1084 ieee transactions on signal processing, vol. 57,...

1084 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 3, MARCH 2009

Range-Doppler Imaging Via a Train of Probing PulsesXing Tan, William Roberts, Jian Li, Fellow, IEEE, and Petre Stoica, Fellow, IEEE

Abstract—We consider range-Doppler imaging via transmit-ting a train of probing pulses. We present two methods forrange-Doppler imaging. The first one is based on the instrumentalvariables (IV) filter and the second one is based on the iterativeadaptive approach (IAA). Numerical results show that bothmethods can suppress interference from neighboring range andDoppler bins. An attractive feature of the IV filter is that it canbe computed offline. IAA has better performance than IV andhas super resolution, but at the cost of a higher computationalcomplexity.

Index Terms—Instrumental variables filter, iterative adap-tive approach, radar imaging, Range-Doppler imaging, superresolution.

I. INTRODUCTION

W E consider the problem of range-Doppler imaging,in active sensing applications such as in radar, sonar,

seismic exploration and biomedical imaging. We focus on radarrange-Doppler imaging, i.e., obtaining the target amplituderesponse [which is proportional to the target radar-crosssection(RCS)] on a two dimensional grid, with one dimension beingthe range bin and the other dimension being the Dopplerfrequency bin. Although we focus on radar range-Dopplerimaging in this paper, the methods we propose can be appliedto other active sensing applications directly.

In practical applications, we wish to design radar systemswith high range resolution and high Doppler resolution. Toachieve high range resolution, unmodulated radar signals musthave very narrow probing pulses. In other words, a very highinstantaneous power is required to maintain a constant pulseenergy. However, it is difficult to design and implement suchantennas or sensors at a reasonable cost. An alternative wayis to use pulse compression (see, e.g., [1]–[5], and [6]), bytransmitting a train of modulated subpulses towards the areaof interest. The subpulse train has a much smaller peak powerthan a single pulse, for the same total transmitted energy. Manyrange compression algorithms have been proposed in the liter-ature to reduce the sidelobe levels. Recently, an instrumental

Manuscript received May 05, 2008; revised October 08, 2008. First pub-lished November 25, 2008; current version published February 13, 2009. Theassociate editor coordinating the review of this manuscript and approving it forpublication was Prof. Antonio Napolitano. This work was supported in part bythe office of Naval Research (ONR) under grant N00014-07-1-0293, the ArmyResearch Office (ARO) by Grant NNX07AO15A, by the European ResearchCouncil (ERC), and by the Swedish Research Council (VR).

X. Tan, W. Roberts, and J. Li are with the Department of Electrical and Com-puter Engineering, University of Florida, Gainesville, FL 32611 USA (e-mail:[email protected]; [email protected]; [email protected]).

P. Stoica is with the Department of Information Technology, Uppsala Uni-versity, Uppsala, Sweden (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2008.2010010

TABLE INOTATIONS USED IN THE TEXT

variables (IV) filter has been devised for range compressionby optimizing several relevant metrics, including the inte-grated-sidelobe-level (ISL), peak-sidelobe-level (PSL), andinverse signal-to-noise ratio (ISNR). Both cases of negligibleand non-negligible Doppler shifts are considered in [2]. Forthe case of negligible Doppler shifts, the sidelobe-level of theIV filter can be made arbitrarily low. However, for the caseof non-negligible Doppler shifts, the IV filter does not havesatisfactory sidelobe reduction performance and its Dopplerresolution is poor, due to the transmission of only one probingpulse.

In the present paper, we assume that the radar system trans-mits a probing pulse train and that each pulse in the pulse trainconsists of modulated subpulses. The idea of sending a pulsetrain also has applications in space-time adaptive processing(STAP) (see, e.g., [7]–[10]). We propose two methods forrange-Doppler imaging. The first method is data independentand makes use of an instrumental variables (IV) filter (or some-times called mismatched filter, see, e.g., [11]–[17]) to filter outthe interferences from adjacent range and Doppler bins. Theresulting range-Doppler images formed by the IV filter havemuch higher Doppler resolution than those in [2] due to theuse of a probing pulse train instead of a single pulse. However,since the IV filter is a data-independent method, its Doppler res-olution is not as high as that of the data-adaptive methods. Forapplications which require a precise estimation of the Dopplershift (i.e., the target moving speed relative to the radar) andsuper Doppler resolution, we present a data-adaptive method,called the iterative adaptive approach (IAA) for range-Dopplerimaging. IAA was first proposed in [18] for source localizationand sensing. We extend the IAA algorithm in [18] to the caseof using the probing pulse train for range-Doppler imaging.For the same transmitted probing pulse train, IAA provides ahigher Doppler resolution than the IV filter.

This paper is organized as follows. In Section II, we formu-late the problem of interest. In Section III, we derive the IVfilter based on the minimum average ISL design. In Section IV,we present the IAA method for range-Doppler imaging. InSection V, we demonstrate the effectiveness of these twomethods via numerical examples. The final section presents theconclusions of this paper.

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TAN et al.: RANGE-DOPPLER IMAGING 1085

Fig. 1. An illustration of the pulse train signal.

TABLE IIACRONYMS FREQUENTLY USED IN THE TEXT

We denote vectors by boldface lowercase letters and matricesby boldface uppercase letters. The other notations that we usethroughout this paper are presented in Table I. The frequentlyused acronyms are summarized in Table II.

II. DATA MODEL AND PROBLEM FORMULATION

Consider a radar system transmitting a train of probingpulses with each pulse consisting of subpulses (see Fig. 1). Also consider the carrier frequency , the subpulse band-width and the pulse repetition frequency (we assume that

). These parameters determine the range res-olution, the maximum unambiguous range, the Doppler reso-lution and the maximum unambiguous Doppler frequency asfollows (see, e.g., [5] for the detailed descriptions and mathe-matical expressions):

i) The range resolution is the minimum range difference oftwo targets required by the radar system to distinguishthem. The range resolution is determined by ,where is the speed of propagation and is the band-width of the probing subpulses.

ii) The maximum unambiguous range is the maximum rangeof a target for unambiguous detection. The maximum un-ambiguous range is . In this paper, we assume thatwe do not encounter range ambiguities, i.e., the pulse rep-etition interval is sufficiently large so that any return dueto the previous transmitted pulse arrives at the receiverbefore the returns due to the current pulse. In many ap-plications, the targets are located within the maximumunambiguous range, and hence neglecting range ambigu-ities will not pose any problems. In some other applica-tions, the targets are far from the radar. In that case, onecan send two pulse trains with different pulse repetitionintervals and determine the range of the target by using,e.g., the Chinese remainder theorem (see [19, Ch. 17]).

iii) The Doppler resolution is the minimum difference in theDoppler frequencies of two moving targets required by

the radar system to distinguish them. The best possibleDoppler resolution of the data-independent approachesequals . There are data-dependent methods, suchas IAA, that can provide super resolution, meaning thatthey can distinguish two targets whose Doppler frequencydifference is less than .

iv) The maximum unambiguous Doppler frequency isthe maximum difference in Doppler frequency of twomoving targets for unambiguous Doppler frequencyestimation. It is determined by .

The data model we use is similar to that in [2]–[4], exceptthat we plot range-Doppler images on a two-dimensional grid.When the grid is fine enough, the parameter estimates of off-gridtargets are sufficiently accurate. We label the range bins illu-minated by the radar as . We assume that theDoppler frequency shift interval of interest is , andthat the Doppler frequency shifts of all targets are contained inthis interval to avoid ambiguity. The Doppler frequency shiftinterval of interest is divided into a grid of Doppler bins

. The Doppler phase shift over one subpulse forthe th Doppler frequency bin is denoted as ,and the Doppler phase shift interval of interest is ,where .

Let denote the modulated code sequence. Letdenote the received data sequence for the th probing pulse.Since includes the returned echoes from all range bins, thelength of is . When is large, using the entirereceived data sequence is computationally prohibitive. Instead,we apply an observation window to the received data sequence

to obtain a data segment related only to a range bin ofcurrent interest. Let the length of the observation window be

, where is a user parameter. Consider the rangebin of current interest. Its observation window has the returnsfrom the th range bin in the middle (see Fig. 2). Similarly to[2 eq. (9) ], the received data sequence inside this observationwindow has the form

(1)where is the complex amplitude of the returned signalfrom the th range bin and the th Doppler bin. Eachis proportional to the radar-cross-section of the correspondingtarget and is assumed constant during pulse repetitions. Notethat only a small portion of take on nonzero values, since



Fig. 2. An illustration of the data model.

the target distribution in a range-Doppler image is not dense. Werarely have a situation where targets are packed in one range binand moving at various possible speeds relative to the radar. Thevariable is defined as the pulse repetition interval divided bythe duration of a single subpulse. The vector is defined as

(2)

where , , is the code sequence. Furthermore,is the noise vector. Since both IV and IAA are nonpara-

metric methods, we do not have to specify the statistical distri-bution of the noise vector. (In our numerical examples, we as-sume that the noise vector is a complex Gaussian random vectorwith zero-mean and covariance matrix equal to .) We define the

matrix as follows:

(3)

where

. . . (4)

We now form the data vector by stacking the vectorson top of each other to obtain

(5)

where

(6)

Note that can be viewed as a steering vector,a phrase commonly used in the array signal processing litera-ture.

Our problem of interest is to compute for the rangebin and Doppler frequency bin from the received signals

. In other words, we aim at forming range-Dopplerimages from by plotting versus and . Thedifficulty lies in the fact that signals reflected from differentrange bins and Doppler bins overlap with each other as shownin (5) and in Fig. 2. To accurately estimate in the presenceof interferences in the neighboring range and Doppler bins, wepropose a data-independent IV filter and a data-dependent IAAalgorithm.

III. DATA-INDEPENDENT IV FILTER DESIGN

We derive below the data-independent IV filter basedon the data model (5) to obtain for the current range binand current Doppler frequency bin and filter out the interfer-ences from the neighboring range and Doppler bins. The pass-band of the IV filter in the current range bin (for ) is

, where and is a user parameterthat controls the width of the passband for . The stopbandof the IV filter is forthe current range bin (for ) andfor the neighboring range bins (for ), where



. If is given, we estimate by (see [12, eq.(13)])

(7)

We define the sidelobe-level (SL), integrated-sidelobe-level(ISL), and sidelobe-level of the range bin of current interest at

(SL0), respectively, as

(8)

(9)

(10)

The ISL metric measures the ability of the IV filter to suppressinterferences in neighboring range bins. The SL0 metric mea-sures the ability of the IV filter to suppress interferences in thestopband of the current range bin of interest .

We design the IV filter by minimizing the average ISL in thestopband (this criterion is similar to [2, eq. (50)])

(11)

where , . Solving this opti-mization problem (see Appendix A for the detailed derivations),we obtain

(12)

where is a constant and

(13)

TABLE IIITHE IAA ALGORITHM

Then, by (7), we can estimate as follows:

(14)

Some further simplification of the above equation is nec-essary to reduce the computational complexity, especially insituations where the dimension of is large. We denotethe intrapulse Doppler shift as the Doppler shift over onesubpulse. When the intrapulse Doppler shift is negligible(i.e., , where is the maximumpossible speed of a target), can be approximated as

(15)

(We will justify this approximation by numerical examples inSection V). With this assumption, we can simplify the IV filterto obtain (see Appendix B for the detailed derivations)

(16)

where( and

are defined in (37) in Appendix B and is defined in (42) inAppendix B), and is defined in Table I. As a result, whenthe intrapulse Doppler effect can be neglected, can beviewed as the Doppler shifted version of .

We conclude this section by making two comments on the IVfilter design:

1) The computational complexity of calculating the inverse ofis high, as is an

matrix. In this paper, we provide an efficient methodto calculate under the assumption that the intrapulseDoppler effect is negligible. Thus, we have circumventedcalculating directly and reduced the amount ofcalculations significantly by only calculating the inverseof , which is an matrix and theinverse of , which is an

matrix. Furthermore, we onlyneed to calculate , as can be obtained simply byusing (16). Note also that all computations needed to obtainthe IV filter can be done offline.

2) The parameter is critical in determining the perfor-mance of the IV filter. Furthermore, must be larger thanthe Doppler resolution, namely ,where is the phase shift corresponding to the Doppler



TABLE IVTHE IAA ALGORITHM

resolution over one subpulse. As becomes larger, thesidelobe-level in the stopband becomes lower, but the pass-band for the current range bin becomes wider, which meansthat the IV filter has to trade off poorer Doppler resolutionfor lower sidelobe-levels, as all data-independent filters do.

IV. THE ITERATIVE ADAPTIVE APPROACH

IAA was first introduced in [18] for source localization inarray signal processing. IAA is a non-parametric and user pa-rameter free approach derived based on a weighted least squarescriterion [18]. IAA aims to estimate the coefficients inthe following linear model:

(17)

where are the steering vectors and is a noise vector.The IAA algorithm is summarized in Table III. IAA usually con-verges in 10 to 15 iterations and the convergence property ofIAA is discussed in [18].

In this section, we extend IAA to the multi-pulse probing sce-nario based on the model (5). Note that the steering vector in (5)is instead of in (17). For the purpose ofcomputational simplicity, we assume that throughoutthis section, i.e., the length of the observation window equals

. The multi-pulse IAA is presented in Table IV, where theintrapulse Doppler effect is considered without any difficulties.

IAA requires the iterative calculation of the following twoequations:

(18)

and

(19)

The computational complexity of (18) is on the order of. The computational complexity of

(19) comes largely from the inversion of , which has acomputational complexity on the order of . Note

that we must calculate for each range bin . Further-more, if the algorithm converges within iterations, thenthe total computational complexity of IAA is on the order of

.The computational complexity of (18) and (19) can be re-

duced by a factor of if we exploit the block Toeplitz structureof the matrix

......

......

...

(20)

with being a submatrix of with dimension .Note that the first block column of in (20) is

. The first block column of the rightside of (18) is

. Thus, we have

...

(21)

Once we know the first block column , wecan reconstruct the rest of the block columns of by usingthe Toeplitz structure of in (20).

Equation (19) can be simplified by using the block Toeplitzmatrix inversion, effectively reducing the computational com-plexity of (19) by a factor of . A procedure of block Toeplitzinversion is summarized in Appendix C. See [20] for the proofof this algorithm.

In most cases, the IAA algorithm exhibits sharp peaks at lo-cations (range bin and Doppler bin) where the targets appear.However, there are occasions when signals reflected by targetsare hard to discern. For example, the signal strength could beweak compared to the noise level. Moreover, two signals couldbe close in Doppler frequency and then it may be hard to dis-tinguish them. In those cases, we need a model order selectiontool to determine the number of targets in the range bins of in-terest. A model order selection tool is a tool for determiningthe dimension of the parameter spaces, that is, the number ofunknowns (see, e.g., [21]). The Bayesian Information Criterion



Fig. 3. The sidelobe levels of (a) the original IV filter [see (12)], and (b) the IVfilter which neglects the intrapulse Doppler shift [see (16)].

(BIC) serves this purpose well. Compared to other model orderselection rules, such as the Akaike information criterion (AIC),which has a problem of overfitting, BIC provides a more reli-able estimate of the model order (see [23, ref. 21, 22, App. C]).

In this paper, for computational simplicity reasons, we usethe approximate sequential search method given in [18] insteadof the original BIC method. Let denote the rangebins and Doppler bins corresponding to the largest peaks of

, arranged in decreasing order

(22)

Let

(23)

Fig. 4. ISL and SL0 for� � �, 8, 16, 32, when� � �� and � � �� .

where denotes the received data sequence for the thprobing pulse. Let

(24)

We define the in the same way as in (3) except that is anmatrix.

is defined as follows:

(25)

Note that contains two terms. The first term is a leastsquares data fitting term which decreases as increases. Thesecond term is a model order penalization term which increasesas increases. The estimated number of targets is chosen tominimize .

V. NUMERICAL EXAMPLES

A. Performance of the IV Filter Approach

We assume that the radar system has a central frequency ofand bandwidth of . For simplicity,

we assume that each subpulse is binary phase-modulated. We



Fig. 5. ISL and SL0 for � � ��, 20, 40, when � � �� and � � �� .

fix for our simulations and we use the same codesequences as in [2], i.e.

(26)

In the first example, we show that the assumption (15) is valideven when the speed of the target is high. In this example, weassume that , , and .In Fig. 3(a) and (b), we plot the sidelobe-levelfor and forthe original IV filter (12) and the IV filter with intrapulseDoppler neglected (16). These two figures show that there isvery little performance loss (below ) when we neglectthe intrapulse Doppler shift. This result is not surprising, as

when .In the following examples, we set and consider

the effect of the number of zeros , the number of probingpulses , and on ISL and SL0. We assume that the intra-pulse Doppler is negligible and we use the IV filter (16).

In Fig. 4, we show the ISL and SL0 as functions of theDoppler frequency (for , 8, 16, 24, and

). From Fig. 4(a), we see that as increases, theISL decreases significantly. In other words, as increases, theIV filter suppresses the interferences from neighboring rangebins better. From Fig. 4(b), we can see that as increases, theSL0 metric has slightly lower sidelobe-level in the stopband.This indicates that as increases, the IV filter suppressesinterferences in the stopband better.

Fig. 6. ISL and SL0 for � � � � �� , when � � �� and � � ��.

TABLE VGROUND TRUTH 1

In Fig. 5, we show the ISL and SL0 metrics as functions ofthe Doppler frequency (for , , 20, 40, and

). From Fig. 5(b), we can see that the width of thepassband becomes smaller, and, thus, the IV filter has betterDoppler resolution when is large. Fig. 5(a) shows that as

increases, the IV filter suppresses the interferences in theneighboring range bins better.

In Fig. 6, we show the ISL and SL0 as functions ofthe Doppler frequency (for , and

). From this figure, we can see that asincreases, the sidelobe-level of the stopband becomes

smaller, but the passband becomes wider. Thus, the parameterprovides a tradeoff between the sidelobe-level in the stop-

band and the width of the passband.

B. Performance of IAA

In what follows, we assume that the number of range binsilluminated by the radar probing signals is 60 (i.e., ).Let the system parameters be , ,

, , , and . The Doppler



TABLE VIGROUND TRUTH 2

TABLE VIIGROUND TRUTH 3

resolution in this scenario is 100 Hz, which corresponds toa target moving speed of 15 m/s. The noise is assumed tobe a white circularly symmetric complex Gaussian randomvector with zero mean and covariance matrix . The SNRof a target at range bin and Doppler bin is defined to be

. In all the examples shown below,the scanning grid of the Doppler frequency domain is uniformin the range from 1000 to 1000 Hz with 25–Hz incrementbetween adjacent grid points. We fix the number of iterationsto be 10 for the IAA algorithm, as IAA provides good resultswith 10 iterations in all the numerical examples shown below(more iterations do not improve the results further).

1) Comparing Among MF, IV and IAA: In this example, wecompare the range-Doppler images formed by MF, IV, and IAA(without BIC). We consider five targets in the range bins of in-terest. The first one and the second one are both in range bin 25.The first one has a moving speed of 37.5 m/s, which corre-sponds to a Doppler frequency of 250 Hz. The second one isstill. The third one is assumed to be in range bin 30 with movingspeed 0. The fourth one and the fifth one are both in range bin35. The fourth one is still and the fifth one has a moving speed of37.5 m/s, which corresponds to a Doppler frequency of 250 Hz.Their corresponding values are set to be 30, 10, 30, 10, and30 dB, respectively. The ground truth is summarized in Table V.

From Fig. 7, we can see clearly that IAA performs the bestand that both IAA and the IV filter significantly outperform MF.The IV filter can suppress the interference from neighboringrange bins while MF suffers from the interference from neigh-boring range bins. However, the IV filter cannot resolve two tar-gets closely spaced in Doppler frequency in the same range bin.IAA outperforms both the MF and the IV filter, because it showsall five targets clearly. In fact, the range-Doppler image formedby IAA is very close to a line spectrum. There are sharp peaksat the locations of the targets and near-zero values at the otherlocations.

2) Super Resolution: In this example, we show that IAA withBIC has super resolution. We assume the same scenario as theabove example, except that the targets are moving at differentspeeds. We assume that the first target has a moving speed of

Fig. 7. Range-Doppler images formed by (a) MF; (b) IV filter; (c) IAA withMF as the initial estimator; and (d) IAA with the IV filter as the initial estimator.The ground truth is summarized in Table V.

7.5 m/s, which corresponds to a Doppler frequency of 50 Hz.The second one, the third one and the fourth one are all still. The



fifth one has a moving speed of 7.5 m/s, which corresponds toa Doppler frequency of 50 Hz. The ground truth is summarizedin Table VI.

Fig. 8 clearly shows that IAA and IAA with BIC can detectall five targets, even if the difference in their Doppler frequen-cies is smaller than the Doppler resolution of data-independentmethods. IAA with BIC yields the best range-Doppler image vi-sually. In this case, the valley between two peaks is no longernegligible in the IAA image.

3) Small SNR: In this example, we show that IAA with BICcan detect all the targets when the SNR is low. Again, we con-sider five targets in the system. The locations and the speeds ofthe five targets are the same as those in example V-B1 shown inFig. 7. Their corresponding values are set to be 10, 10, 10,

10, and 10 dB, respectively. The ground truth is summarizedin Table VI.

Fig. 9 clearly shows that IAA with BIC can detect all fivetargets quite accurately even if the signal power is 10 dB belowthe noise power, for some of the targets.

In the three examples above, we can reduce the computationtime of IAA by a factor of more than 5 by exploiting the blockToeplitz structure of .

VI. CONCLUSION

The main contributions of this paper can be summarized asfollows:

i) We have developed a data model (5) for parameter estima-tion by using a pulse train radar. This model has two prop-erties. First, it is a linear model, as the received signal is alinear combination of the returned signals from differentrange and Doppler bins. Linear models are relatively easyto handle. Particularly, we can use the IV filter to suppressinterferences. Second, (5) is also a sparse signal model.Since IAA is useful for this type of models, we can useIAA to achieve high resolution range-Doppler imaging.

ii) We have employed an IV filter to form range-Dopplerimages. This IV filter has been derived based on a min-imum average ISL criterion. Our aim is to suppress thesidelobe-level in the stopband for a given passband (i.e.,

for the range bin of current interest).In most cases, the intrapulse Doppler in one pulse can beneglected. Under this assumption, we have developed anefficient method for the fast computation of the IV filter.An important feature of the IV filter is that it can be com-puted offline.

iii) Based on the sparse signal model derived in Section II, wehave extended IAA, a data-adaptive approach, to range-Doppler imaging via a train of probing pulses. In this ap-proach, we have taken into account the intrapulse Dopplereffect without any difficulties. By exploiting the blockToeplitz structure of the matrix , we have simplifiedthe computation of IAA. In addition, we have used BICto determine the number of targets in the range bins ofinterest.

iv) The numerical examples of the paper have shown that theIV filter is superior to the conventional MF, as it can sup-press the interference from neighboring range bins better.The Doppler resolution of the IV filter increases with the

Fig. 8. Range-Doppler images formed by (a) MF; (b) IV filter; (c) IAA with MFas the initial estimator; and (d) IAA with BIC with MF as the initial estimator.The ground truth is summarized in Table VI.

number of probing pulses. The numerical examples havealso shown that the IAA algorithm is better than both the



MF and the IV filter. The range-Doppler images formedby IAA contain sharp peaks at target locations. IAA withBIC provides super resolution images, while traditionalmethods that achieve super resolution (such as Capon)cannot be applied since there is only one arbitrary snap-shot available for range-Doppler imaging. Furthermore,IAA with BIC can detect targets quite accurately even ifthe SNR is very low.

APPENDIX ATHE DERIVATION OF THE IV FILTER

We can simplify (11) as follows:

(27)

where is defined in (13) and can be written as

(28)

in which

and

(29)

The th elementof is shown in (30) at the bottom of the next page, andthe th elementof as shown in (31) at the bottom of the next page.

Similar to [2], we apply the Cauchy–Schwartz inequality toobtain

(32)

where is a Hermitian square root of the matrix .

Fig. 9. Range-Doppler images formed by (a) MF; (b) IV filter; (c) IAA with MFas the initial estimator; and (d) IAA with BIC with MF as the initial estimator.The ground truth is summarized in Table VII.



TABLE VIIIBLOCK TOEPLITZ MATRIX INVERSION

Then (27) has a lower bound, given by

(33)

The minimum value of is attained when

(34)

where is a constant. Note that if we replace with in (12),then the IV filter becomes the standard matched filter (MF)

(35)

APPENDIX BTHE SIMPLIFICATION OF THE IV FILTER

When intrapulse Doppler is negligible [see (15)], we can sim-plify (28) as follows:

(36)

or

and(30)

or

and .(31)



where

and

(37)

The th element of is

(38)

and the th element of is

.(39)

Note that and have the following properties:

(40)

Let

(41)

and

(42)

Then we can write as

(43)

where we have used the Matrix Inversion Lemma in the secondequality above.

The IV filter is

(44)



where we have used the Matrix Inversion Lemma again in thelast equality. By using (40), we get

(45)

where

APPENDIX CBLOCK TOEPLITZ MATRIX INVERSION

We use similar notations as those in the paper [20]. Let theblock Toeplitz matrix be . It containsblocks and each block is a submatrix. Let

and ,where we denote by the block transposition, i.e., the replace-ment of the th block by the th in a block matrix.We denote by the reversal of ordering in block rows and by

the transposition of elements of each block. For example,

and .With notations defined above, we can summarize the proce-dures for block Toeplitz matrix inversion in Table VIII.

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[17] N. Levanon, “Cross-correlation of long binary signals with longermismatched filters,” IEE Proc.—Radar, Sonar, Navig., vol. 152, pp.377–382, Dec. 2005.

[18] T. Yardibi, J. Li, P. Stoica, M. Xue, and A. B. Baggeroer, “Sourcelocalization and sensing: A nonparametric iterative adaptive approachbased on weighted least squares,” IEEE Trans. Aerosp. Electron. Syst.,2007.

[19] M. I. Skolnik, Radar Handbook. New York: McGraw-Hill, 1990.[20] H. Akaike, “Block Toeplitz matrix inversion,” SIAM J. Appl. Math.,

vol. 24, no. 2, pp. 234–241, 1973.[21] P. Stoica and Y. Selén, “Model-order selection: A review of informa-

tion criterion rules,” IEEE Signal Process. Mag., vol. 21, pp. 36–47,Jul. 2004.

[22] G. Schwarz, “Estimating the dimension of a model,” Ann. Statist., vol.6, pp. 461–464, May 1978.

[23] P. Stoica and R. L. Moses, Spectral Analysis of Signals. Upper SaddleRiver, NJ: Prentice-Hall, 2005.

Xing Tan received the B.S. and M.S. degrees fromShanghai Jiaotong University, China, in 2003 and2005, respectively.

He is currently pursuing the Ph.D. degree in elec-trical and computer engineering from the Universityof Florida, Gainesville. His research interests includesparse Bayesian Learning, array signal processing,signal processing for communication, and channelcoding.

William Roberts received the B.Sc. degree in 2006and the M.Sc. degree in 2007, both in electrical engi-neering, from the University of Florida, Gainesville.

He is currently pursuing the Ph.D. degree with theDepartment of Electrical Engineering, University ofFlorida.

Mr. Roberts received the SMART fellowship fromthe Department of Defense in 2008. His research in-terests include signal processing and radar systems.



Jian Li (S’87–M’91–SM’97–F’05) received theM.Sc. and Ph.D. degrees in electrical engineeringfrom The Ohio State University, Columbus, in 1987and 1991, respectively.

From April 1991 to June 1991, she was an AdjunctAssistant Professor with the Department of ElectricalEngineering, The Ohio State University. From July1991 to June 1993, she was an Assistant Professorwith the Department of Electrical Engineering, Uni-versity of Kentucky, Lexington. Since August 1993,she has been with the Department of Electrical and

Computer Engineering, University of Florida, Gainesville, where she is cur-rently a Professor. In fall 2007, she was on sabbatical leave at the MassachusettsInstitute of Technology, Cambridge. Her current research interests include spec-tral estimation, statistical and array signal processing, and their applications.

Dr. Li is a Fellow of IET. She is a member of Sigma Xi and Phi Kappa Phi. Shereceived the 1994 National Science Foundation Young Investigator Award andthe 1996 Office of Naval Research Young Investigator Award. She was an Exec-utive Committee Member of the 2002 International Conference on Acoustics,Speech, and Signal Processing, Orlando, Florida, May 2002. She was an As-sociate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING from 1999to 2005, an Associate Editor of the IEEE Signal Processing Magazine from2003 to 2005, and a member of the Editorial Board of the European Associ-ation for Signal Processing (EURASIP)Signal Processing from 2005 to 2007.She has been a member of the Editorial Board of Digital Signal Processing—AReview Journal, a publication of Elsevier, since 2006. She is presently a memberof the Sensor Array and Multichannel (SAM) Technical Committee of IEEESignal Processing Society. She is a coauthor of the papers that have receivedthe First and Second Place Best Student Paper Awards, respectively, at the 2005and 2007 Annual Asilomar Conferences on Signals, Systems, and Computersin Pacific Grove, CA. She is also a coauthor of the paper that has received theM. Barry Carlton Award for the best paper published in IEEE TRANSACTIONS

ON AEROSPACE AND ELECTRONIC SYSTEMS in 2005.

Petre Stoica (F’94) received the D.Sc. degree inautomatic control from the Polytechnic Instituteof Bucharest (BPI), Bucharest, Romania, in 1979and an honorary doctorate degree in science fromUppsala University (UU), Uppsala, Sweden, in 1993.

He is a professor of systems modeling with theDivision of Systems and Control, the Departmentof Information Technology, UU. Previously, hewas a professor of system identification and signalprocessing with the Faculty of Automatic Controland Computers, BPI. He held longer visiting posi-

tions with Eindhoven University of Technology, Eindhoven, The Netherlands;Chalmers University of Technology, Gothenburg, Sweden (where he held aJubilee Visiting Professorship); UU; The University of Florida, Gainesville;and Stanford University, Stanford, CA. His main scientific interests are in theareas of system identification, time series analysis and prediction, statisticalsignal and array processing, spectral analysis, wireless communications, andradar signal processing. He has published nine books, 10 book chapters, andsome 500 papers in archival journals and conference records. The most recentbook he coauthored, with R. Moses, is Spectral Analysis of Signals (EnglewoodCliffs, NJ: Prentice-Hall, 2005).

Dr. Stoica is on the editorial boards of six journals: Journal of Forecasting,Signal Processing, Circuits, Signals, and Signal Processing, Digital SignalProcessing-CA Review Journal, Signal Processing Magazine, and Multidi-mensional Systems and Signal Processing. He was a coguest editor for severalspecial issues on system identification, signal processing, spectral analysis,and radar for some of the aforementioned journals, as well as for the IEEPROCEEDINGS. He was corecipient of the IEEE ASSP Senior Award for apaper on statistical aspects of array signal processing. He was also recipient ofthe Technical Achievement Award of the IEEE Signal Processing Society. In1998, he was the recipient of a Senior Individual Grant Award of the SwedishFoundation for Strategic Research. He was also corecipient of the 1998EURASIP Best Paper Award for Signal Processing for a work on parameterestimation of exponential signals with time-varying amplitude, a 1999 IEEESignal Processing Society Best Paper Award for a paper on parameter and rankestimation of reduced-rank regression, a 2000 IEEE Third Millennium Medal,and the 2000 W. R. G. Baker Prize Paper Award for a paper on maximumlikelihood methods for radar. He was a member of the international programcommittees of many topical conferences. From 1981 to 1986, he was a Directorof the International Time-Series Analysis and Forecasting Society, and he wasalso a member of the IFAC Technical Committee on Modeling, Identification,and Signal Processing. He is also a member of the Royal Swedish Academy ofEngineering Sciences, an honorary member of the Romanian Academy, and afellow of the Royal Statistical Society.


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