NY USAApril 2006
[6] P Antonik M CWicks H D Griffiths and C J Baker ldquoMulti-mission multi-mode waveform diversityrdquo in Proceedings of theIEEE Radar Conference (RADAR rsquo06) pp 580ndash582 Verona NYUSA April 2006
[7] P Antonik H D Griffiths and C J Baker ldquoRange depen-dent beamforming using element level waveform diversityrdquo inProceedings of the International Waveform Diversity and DesignConference pp 1ndash4 Las Vegas Nev USA January 2006
[8] P Baizert T B HaleM A Temple andM CWicks ldquoForward-looking radar GMTI benefits using a linear frequency diversearrayrdquo Electronics Letters vol 42 no 22 pp 1311ndash1312 2006
[9] B W Jung R S Adve and J Chun ldquoFrequency diversity inmultistatic radarsrdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo08) pp 1ndash6 Rome Italy May 2008
[10] M Secmen S Demir A Hizal and T Eker ldquoFrequencydiverse array antenna with periodic time modulated pattern inrange and anglerdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo07) pp 427ndash430 Boston Mass USA April 2007
[11] J Huang K F Tong and C J Baker ldquoFrequency diversearray with beam scanning featurerdquo in Proceedings of the IEEEAntennas and Propagation Conference (AP-S rsquo08) pp 1ndash4 SanDiego Calif USA July 2008
[12] T Higgins and S D Blunt ldquoAnalysis of range-angle coupledbeamforming with frequency-diverse chirpsrdquo in Proceedings ofthe International Waveform Diversity and Design Conference(WDD rsquo09) pp 140ndash144 Orlando Fla USA February 2009
[13] J Farooq M A Temple and M A Saville ldquoApplication of fre-quency diverse arrays to synthetic aperture radar imagingrdquo inProceedings of the International Conference on Electromagneticsin Advanced Applications (ICEAA rsquo07) pp 447ndash449 TorinoItaly September 2007
[14] J Farooq M A Temple and M A Saville ldquoExploiting fre-quency diverse array processing to improve SAR image reso-lutionrdquo in Proceedings of the IEEE Radar Conference (RADARrsquo08) pp 1ndash5 Rome Italy May 2008
[15] W QWang ldquoPhased-MIMO radar with frequency diversity forrangedependent beamformingrdquo IEEE Sensors Journal vol 13no 8 pp 1320ndash1328 2013
[16] A L Swindlehurst and P Stoica ldquoMaximum likelihood meth-ods in radar array signal processingrdquo Proceedings of the IEEEvol 86 no 2 pp 421ndash441 1998
[17] J Ward ldquoCramer-Rao bounds for target angle and Dopplerestimation with space-time adaptive processing radarrdquo in Pro-ceedings of the 29th Asilomar Conference on Signals Systems andComputers pp 1198ndash1202 Pacific Grove Calif USA November1995
[18] A Dogandzic and A Nehorai ldquoCramer-Rao bounds for esti-mating range velocity and direction with an active arrayrdquo IEEETransactions on Signal Processing vol 49 no 6 pp 1122ndash11372001
[19] A Dogandzic and A Nehorai ldquoEstimating range velocity anddirection with a radar arrayrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo99) pp 2773ndash2776 Phoenix Ariz USA March 1999
[20] J J Zhang G Maalouli A P Suppappola and D MorrellldquoCramer-Rao lower bounds for the joint estimation of targetattributes using MIMO radarrdquo in Proceedings of the Interna-tional Waveform Diversity and Design Conference (WDD rsquo09)pp 103ndash107 Orlando Fla USA February 2009
[21] A Hassanien S A Vorobyov and A B Gershman ldquoMovingtarget parameters estimation in noncoherent MIMO radarsystemsrdquo IEEE Transactions on Signal Processing vol 60 no 5pp 2354ndash2361 2012
[22] Q He R S Blum and A M Haimovich ldquoNoncoherent MIMOradar for location and velocity estimation more antennasmeans better performancerdquo IEEE Transactions on Signal Pro-cessing vol 58 no 7 pp 3661ndash3680 2010
[23] S M Kay Fundamentals of Statistical Signal Processing Estima-tion Theory vol 1 Pearson London UK 2nd edition 2011
[24] H L van TreesOptimum Array Processing JohnWiley amp SonsNew York NY USA 2002
[25] D Wilcox and M Sellathurai ldquoOn MIMO radar subarrayedtransmit beamformingrdquo IEEE Transactions on Signal Processingvol 60 no 4 pp 2076ndash2081 2012
[26] WQWang andH Z Shao ldquoRange-angle localization of targetsby a double-pulse frequency diverse array radarrdquo IEEE Journalon Selected Topics in Signal Processing vol 8 no 1 pp 106ndash1142014
[27] S Gogineni and A Nehorai ldquoTarget estimation using sparsemodeling for distributed MIMO radarrdquo IEEE Transactions onSignal Processing vol 59 no 11 pp 5315ndash5325 2011
[28] T Li andA Nehorai ldquoMaximum likelihood direction finding inspatially colored noise fields using sparse sensor arraysrdquo IEEETransactions on Signal Processing vol 59 no 3 pp 1048ndash10622011
[29] S Sen ldquoOFDMradar space-time adaptive processing by exploit-ing spatio-temporal sparsityrdquo IEEE Transactions on SignalProcessing vol 61 no 1 pp 118ndash130 2013
[30] J J Blanz A Papathanassiou M Haardt I Furio and P WBaier ldquoSmart antennas for combined DOA and joint channelestimation in time-slotted CDMA mobile radio systems withjoint detectionrdquo IEEE Transactions onVehicular Technology vol49 no 2 pp 293ndash306 2000
[31] P N Pathirana S C K Herath and A V Savkin ldquoMultitargettracking via space transformations using a single frequencycontinuous wave radarrdquo IEEE Transactions on Signal Processingvol 60 no 10 pp 5217ndash5229 2012
International Journal of Antennas and Propagation 15
[32] P H Leong T D Abhayapala and T A Lamahewa ldquoMultipletarget localization using wideband echo chirp signalsrdquo IEEETransactions on Signal Processing vol 61 no 16 pp 4077ndash40892013
[33] L Zhuang X Liu and W Yu ldquoPrecisely beam steering forfrequency diverse arrays based on frequency offset selectionrdquoin Proceedings of the International Radar Conference (RADARrsquo09) pp 1ndash4 Bordeaux France December 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
12 International Journal of Antennas and Propagation
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHz
70001 70001 70001 70002100286514
100286515
100286515
kHz
(c) CRLB for Doppler shift versus SNR
Figure 8 General CRLB results of TS-FDA radar with119872 = 32
119868120579120579minus (119868
120579120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198602minus119860
1119860
H1
119872) (D8a)
119868120578120579minus (119868
120578120573119868119879
120579120573)
1205902
119899
s119872= 2
s 10038161003816100381610038161205731003816100381610038161003816
2
1205902
119899
(1198605minus119860
4119860
H1
119872) (D8b)
119868120578120578
minus (119868120578120573119868119879
120578120573)
1205902
119899
s119872
= 2
10038161003816100381610038161205731003816100381610038161003816
2
s1205902
119899
(119872119860
6
s+ 119860
7minus119872119860
3119860
H3
s2minus119860
4119860
H4
119872)
(D8c)
1198601= minus119895(
2120587119889 cos (120579)sum119872
119898=1(119898 minus 1)
120582
+2120587119889Δ119891 cos (120579)sum119872
119898=1(119898 minus 1)
2
119888)
(D9a)
1198602= 4120587
2
1198892cos2 (120579)
sdot sum
119872
119898=1(119898 minus 1)
2
1205822+(Δ119891)
2
sum119872
119898=1(119898 minus 1)
4
1198882
+2Δ119891sum
119872
119898=1(119898 minus 1)
3
120582119888
(D9b)
International Journal of Antennas and Propagation 13
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(c) CRLB for Doppler shift versus SNR
Figure 9 General CRLB results of TS-FDA radar with119872 = 20
1198603= [int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905 int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
119879
(D9c)
1198604= [
2120587Δ119891sum119872
119898=1(119898 minus 1)
1198880]
119879
(D9d)
1198605= [minus4120587
2119889Δ119891 cos(120579) [
sum119872
119898=1(119898 minus 1)
2
12120582119888
+
Δ119891sum119872
119898=1(119898 minus 1)
3
121198882
] 0]
119879
(D9e)
1198606=[[
[
int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905 int
infin
minusinfin
119905119904H(119905 minus 120591)
119889119904 (119905 minus 120591)
119889119905
119889119905
int
infin
minusinfin
119905119904 (119905 minus 120591)
119889119904(119905 minus 120591)H
119889119905
119889119905 int
infin
minusinfin
1199052|119904 (119905 minus 120591)|
2119889119905
]]
]
(D9f)
1198607=[[
[
41205872
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
0
0 0
]]
]
(D9g)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework described in this paper was supported in part by theNational Natural Science Foundation of China under Grant
14 International Journal of Antennas and Propagation
41101317 the Program for New Century Excellent Talentsin University under Grant NCET-12-0095 Sichuan ProvinceScience Fund for Distinguished Young Scholars under Grant2013JQ0003 and Fundamental Research Fund for theCentralUniversities
References
[1] F Bandiera M Mancino and G Ricci ldquoLocalization strategiesfor multiple point-like radar targetsrdquo IEEE Transactions onSignal Processing vol 60 no 12 pp 6708ndash6712 2012
[2] D R Fuhrmann J P Browning and M Rangaswamy ldquoSignal-ing strategies for the hybrid MIMO phased-array radarrdquo IEEEJournal on Selected Topics in Signal Processing vol 4 no 1 pp66ndash78 2010
[3] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011
[4] M Compagnoni P Bestagini F Antonacci A Sarti and STubaro ldquoLocalization of acoustic sources through the fittingof propagation cones using multiple independent arraysrdquo IEEETransactions on Audio Speech and Language Processing vol 20no 7 pp 1964ndash1975 2012
[5] P Antonik M C Wicks H D Griffiths and C J BakerldquoFrequency diverse array radarsrdquo in Proceedings of the IEEERadar Conference (RADAR rsquo06) pp 215ndash217 Verona NY USAApril 2006
[6] P Antonik M CWicks H D Griffiths and C J Baker ldquoMulti-mission multi-mode waveform diversityrdquo in Proceedings of theIEEE Radar Conference (RADAR rsquo06) pp 580ndash582 Verona NYUSA April 2006
[7] P Antonik H D Griffiths and C J Baker ldquoRange depen-dent beamforming using element level waveform diversityrdquo inProceedings of the International Waveform Diversity and DesignConference pp 1ndash4 Las Vegas Nev USA January 2006
[8] P Baizert T B HaleM A Temple andM CWicks ldquoForward-looking radar GMTI benefits using a linear frequency diversearrayrdquo Electronics Letters vol 42 no 22 pp 1311ndash1312 2006
[9] B W Jung R S Adve and J Chun ldquoFrequency diversity inmultistatic radarsrdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo08) pp 1ndash6 Rome Italy May 2008
[10] M Secmen S Demir A Hizal and T Eker ldquoFrequencydiverse array antenna with periodic time modulated pattern inrange and anglerdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo07) pp 427ndash430 Boston Mass USA April 2007
[11] J Huang K F Tong and C J Baker ldquoFrequency diversearray with beam scanning featurerdquo in Proceedings of the IEEEAntennas and Propagation Conference (AP-S rsquo08) pp 1ndash4 SanDiego Calif USA July 2008
[12] T Higgins and S D Blunt ldquoAnalysis of range-angle coupledbeamforming with frequency-diverse chirpsrdquo in Proceedings ofthe International Waveform Diversity and Design Conference(WDD rsquo09) pp 140ndash144 Orlando Fla USA February 2009
[13] J Farooq M A Temple and M A Saville ldquoApplication of fre-quency diverse arrays to synthetic aperture radar imagingrdquo inProceedings of the International Conference on Electromagneticsin Advanced Applications (ICEAA rsquo07) pp 447ndash449 TorinoItaly September 2007
[14] J Farooq M A Temple and M A Saville ldquoExploiting fre-quency diverse array processing to improve SAR image reso-lutionrdquo in Proceedings of the IEEE Radar Conference (RADARrsquo08) pp 1ndash5 Rome Italy May 2008
[15] W QWang ldquoPhased-MIMO radar with frequency diversity forrangedependent beamformingrdquo IEEE Sensors Journal vol 13no 8 pp 1320ndash1328 2013
[16] A L Swindlehurst and P Stoica ldquoMaximum likelihood meth-ods in radar array signal processingrdquo Proceedings of the IEEEvol 86 no 2 pp 421ndash441 1998
[17] J Ward ldquoCramer-Rao bounds for target angle and Dopplerestimation with space-time adaptive processing radarrdquo in Pro-ceedings of the 29th Asilomar Conference on Signals Systems andComputers pp 1198ndash1202 Pacific Grove Calif USA November1995
[18] A Dogandzic and A Nehorai ldquoCramer-Rao bounds for esti-mating range velocity and direction with an active arrayrdquo IEEETransactions on Signal Processing vol 49 no 6 pp 1122ndash11372001
[19] A Dogandzic and A Nehorai ldquoEstimating range velocity anddirection with a radar arrayrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo99) pp 2773ndash2776 Phoenix Ariz USA March 1999
[20] J J Zhang G Maalouli A P Suppappola and D MorrellldquoCramer-Rao lower bounds for the joint estimation of targetattributes using MIMO radarrdquo in Proceedings of the Interna-tional Waveform Diversity and Design Conference (WDD rsquo09)pp 103ndash107 Orlando Fla USA February 2009
[21] A Hassanien S A Vorobyov and A B Gershman ldquoMovingtarget parameters estimation in noncoherent MIMO radarsystemsrdquo IEEE Transactions on Signal Processing vol 60 no 5pp 2354ndash2361 2012
[22] Q He R S Blum and A M Haimovich ldquoNoncoherent MIMOradar for location and velocity estimation more antennasmeans better performancerdquo IEEE Transactions on Signal Pro-cessing vol 58 no 7 pp 3661ndash3680 2010
[23] S M Kay Fundamentals of Statistical Signal Processing Estima-tion Theory vol 1 Pearson London UK 2nd edition 2011
[24] H L van TreesOptimum Array Processing JohnWiley amp SonsNew York NY USA 2002
[25] D Wilcox and M Sellathurai ldquoOn MIMO radar subarrayedtransmit beamformingrdquo IEEE Transactions on Signal Processingvol 60 no 4 pp 2076ndash2081 2012
[26] WQWang andH Z Shao ldquoRange-angle localization of targetsby a double-pulse frequency diverse array radarrdquo IEEE Journalon Selected Topics in Signal Processing vol 8 no 1 pp 106ndash1142014
[27] S Gogineni and A Nehorai ldquoTarget estimation using sparsemodeling for distributed MIMO radarrdquo IEEE Transactions onSignal Processing vol 59 no 11 pp 5315ndash5325 2011
[28] T Li andA Nehorai ldquoMaximum likelihood direction finding inspatially colored noise fields using sparse sensor arraysrdquo IEEETransactions on Signal Processing vol 59 no 3 pp 1048ndash10622011
[29] S Sen ldquoOFDMradar space-time adaptive processing by exploit-ing spatio-temporal sparsityrdquo IEEE Transactions on SignalProcessing vol 61 no 1 pp 118ndash130 2013
[30] J J Blanz A Papathanassiou M Haardt I Furio and P WBaier ldquoSmart antennas for combined DOA and joint channelestimation in time-slotted CDMA mobile radio systems withjoint detectionrdquo IEEE Transactions onVehicular Technology vol49 no 2 pp 293ndash306 2000
[31] P N Pathirana S C K Herath and A V Savkin ldquoMultitargettracking via space transformations using a single frequencycontinuous wave radarrdquo IEEE Transactions on Signal Processingvol 60 no 10 pp 5217ndash5229 2012
International Journal of Antennas and Propagation 15
[32] P H Leong T D Abhayapala and T A Lamahewa ldquoMultipletarget localization using wideband echo chirp signalsrdquo IEEETransactions on Signal Processing vol 61 no 16 pp 4077ndash40892013
[33] L Zhuang X Liu and W Yu ldquoPrecisely beam steering forfrequency diverse arrays based on frequency offset selectionrdquoin Proceedings of the International Radar Conference (RADARrsquo09) pp 1ndash4 Bordeaux France December 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 13
10minus1
10minus2
10minus3
10minus4
CRLB
of a
ngle
estim
atio
n (d
eg)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(a) CRLB for estimating angle versus SNR
102
101
100
CRLB
of r
ange
estim
atio
n (m
)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(b) CRLB for estimating range versus SNR
102
101
100
10minus1
CRLB
of D
oppl
er sh
ift es
timat
ion
(rad
s)
20151050minus5minus10
SNR (dB)
Δf1 = minus30kHz Δf2 = 30kHzΔf1 = minus15kHz Δf2 = 30kHz
Δf1 = 0kHz Δf2 = 30
Δf1 = 15kHz Δf2 = 40kHzkHz
(c) CRLB for Doppler shift versus SNR
Figure 9 General CRLB results of TS-FDA radar with119872 = 20
1198603= [int
infin
minusinfin
119904 (119905 minus 120591)119889119904(119905 minus 120591)
H
119889119905119889119905 int
infin
minusinfin
119905|119904 (119905 minus 120591)|2
119889119905]
119879
(D9c)
1198604= [
2120587Δ119891sum119872
119898=1(119898 minus 1)
1198880]
119879
(D9d)
1198605= [minus4120587
2119889Δ119891 cos(120579) [
sum119872
119898=1(119898 minus 1)
2
12120582119888
+
Δ119891sum119872
119898=1(119898 minus 1)
3
121198882
] 0]
119879
(D9e)
1198606=[[
[
int
infin
minusinfin
10038161003816100381610038161003816100381610038161003816
119889119904 (119905 minus 120591)
119889119905
10038161003816100381610038161003816100381610038161003816
2
119889119905 int
infin
minusinfin
119905119904H(119905 minus 120591)
119889119904 (119905 minus 120591)
119889119905
119889119905
int
infin
minusinfin
119905119904 (119905 minus 120591)
119889119904(119905 minus 120591)H
119889119905
119889119905 int
infin
minusinfin
1199052|119904 (119905 minus 120591)|
2119889119905
]]
]
(D9f)
1198607=[[
[
41205872
(Δ119891)2
1198882
119872
sum
119898=1
(119898 minus 1)2
0
0 0
]]
]
(D9g)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thework described in this paper was supported in part by theNational Natural Science Foundation of China under Grant
14 International Journal of Antennas and Propagation
41101317 the Program for New Century Excellent Talentsin University under Grant NCET-12-0095 Sichuan ProvinceScience Fund for Distinguished Young Scholars under Grant2013JQ0003 and Fundamental Research Fund for theCentralUniversities
References
[1] F Bandiera M Mancino and G Ricci ldquoLocalization strategiesfor multiple point-like radar targetsrdquo IEEE Transactions onSignal Processing vol 60 no 12 pp 6708ndash6712 2012
[2] D R Fuhrmann J P Browning and M Rangaswamy ldquoSignal-ing strategies for the hybrid MIMO phased-array radarrdquo IEEEJournal on Selected Topics in Signal Processing vol 4 no 1 pp66ndash78 2010
[3] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011
[4] M Compagnoni P Bestagini F Antonacci A Sarti and STubaro ldquoLocalization of acoustic sources through the fittingof propagation cones using multiple independent arraysrdquo IEEETransactions on Audio Speech and Language Processing vol 20no 7 pp 1964ndash1975 2012
[5] P Antonik M C Wicks H D Griffiths and C J BakerldquoFrequency diverse array radarsrdquo in Proceedings of the IEEERadar Conference (RADAR rsquo06) pp 215ndash217 Verona NY USAApril 2006
[6] P Antonik M CWicks H D Griffiths and C J Baker ldquoMulti-mission multi-mode waveform diversityrdquo in Proceedings of theIEEE Radar Conference (RADAR rsquo06) pp 580ndash582 Verona NYUSA April 2006
[7] P Antonik H D Griffiths and C J Baker ldquoRange depen-dent beamforming using element level waveform diversityrdquo inProceedings of the International Waveform Diversity and DesignConference pp 1ndash4 Las Vegas Nev USA January 2006
[8] P Baizert T B HaleM A Temple andM CWicks ldquoForward-looking radar GMTI benefits using a linear frequency diversearrayrdquo Electronics Letters vol 42 no 22 pp 1311ndash1312 2006
[9] B W Jung R S Adve and J Chun ldquoFrequency diversity inmultistatic radarsrdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo08) pp 1ndash6 Rome Italy May 2008
[10] M Secmen S Demir A Hizal and T Eker ldquoFrequencydiverse array antenna with periodic time modulated pattern inrange and anglerdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo07) pp 427ndash430 Boston Mass USA April 2007
[11] J Huang K F Tong and C J Baker ldquoFrequency diversearray with beam scanning featurerdquo in Proceedings of the IEEEAntennas and Propagation Conference (AP-S rsquo08) pp 1ndash4 SanDiego Calif USA July 2008
[12] T Higgins and S D Blunt ldquoAnalysis of range-angle coupledbeamforming with frequency-diverse chirpsrdquo in Proceedings ofthe International Waveform Diversity and Design Conference(WDD rsquo09) pp 140ndash144 Orlando Fla USA February 2009
[13] J Farooq M A Temple and M A Saville ldquoApplication of fre-quency diverse arrays to synthetic aperture radar imagingrdquo inProceedings of the International Conference on Electromagneticsin Advanced Applications (ICEAA rsquo07) pp 447ndash449 TorinoItaly September 2007
[14] J Farooq M A Temple and M A Saville ldquoExploiting fre-quency diverse array processing to improve SAR image reso-lutionrdquo in Proceedings of the IEEE Radar Conference (RADARrsquo08) pp 1ndash5 Rome Italy May 2008
[15] W QWang ldquoPhased-MIMO radar with frequency diversity forrangedependent beamformingrdquo IEEE Sensors Journal vol 13no 8 pp 1320ndash1328 2013
[16] A L Swindlehurst and P Stoica ldquoMaximum likelihood meth-ods in radar array signal processingrdquo Proceedings of the IEEEvol 86 no 2 pp 421ndash441 1998
[17] J Ward ldquoCramer-Rao bounds for target angle and Dopplerestimation with space-time adaptive processing radarrdquo in Pro-ceedings of the 29th Asilomar Conference on Signals Systems andComputers pp 1198ndash1202 Pacific Grove Calif USA November1995
[18] A Dogandzic and A Nehorai ldquoCramer-Rao bounds for esti-mating range velocity and direction with an active arrayrdquo IEEETransactions on Signal Processing vol 49 no 6 pp 1122ndash11372001
[19] A Dogandzic and A Nehorai ldquoEstimating range velocity anddirection with a radar arrayrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo99) pp 2773ndash2776 Phoenix Ariz USA March 1999
[20] J J Zhang G Maalouli A P Suppappola and D MorrellldquoCramer-Rao lower bounds for the joint estimation of targetattributes using MIMO radarrdquo in Proceedings of the Interna-tional Waveform Diversity and Design Conference (WDD rsquo09)pp 103ndash107 Orlando Fla USA February 2009
[21] A Hassanien S A Vorobyov and A B Gershman ldquoMovingtarget parameters estimation in noncoherent MIMO radarsystemsrdquo IEEE Transactions on Signal Processing vol 60 no 5pp 2354ndash2361 2012
[22] Q He R S Blum and A M Haimovich ldquoNoncoherent MIMOradar for location and velocity estimation more antennasmeans better performancerdquo IEEE Transactions on Signal Pro-cessing vol 58 no 7 pp 3661ndash3680 2010
[23] S M Kay Fundamentals of Statistical Signal Processing Estima-tion Theory vol 1 Pearson London UK 2nd edition 2011
[24] H L van TreesOptimum Array Processing JohnWiley amp SonsNew York NY USA 2002
[25] D Wilcox and M Sellathurai ldquoOn MIMO radar subarrayedtransmit beamformingrdquo IEEE Transactions on Signal Processingvol 60 no 4 pp 2076ndash2081 2012
[26] WQWang andH Z Shao ldquoRange-angle localization of targetsby a double-pulse frequency diverse array radarrdquo IEEE Journalon Selected Topics in Signal Processing vol 8 no 1 pp 106ndash1142014
[27] S Gogineni and A Nehorai ldquoTarget estimation using sparsemodeling for distributed MIMO radarrdquo IEEE Transactions onSignal Processing vol 59 no 11 pp 5315ndash5325 2011
[28] T Li andA Nehorai ldquoMaximum likelihood direction finding inspatially colored noise fields using sparse sensor arraysrdquo IEEETransactions on Signal Processing vol 59 no 3 pp 1048ndash10622011
[29] S Sen ldquoOFDMradar space-time adaptive processing by exploit-ing spatio-temporal sparsityrdquo IEEE Transactions on SignalProcessing vol 61 no 1 pp 118ndash130 2013
[30] J J Blanz A Papathanassiou M Haardt I Furio and P WBaier ldquoSmart antennas for combined DOA and joint channelestimation in time-slotted CDMA mobile radio systems withjoint detectionrdquo IEEE Transactions onVehicular Technology vol49 no 2 pp 293ndash306 2000
[31] P N Pathirana S C K Herath and A V Savkin ldquoMultitargettracking via space transformations using a single frequencycontinuous wave radarrdquo IEEE Transactions on Signal Processingvol 60 no 10 pp 5217ndash5229 2012
International Journal of Antennas and Propagation 15
[32] P H Leong T D Abhayapala and T A Lamahewa ldquoMultipletarget localization using wideband echo chirp signalsrdquo IEEETransactions on Signal Processing vol 61 no 16 pp 4077ndash40892013
[33] L Zhuang X Liu and W Yu ldquoPrecisely beam steering forfrequency diverse arrays based on frequency offset selectionrdquoin Proceedings of the International Radar Conference (RADARrsquo09) pp 1ndash4 Bordeaux France December 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
14 International Journal of Antennas and Propagation
41101317 the Program for New Century Excellent Talentsin University under Grant NCET-12-0095 Sichuan ProvinceScience Fund for Distinguished Young Scholars under Grant2013JQ0003 and Fundamental Research Fund for theCentralUniversities
References
[1] F Bandiera M Mancino and G Ricci ldquoLocalization strategiesfor multiple point-like radar targetsrdquo IEEE Transactions onSignal Processing vol 60 no 12 pp 6708ndash6712 2012
[2] D R Fuhrmann J P Browning and M Rangaswamy ldquoSignal-ing strategies for the hybrid MIMO phased-array radarrdquo IEEEJournal on Selected Topics in Signal Processing vol 4 no 1 pp66ndash78 2010
[3] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011
[4] M Compagnoni P Bestagini F Antonacci A Sarti and STubaro ldquoLocalization of acoustic sources through the fittingof propagation cones using multiple independent arraysrdquo IEEETransactions on Audio Speech and Language Processing vol 20no 7 pp 1964ndash1975 2012
[5] P Antonik M C Wicks H D Griffiths and C J BakerldquoFrequency diverse array radarsrdquo in Proceedings of the IEEERadar Conference (RADAR rsquo06) pp 215ndash217 Verona NY USAApril 2006
[6] P Antonik M CWicks H D Griffiths and C J Baker ldquoMulti-mission multi-mode waveform diversityrdquo in Proceedings of theIEEE Radar Conference (RADAR rsquo06) pp 580ndash582 Verona NYUSA April 2006
[7] P Antonik H D Griffiths and C J Baker ldquoRange depen-dent beamforming using element level waveform diversityrdquo inProceedings of the International Waveform Diversity and DesignConference pp 1ndash4 Las Vegas Nev USA January 2006
[8] P Baizert T B HaleM A Temple andM CWicks ldquoForward-looking radar GMTI benefits using a linear frequency diversearrayrdquo Electronics Letters vol 42 no 22 pp 1311ndash1312 2006
[9] B W Jung R S Adve and J Chun ldquoFrequency diversity inmultistatic radarsrdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo08) pp 1ndash6 Rome Italy May 2008
[10] M Secmen S Demir A Hizal and T Eker ldquoFrequencydiverse array antenna with periodic time modulated pattern inrange and anglerdquo in Proceedings of the IEEE Radar Conference(RADAR rsquo07) pp 427ndash430 Boston Mass USA April 2007
[11] J Huang K F Tong and C J Baker ldquoFrequency diversearray with beam scanning featurerdquo in Proceedings of the IEEEAntennas and Propagation Conference (AP-S rsquo08) pp 1ndash4 SanDiego Calif USA July 2008
[12] T Higgins and S D Blunt ldquoAnalysis of range-angle coupledbeamforming with frequency-diverse chirpsrdquo in Proceedings ofthe International Waveform Diversity and Design Conference(WDD rsquo09) pp 140ndash144 Orlando Fla USA February 2009
[13] J Farooq M A Temple and M A Saville ldquoApplication of fre-quency diverse arrays to synthetic aperture radar imagingrdquo inProceedings of the International Conference on Electromagneticsin Advanced Applications (ICEAA rsquo07) pp 447ndash449 TorinoItaly September 2007
[14] J Farooq M A Temple and M A Saville ldquoExploiting fre-quency diverse array processing to improve SAR image reso-lutionrdquo in Proceedings of the IEEE Radar Conference (RADARrsquo08) pp 1ndash5 Rome Italy May 2008
[15] W QWang ldquoPhased-MIMO radar with frequency diversity forrangedependent beamformingrdquo IEEE Sensors Journal vol 13no 8 pp 1320ndash1328 2013
[16] A L Swindlehurst and P Stoica ldquoMaximum likelihood meth-ods in radar array signal processingrdquo Proceedings of the IEEEvol 86 no 2 pp 421ndash441 1998
[17] J Ward ldquoCramer-Rao bounds for target angle and Dopplerestimation with space-time adaptive processing radarrdquo in Pro-ceedings of the 29th Asilomar Conference on Signals Systems andComputers pp 1198ndash1202 Pacific Grove Calif USA November1995
[18] A Dogandzic and A Nehorai ldquoCramer-Rao bounds for esti-mating range velocity and direction with an active arrayrdquo IEEETransactions on Signal Processing vol 49 no 6 pp 1122ndash11372001
[19] A Dogandzic and A Nehorai ldquoEstimating range velocity anddirection with a radar arrayrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo99) pp 2773ndash2776 Phoenix Ariz USA March 1999
[20] J J Zhang G Maalouli A P Suppappola and D MorrellldquoCramer-Rao lower bounds for the joint estimation of targetattributes using MIMO radarrdquo in Proceedings of the Interna-tional Waveform Diversity and Design Conference (WDD rsquo09)pp 103ndash107 Orlando Fla USA February 2009
[21] A Hassanien S A Vorobyov and A B Gershman ldquoMovingtarget parameters estimation in noncoherent MIMO radarsystemsrdquo IEEE Transactions on Signal Processing vol 60 no 5pp 2354ndash2361 2012
[22] Q He R S Blum and A M Haimovich ldquoNoncoherent MIMOradar for location and velocity estimation more antennasmeans better performancerdquo IEEE Transactions on Signal Pro-cessing vol 58 no 7 pp 3661ndash3680 2010
[23] S M Kay Fundamentals of Statistical Signal Processing Estima-tion Theory vol 1 Pearson London UK 2nd edition 2011
[24] H L van TreesOptimum Array Processing JohnWiley amp SonsNew York NY USA 2002
[25] D Wilcox and M Sellathurai ldquoOn MIMO radar subarrayedtransmit beamformingrdquo IEEE Transactions on Signal Processingvol 60 no 4 pp 2076ndash2081 2012
[26] WQWang andH Z Shao ldquoRange-angle localization of targetsby a double-pulse frequency diverse array radarrdquo IEEE Journalon Selected Topics in Signal Processing vol 8 no 1 pp 106ndash1142014
[27] S Gogineni and A Nehorai ldquoTarget estimation using sparsemodeling for distributed MIMO radarrdquo IEEE Transactions onSignal Processing vol 59 no 11 pp 5315ndash5325 2011
[28] T Li andA Nehorai ldquoMaximum likelihood direction finding inspatially colored noise fields using sparse sensor arraysrdquo IEEETransactions on Signal Processing vol 59 no 3 pp 1048ndash10622011
[29] S Sen ldquoOFDMradar space-time adaptive processing by exploit-ing spatio-temporal sparsityrdquo IEEE Transactions on SignalProcessing vol 61 no 1 pp 118ndash130 2013
[30] J J Blanz A Papathanassiou M Haardt I Furio and P WBaier ldquoSmart antennas for combined DOA and joint channelestimation in time-slotted CDMA mobile radio systems withjoint detectionrdquo IEEE Transactions onVehicular Technology vol49 no 2 pp 293ndash306 2000
[31] P N Pathirana S C K Herath and A V Savkin ldquoMultitargettracking via space transformations using a single frequencycontinuous wave radarrdquo IEEE Transactions on Signal Processingvol 60 no 10 pp 5217ndash5229 2012
International Journal of Antennas and Propagation 15
[32] P H Leong T D Abhayapala and T A Lamahewa ldquoMultipletarget localization using wideband echo chirp signalsrdquo IEEETransactions on Signal Processing vol 61 no 16 pp 4077ndash40892013
[33] L Zhuang X Liu and W Yu ldquoPrecisely beam steering forfrequency diverse arrays based on frequency offset selectionrdquoin Proceedings of the International Radar Conference (RADARrsquo09) pp 1ndash4 Bordeaux France December 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 15
[32] P H Leong T D Abhayapala and T A Lamahewa ldquoMultipletarget localization using wideband echo chirp signalsrdquo IEEETransactions on Signal Processing vol 61 no 16 pp 4077ndash40892013
[33] L Zhuang X Liu and W Yu ldquoPrecisely beam steering forfrequency diverse arrays based on frequency offset selectionrdquoin Proceedings of the International Radar Conference (RADARrsquo09) pp 1ndash4 Bordeaux France December 2009
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of