research article array processing for radar: achievements...

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2013 Article ID 261230 21 pageshttpdxdoiorg1011552013261230

Research ArticleArray Processing for Radar Achievements and Challenges

Ulrich Nickel

Fraunhofer Institute for Communication Information Processing and Ergonomics (FKIE) Fraunhoferstrasse 2053343 Wachtberg Germany

Correspondence should be addressed to Ulrich Nickel ulrichnickelfkiefraunhoferde

Received 26 March 2013 Accepted 26 July 2013

Academic Editor Hang Hu

Copyright copy 2013 Ulrich NickelThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Array processing for radar is well established in the literature but only few of these algorithms have been implemented in realsystems The reason may be that the impact of these algorithms on the overall system must be well understood For a successfulimplementation of array processing methods exploiting the full potential the desired radar task has to be considered and allprocessing necessary for this task has to be eventually adapted In this tutorial paper we point out several viewpoints whichare relevant in this context the restrictions and the potential provided by different array configurations the predictability of thetransmission function of the array the constraints for adaptive beamforming the inclusion of monopulse detection and trackinginto the adaptive beamforming concept and the assessment of superresolution methods with respect to their application in a radarsystem The problems and achieved results are illustrated by examples from previous publications

1 Introduction

Array processing is well established for radar Publications ofthis topic have appeared for decades and one might questionwhat kind of advances we may still expect now On the otherhand if we look at existing radar systems we will find veryfew methods implemented from the many ideas discussed inthe literature The reason may be that all processing elementsof a radar system are linked and it is not very useful tosimply implement an isolated algorithm The performanceand the property of any algorithm will have an influence onthe subsequent processing steps and on the radar operationalmodes Predictability of the systemperformancewith the newalgorithms is a key issue for the radar designer Advancedarray processing for radar will therefore require to takethese interrelationships into account and to adapt the relatedprocessing in order to achieve the maximum possibleimprovementThe standard handbooks on radar [1 2] do notmention this problem The book of Wirth [3] is an exceptionand mentions a number of the array processing techniquesdescribed below

In this tutorial paper viewpoints are presented which arerelevant for the implementation of array processingmethodsWe do not present any new sophisticated algorithms but forthe established algorithms we give examples of the relations

between array processing and preceding and subsequentradar processing We point out the problems that have to beencountered and the solutions that need to be developed Westart with spatial sampling that is the antenna array that hasto be designed to fulfill all requirements of the radar systemA modern radar is typically a multitasking system So thedesign of the array antenna has to fulfill multiple purposes ina compromise In Section 3 we briefly review the approachesfor deterministic pattern shaping which is the standardapproach of antenna-based interference mitigation It has theadvantage of requiring little knowledge about the interferencescenario but very precise knowledge about the array trans-fer function (ldquothe array manifoldrdquo) Adaptive beamforming(ABF) is presented in Section 4 This approach requires littleknowledge about the array manifold but needs to estimatethe interference scenario from some training data Superres-olution for best resolution of multiple targets is sometimesalso subsumed under adaptive beamforming as it resolveseverything interference and targets These methods are con-sidered in Section 5 We consider superresolution methodshere solely for the purpose of improved parameter estimationIn Section 6 we briefly mention the canonical extensionof ABF and superresolution to space-time array processingSection 7 is the final and most important contribution Here

2 International Journal of Antennas and Propagation

we point out how direction estimation must be modified ifadaptive beams are used and how the radar detector thetracking algorithm and the track management should beadapted for ABF

2 Design Factors for Arrays

Array processing starts with the array antenna This is hard-ware and is a selected construction that cannot be alteredIt must therefore be carefully designed to fulfill all require-ments Digital array processing requires digital array outputsThe number and quality of these receivers (eg linearity andnumber of ADC bits) determine the quality and the cost ofthe whole system It may be desirable to design a fully digitalarray with AD-converters at each antenna element Howeverweight and cost will often lead to a system with reducednumber of digital receivers On the other hand becauseof the 11198774-decay of the received power radar needs antennaswith high gain and high directional discrimination whichmeans arrays with many elements There are different solu-tions to solve this contradiction

(i) Thinned arrays the angular discrimination of anarray with a number of elements can be improvedby increasing the separation of the elements and thusincreasing the aperture of the antenna Note that thethinned array has the same gain as the correspondingfully filled array

(ii) Subarrays element outputs are summed up in ananalog manner into subarrays which are then AD-converted and processed digitally The size and theshape of the subarrays are an important design cri-terion The notion of an array with subarrays is verygeneral and includes the case of steerable directionalarray elements

21 Impact of the Dimensionality of the Array Antenna ele-ments may be arranged on a line (1-dimensional array) on aplane (a ring or a 2-dimensional planar array) on a curvedsurface (conformal array) or within a volume (3D-arrayalso called Crowrsquos Nest antenna [3 Section 461]) A 1-dimensional array can only measure one independent angle2D and 3D arrays can measure the full polar coordinates inR3

Antenna element design and the need for fixing elementsmechanically lead to element patterns which are never omni-directional The elements have to be designed with patternsthat allow a unique identification of the direction Typicallya planar array can only observe a hemispherical half spaceTo achieve full spherical coverage several planar arrays canbe combined (multifacetted array) or a conformal or volumearray may be used

211 Arrays with Equal Patterns For linear planar and vol-ume arrays elements with nearly equal patterns can berealized These have the advantage that the knowledge of theelement pattern is for many array processing methods notnecessary An equal complex value can be interpreted as amodified target complex amplitude which is often a nuisance

parameter More important is that if the element patternsare really absolutely equal any cross-polar components of thesignal are in all channels equal and fulfill the array model inthe sameway as the copolar components that is they produceno error effect

212 Arrays with Unequal Array Patterns This occurs typi-cally by tilting the antenna elements as is done for conformalarrays For a planar array this tilt may be used to realize anarray with polarization diversity Single polarized elementsare then mounted with orthogonal alignment at differentpositions Such an array can provide some degree of dualpolarization receptionwith single channel receivers (contraryto more costly fully polarimetric arrays with receivers forboth polarizations for each channel)

Common to arrays with unequal patterns is that we haveto know the element patterns for applying array processingmethods The full element pattern function is also called thearray manifold In particular the cross-polar (or short 119909-pol)component has a different influence for each element Thismeans that if this component is not known and if the 119909-polcomponent is not sufficiently attenuated it can be a signifi-cant source of error

22 Thinned Arrays To save the cost of receiving modulessparse arrays are considered that is with fewer elements thanthe full populated 1205822 grid Because such a ldquothinned arrayrdquospans the same aperture it has the same beamwidth Hencethe angular accuracy and resolution are the same as the fullyfilled array Due to the gaps ambiguities or at least high side-lobesmay arise In early publications like [1] it was advocatedto simply take out elements of the fully filled array It wasearly recognized that this kind of thinning does not implyldquosufficiently randomrdquo positions Random positions on a 12058216grid as used in [3 Chapter 17] can provide quite acceptablepatterns Note that the array gain of a thinned array with 119873elements is always 119873 and the average sidelobe level is 1119873Today we know from the theory of compressed sensing thata selection of sufficiently random positions can produce aunique reconstruction of a not too large number of impingingwave fields with high probability [5]

23 Arrays with Subarrays If a high antenna gain with lowsidelobes is desired one has to go back to the fully filledarray For large arrays with thousands of elements the largenumber of digital channel constitutes a significant cost factorand a challenge for the resulting data rate Therefore oftensubarrays are formed and all digital (adaptive) beamformingand sophisticated array processing methods are applied tothe subarray outputs Subarraying is a very general conceptAt the elements we may have phase shifters such that allsubarrays are steered into a given direction and wemay applysome attenuation (tapering) to influence the sidelobe levelThe sum of the subarrays then gives the sum beam outputThe subarrays can be viewed as a superarray with elementshaving different patterns steered into the selected directionThe subarrays should have unequal size and shape to avoidgrating effects for subsequent array processing because thesubarray centers constitute a sparse array (for details see [6])

International Journal of Antennas and Propagation 3

Analogue

Digital

sumADC withbandpass

matched filtering

(1)

(2)

(3)middot middot middot

w w w w w w w w w w w w w

m m m(n)

Figure 1 Principle of forming subarrays

The principle is indicated in Figure 1 and properties andoptions are described in [6 7] In particular one can alsocombine new subarrays at the digital level or distribute thedesired tapering over the analog level (1) and various digitallevels (2) sdot sdot sdot (119899)

Beamforming using subarrays can be mathematicallydescribed by a simple matrix operation Let the complexarray element outputs be denoted by z The subarray formingoperation is described by a subarray forming matrix T bywhich the element outputs are summed up as z = T119867z For 119871subarrays and119873 antenna elements T is of size119873times119871 Vectorsand matrices at the subarray outputs are denoted by the tildeSuppose we steer the array into a look direction (119906

0 V0) =

u0by applying phase shifts 119886

119894(u0) = 119890

11989521205871198910(1199091198941199060+119910119894V0)119888 andapply additional amplitude weighting 119892

119894at the elements (real

vector of length 119873) for a sum beam with low sidelobesthen we have a complex weighting 119892

119894119886119894(u0) which can be

included in the elements of thematrixT Here1198910denotes the

centre frequency 119909119894 119910119894denote the coordinates of the 119894th array

element 119888 denotes the velocity of light and 119906 V denote thecomponents of the unit direction vector in the planar antenna(119909 119910)-coordinate system The beams are formed digitallywith the subarray outputs by applying a final weighting119894(119894 = 1 119871) as

119910 = m119867z (1)

In the simplest case m consists of only ones The antennapattern of such a sum beam can then be written as

119891 (u) = m119867T119867a (u) = m119867a (u) (2)

where a(u) = (119886119894(u))

119894=1sdotsdotsdot119873and a(u) = T119867a(u) denotes

the plane wave response at the subarray outputs All kindsof beams (sum azimuth and elevation difference guardchannel etc) can be formed from these subarray outputsWecan also scan the beam digitally at subarray level into anotherdirection [7]

Figure 2 2D generic array with 902 elements grouped into 32subarrays

Figure 2 shows a typical planar arraywith 902 elements ona triangular gridwith 32 subarraysThe shape of the subarrayswas optimized by the technique of [6] such that the differencebeams have low sidelobes when a minus40 dB Taylor weighting isapplied at the elements We will use this array in the sequelfor presenting examples

An important feature of digital beamforming with subar-rays is that the weighting for beamforming can be distributedbetween the element level (the weighting incorporated in thematrix T) and the digital subarray level (the weighting m)This yields some freedom in designing the dynamic range ofamplifiers at the elements and the level of the AD-converterinputThis freedom also allows to normalize the power of thesubarray outputs such that T119867T = I As will be shown inSection 4 this is also a reasonable requirement for adaptiveinterference suppression to avoid pattern distortions


minus1 minus08 minus06 minus04 minus02 0 02minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0(d

B)

Conventional BFminus30 dB Taylor = 5

minus15 dB Gaussian35 dB Chebyshevn

sin (120579)

(a) Low sidelobe patterns of equal beamwidth


minus15 dB Gaussian35 dB Chebyshev

0 5 10 15 20 25 30 35 4001

02

03

04

05

06

07

08

09

1

n

(b) Taper functions for low sidelobes

Figure 3 Low sidelobes by amplitude tapering

24 Space-Time Arrays Coherent processing of a time series1199111 119911

119870can be written as a beamforming procedure as in

(1) For a time series of array snapshots z1 z

119870 we have

therefore a double beamforming procedure of the space-timedata matrix Z = (z

1 z

119870) of the form

119878 = m119867

119904Zm

119905 (3)

where m119904 m

119905denote the weight vectors for spatial and

temporal beamforming respectively Using the rule of Kro-necker products (3) can be written as a single beamformingoperation

119878 = (m119905otimesm

119904)119867 vec Z (4)

where vecZ is a vector obtained by stacking all columns ofthe matrix Z on top This shows that mathematically it doesnot matter whether the data come from spatial or tem-poral sampling Coherent processing is in both cases abeamforming-type operation with the correspondinglymod-ified beamforming vector Relation (4) is often exploitedwhen spatial and temporal parameters are dependent (egdirection and Doppler frequency as in airborne radar seeSection 6)

3 Antenna Pattern Shaping

Conventional beamforming is the same as coherent integra-tion of the spatially sampled data that is the phase differencesof a plane wave signal at the array elements are compensatedand all elements are coherently summed up This resultsin a pronounced main beam when the phase differencesmatch with the direction of the plane wave and result insidelobes otherwiseThe beam shape and the sidelobes can beinfluenced by additional amplitude weighting

Let us consider the complex beamforming weights 119908119894=

1198921198941198901198952120587119891r119879

119894u119888 119894 = 1 sdot sdot sdot 119873 The simplest way of pattern shaping

is to impose some bell-shaped amplitude weighting over theaperture like 119892

119894= cos](120587119909

119894119860) + 120572 (for suitable constants ]

120572) or 119892119894= 119890

minus]1199092119894 The foundation of these weightings is quite

heuristical The Taylor weighting is optimized in the sensethat it leaves the conventional (uniformly weighted) patternundistorted except for a reduction of the first 119899 sidelobesbelow a prescribed level The Dolph-Chebyshev weightingcreates a pattern with all sidelobes equal to a prescribed levelFigure 3 shows examples of such patterns for a uniform lineararray with 40 elements The taper functions for low sidelobeswere selected such that the 3 dB beamwidth of all patternsis equal The conventional pattern is plotted for referenceshowing how tapering increases the beamwidth Which ofthese taperings may be preferred depends on the emphasison close in and far off sidelobes Another point of interest isthe dynamic range of the weights and the SNR loss because atthe array elements only attenuations can be applied One cansee that the Taylor tapering has the smallest dynamic rangeFor planar arrays the efficiency of the taperings is slightlydifferent

The rationale for low sidelobes is that we want to min-imize some unknown interference power coming over thesidelobes This can be achieved by solving the followingoptimization problem [8]

minw int

Ω

10038161003816100381610038161003816w119867a (u)10038161003816100381610038161003816

2

119901 (u) 119889u

subject to w119867a0= 1 or equivalently

minw wCw

st w119867a0= 1

with C = int

Ω

a (u) a(u)119867119901 (u) 119889u

(5)


1

05

0

0

minus05

minus1minus1 minus05 05 1

minus20

minus40

minus60

minus80

minus100

u

1

05

0

0

minus05


u

1

05

0

0

minus05


minus20

minus40

minus60

minus80

minus100

u

1

05

0

0

minus05


u

Main beam elevation +287∘ Main beam elevation minus287∘


Figure 4 Antenna patterns of planar array with reduced sidelobes at lower elevations for different beam pointing directions (by courtesy ofW Burger of Fraunhofer (FHR))

Ω denotes the angular sector where we want to influence thepattern for example the whole visible region 1199062+V2 lt 1 and119901 is aweighting functionwhich allows to put different empha-sis on the criterion in different angular regions The solu-tion of this optimization is

w =Cminus1a

0

a1198670Cminus1a

0

(6)

For the choice of the function 119901 we remark that for a globalreduction of the sidelobes when Ω = u isin R2

| 1199062+ V2 le 1

one should exclude the main beam from theminimization bysetting 119901 = 0 on this set of directions (in fact a slightly largerregion is recommended eg the null-to-null width) to allowa certain mainbeam broadening One may also form discretenulls in directions u

1 u

119872by setting 119901(u) = sum

119872

119896=1120575(u minus

u119896) The solution of (5) then can be shown to be

w =Pa

0

a1198670Pa

0

with P = I minus A (A119867A)minus1

A119867

A = (a (u1) a (u

119872))

(7)

This is just the weight for deterministic nulling To avoidinsufficient suppression due to channel inaccuracies onemayalso create small extended nulls using the matrixC The formof these weights shows the close relationship to the adaptivebeamforming weights in (11) and (17)

An example for reducing the sidelobes in selected areaswhere interference is expected is shown in Figure 4This is anapplication from an airborne radar where the sidelobes in thenegative elevation space have been lowered to reduce groundclutter

4 Adaptive Interference Suppression

Deterministic pattern shaping is applied if we have roughknowledge about the interference angular distribution In thesidelobe region this method can be inefficient because theantenna response to a plane wave (the vector a(u)) must beexactly known which is in reality seldom the case Typicallymuch more suppression is applied than necessary with theprice paid by the related beam broadening and SNR lossAdaptive interference suppression needs no knowledge of thedirectional behavior and suppresses the interference only as


much as necessary The proposition for this approach is thatwe are able to measure or learn in some way the adaptivebeamforming (ABF) weights

In the sequel we formulate the ABF algorithms forsubarray outputs as described in (1) This includes elementspace ABF for subarrays containing only one element

41 Adaptive Beamforming Algorithms Let us first supposethat we know the interference situation that is we know theinterference covariance matrix Q What is the optimumbeamforming vectorw From the Likelihood Ratio test crite-rion we know that the probability of detection is maximizedif we choose the weight vector that maximizes the signal-to-noise-plus-interference ratio (SNIR) for a given (expected)signal a

0

maxw

10038161003816100381610038161003816w119867a

0

10038161003816100381610038161003816

2

119864 1003816100381610038161003816w119867n

1003816100381610038161003816

2

= maxw

10038161003816100381610038161003816w119867a

0

10038161003816100381610038161003816

2

w119867Qw (8)

The solution of this optimization is

w = 120583Qminus1a0

with Q = 119864 nn119867 (9)

120583 is a free normalization constant and n denotes interferenceand receiver noise This weighting has a very intuitive inter-pretation If we decomposeQminus1

= LL119867 and apply this weightto the data we havew119867z = a119867

0Qminus1z = a119867

0L119867Lz = (La

0)119867(Lz)

This reveals that ABF does nothing else but a pre-whitenand match operation if z contains only interference that is119864zz119867 = Q then 119864(Lz)(Lz)119867 = I the prewhitening oper-ation the operation of L on the (matched) signal vector a

0

restores just the matching necessary with the distortion fromthe prewhitening operation

This formulation for weight vectors applied at the arrayelements can be easily extended to subarrays with digitaloutputs As mentioned in Section 23 a subarrayed array canbe viewed as a superarray with directive elements positionedat the centers of the subarrays This means that we have onlyto replace the quantities a n by a = T119867a n = T119867n Howeverthere is a difference with respect to receiver noise If thenoise at the elements is white with covariance matrix 120590

2Iit will be at subarray outputs with covariance matrix Q =

1205902T119867T Adaptive processing will turn this into white noise

Furthermore if we apply at the elements some weighting forlow sidelobes which are contained in the matrix T ABF willreverse this operation by the pre-whiten and match principleand will distort the low sidelobe pattern This can be avoidedby normalizing the matrix T such that TT119867 = I This can beachieved by normalizing the element weight as mentioned inSection 23 (for nonoverlapping subarrays)

Sometimes interference suppression is realized by min-imizing only the jamming power subject to additional con-straints for example w119867c

119894= 119896

119894 for suitable vectors c

119894

and numbers 119896119894 119894 = 1 sdot sdot sdot 119903 Although this is an intuitively

reasonable criterion it does not necessarily give the maxi-mum SNIR For certain constraints however both solutions

are equivalent The constrained optimization problem can bewritten in general terms as

minw w119867Q w

st w119867C = k (or w119867c119894= 119896

119894 119894 = 1 sdot sdot sdot 119903)

(10)

and it has the solution

w =

119903

sum

119894=1

120582119894Qminus1c

119894= Qminus1C (C119867Qminus1C)

minus1

k (11)

Examples of special cases are as follows

(i) Single unit gain directional constraint w119867a0= 1 rArr

w = (a1198670Qminus1a

0)minus1Qminus1a

0 This is obviously equivalent

to the SNIR-optimum solution (9) with a specificnormalization

(ii) Gain and derivative constraint w119867a0= 1 w119867a1015840

0=

0 rArr w = 120583Qminus1a0+120581Qminus1a1015840

0with suitable values of the

Lagrange parameters 120583 120582 A derivative constraint isadded to make the weight less sensitive against mis-match of the steering direction

(iii) Gain and norm constraint w119867a0= 1 w119867w = 119888 rArr

w = 120583(Q + 120575I)minus1a0 The norm constraint is added to

make the weight numerically stableThis is equivalentto the famous diagonal loading technique which wewill consider later

(iv) Norm constraint onlyw119867w = 1 rArr w = min119864119881(Q)Without a directional constraint the weight vectorproduces a nearly omnidirectional pattern but withnulls in the interference directions This is also calledthe power inversion weight because the patterndisplays the inverted interference power

As we mentioned before fulfilling the constraints may implya loss in SNIR Therefore several techniques have beenproposed to mitigate the loss The first idea is to allow acompromise between power minimization and constraintsby introducing coupling factors 119887

119894and solve a soft constraint

optimization

minw w119867Qw +

119903

sum

119894=1

119887119894

10038161003816100381610038161003816w119867c

119894minus 119896

119894

10038161003816100381610038161003816

2

or

minw w119867Qw + (w119867C minus k)119867

B (w119867C minus k)

(12)

with B = diag1198871 119887

119903 The solution of the soft-constraint

optimization is

w = (Q + CBC119867)minus1

CBk (13)

One may extend the constrained optimization by addinginequality constraints This leads to additional and improvedrobustness properties A number of methods of this kindhave been proposed for example in [9ndash12] As we are onlypresenting the principles here we do not go into furtherdetails


minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus60

minus50

minus40

minus30

minus20

minus10

0

u

20 20 20

(dB)

(a) Adapted antenna pattern

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

20 2020

SNIR

(dB)

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

u

(b) SNIR

Figure 5 Antenna and normalized SNIR patterns for a three jammer configuration and generic array

The performance of ABF is often displayed by the adaptedantenna pattern A typical adapted antenna pattern with 3jammers of 20 dB SNR is shown in Figure 5(a) for genericarray of Figure 2 This pattern does not show how the actualjamming power and the null depth play together

Plots of the SNIR are better suited for displaying thiseffect The SNIR is typically plotted for varying target direc-tion while the interference scenario is held fixed as seenin Figure 5(b) The SNIR is normalized to the SNR in theclear absence of any jamming and without ABF In otherwords this pattern shows the insertion loss arising from thejamming scenario with applied ABF

The effect of target and steering directionmismatch is notaccounted for in the SNIR plot This effect is displayed bythe scan pattern that is the pattern that arises if the adaptedbeam scans over a fixed target and interference scenario Sucha plot is rarely shown because of the many parameters tobe varied In this context we note that for the case that thetraining data contains the interference and noise alone themain beam of the adapted pattern is fairly broad similar tothe unadapted sum beam and is therefore fairly insensitiveto pointing mismatch How to obtain an interference-alonecovariance matrix is a matter of proper selection of thetraining data as mentioned in the following section

Figure 5 shows the case of an untapered planar antennaThe first sidelobes of the unadapted antenna pattern are atminus17 dB and are nearly unaffected by the adaptation processIf we have an antenna with low sidelobes the peak sidelobelevel is much more affected see Figure 6 Due to the taperingwe have a loss in SNIR of 17 dB compared to the referenceantenna (untapered without ABF and jamming)

42 Estimation of Adaptive Weights In reality the interfer-ence covariance matrix is not known and must be estimatedfrom some training data Z = (z

1 z

119870) To avoid signal

cancellation the training data should only contain the inter-ference alone If we have a continuously emitting interferencesource (noise jammer) one may sample immediately after or

before the transmit pulse (leading or rear dead zone) Onthe other hand if we sample the training data before pulsecompression the desired signal is typically much below theinterference level and signal cancellation is negligible Othertechniques are described in [13] The maximum likelihoodestimate of the covariance matrix is then

QSMI =1

119870

119870

sum

119896=1

z119896z119867119896 (14)

This is called the Sample Matrix Inversion algorithm (SMI)The SMI method is only asymptotically a good estimate Forsmall sample size it is known to be not very stable Formatrixinvertibility we need at least 119870 = 119873 samples According toBrennanrsquos Rule for example [1] one needs 2119870 samples toobtain an average loss in SNIR below 3 dB For smaller samplesize the performance can be considerably worse Howeverby simply adding a multiple of the identity matrix to the SMIestimate a close to optimum performance can be achievedThis is called the loaded sample matrix estimate (LSMI)

QLSMI =1

119870

119870

sum

119896=1

z119896z119867119896+ 120575 sdot I (15)

The drastic difference between SMI and LSMI is shown inFigure 7 for the planar array of Figure 2 for three jammersof 20 dB input JNR with 32 subarrays and only 32 datasnapshots For a ldquoreasonablerdquo choice of the loading factor (arule of thumb is 120575 = 2120590

2sdot sdot sdot 4120590

2 for an untapered antenna)we need only 2119872 snapshots to obtain a 3 dB SNIR loss if119872 denotes the number of jammers (dominant eigenvalues)present [13] So the sample size can be considerably lowerthan the dimension of the matrix The effect of the loadingfactor is that the dynamic range of the small eigenvalues iscompressed The small eigenvalues possess the largest statis-tical fluctuation but have the greatest influence on the weightfluctuation due to the matrix inversion



minus50

minus40

minus30

minus20

minus10

0

u

20 20 20

(dB)


minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

20 20 20

SNIR

(dB)


u

(b) SNIR

Figure 6 Antenna and normalized SNIR patterns for a three jammer configuration for antennawith low sidelobes (minus40 dBTaylor weighting)

minus30

minus25

minus20

minus15

minus10

minus5

0

SNIR

(dB)

2020

20

LSMI0LSMI4

EVP10

Optimum

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1u

Figure 7 SNIR for SMI LSMI (120575 = 41205902) and eigenvector projection

with dim (JSS) = 3

One may go even further and ignore the small eigenvalueestimates completely that is one tries to find an estimateof the inverse covariance matrix based on the dominanteigenvectors and eigenvalues For high SNR we can replacethe inverse covariancematrix by a projectionmatrix Supposewe have 119872 jammers with amplitudes 119887

1(119905) 119887

119872(119905) in

directionsu1 u

119872 If the received data has the form z(119905

119896) =

sum119872

119898=1a(u

119898)119887119898(119905119896) + n(119905

119896) or short z

119896= Ab

119896+ n

119896 then

119864 zz119867 = Q = ABA119867 + I (16)

Here we have normalized the noise power to 1 and B =

119864bb119867 Using the matrix inversion lemma we have

Qminus1= I minus A(Bminus1 + A119867A)

minus1

A119867

997888rarr

Brarrinfin

I minus A(A119867A)minus1

A119867 = PperpA(17)

PperpA is a projection on the space orthogonal to the columnsof A For strong jammers the space spanned by the columnsof A will be the same as the space spanned by the dominanteigenvectors We may therefore replace the estimated inversecovariance matrix by a projection on the complement of thedominant eigenvectors This is called the EVP method Asthe eigenvectors X are orthonormalized the projection canbe written as Pperp

119883= I minus XX119867

Figure 7 shows the performance of the EVP methodin comparison with SMI LSMI Note the little differencebetween LSMI and EVP The results with the three methodsare based on the same realization of the covariance estimate

For EVP we have to know the dimension of the jammersubspace (dimJSS) In complicated scenarios and with chan-nel errors present this value can be difficult to determineIf dimJSS is grossly overestimated a loss in SNIR occursIf dimJSS is underestimated the jammers are not fully sup-pressed One is therefore interested in subspacemethodswithlow sensitivity against the choice of the subspace dimensionThis property is achieved by a ldquoweighted projectionrdquo that isby replacing the projection by

PLMI = I minus XDX119867 (18)

where D is a diagonal weighting matrix and X is a setorthonormal vectors spanning the interference subspacePLMI does not have the mathematical properties of a projec-tion Methods of this type of are called lean matrix inversion(LMI) A number of methods have been proposed that can beinterpreted as an LMImethodwith different weightingmatri-cesDThe LMImatrix can also be economically calculated byan eigenvector-free QR-decomposition method [14]

One of themost efficientmethods for pattern stabilizationwhile maintaining a low desired sidelobe level is the con-strained adaptive pattern synthesis (CAPS) algorithm [15]which is also a subspace method Let m be the vector for


beamforming with low sidelobes in a certain direction In fullgenerality the CAPS weight can be written as

wCAPS =1

m119867Rminus1mQminus1

SMIm minus Xperp(X119867

perpCX

perp)minus1

times X119867perpC(

1


SMIm minusm)

(19)

where the columns of the matrix Xperpspan the space orthog-

onal to [Xm] and X is again a unitary 119871 times 119872 matrixwith columns spanning the interference subspace which isassumed to be of dimension119872 C is a directional weightingmatrix C = int

Ωa(u)a(u)119867119901(u)119889u Ω denotes the set of direc-

tions of interest and 119901(u) is a directional weighting functionIf we use no directional weighting C asymp I the CAPS weightvector simplifies to

wCAPS = m + P[Xm] (

1


SMIm minusm) (20)

where P[Xm] denotes the projection onto the space spanned

by the columns of X andm

43 Determination of the Dimension of Jammer Subspace(dimJSS) Subspace methods require an estimate of thedimension of the interference subspace Usually this isderived from the sample eigenvalues For complicated scenar-ios and small sample size a clear decision ofwhat constitutes adominant eigenvaluemay be difficultThere are two principleapproaches to determine the number of dominant eigenval-ues information theoretic criteria and noise power tests

The information theoretic criteria are often based on thesphericity test criterion see for example [16]

119879 (119898) =(1 (119873 minus 119898))sum

119873

119894=119898+1120582119894

(prod119873

119894=119898+1120582119894)1(119873minus119898)

(21)

where 120582119894denote the eigenvalues of the estimated covariance

matrix ordered in decreasing magnitude The ratio of thearithmetic to geometric mean of the eigenvalues is a measureof the equality of the eigenvalues The information theoreticcriteriaminimize this ratio with a penalty function added forexample the Akaike Information Criterion (AIC) and Min-imum Description Length (MDL) choose dimJSS as theminimum of the following functions

AIC (119898) = 119870 (119873 minus 119898) log [119879 (119898)] + 119898 (2119873 minus 119898)

MDL (119898) = 119870 (119873 minus 119898) log [119879 (119898)] + (1198982) (2119873 minus 119898) log119870

(22)

The noise power threshold tests (WNT) assume that the noisepower 1205902 is known and just check the estimated noise poweragainst this value [16] This leads to the statistic

119871 (119898) =2119870

1205902

119873

sum

119894=119898+1

120582119894 (23)

no

12

34

56

24

68

1012

0

20

40

60

80

100

dimSS

MDL plainAIC plainMDL asymptoticMDL dload

WNT asymptoticWNT dloadTrue target

Figure 8 Comparison of tests for linear arraywith119873 = 14 elements119870 = 14 snapshots with and without asymptotic correction ordiagonal loading (dload) of 11205902

and the decision is found if the test statistic is for the first timebelow the threshold

for 119894 = 1 sdot sdot sdot 119873 do

if 119871 (119898) le 1205942

2119870(119873minus119898)120572 = 119898 STOP

end

(24)

The symbol 1205942119903120572

denotes the 120572-percentage point of the 1205942-distribution with 119903 degrees of freedom The probability tooverestimate dimJSS is then asymptotically bounded by 120572More modern versions of this test have been derived forexample [17]

For small sample size AIC and MDL are known forgrossly overestimating the number of sources In additionbandwidth and array channels errors lead to a leakage ofthe dominant eigenvalues into the small eigenvalues [18]Improved eigenvalue estimates for small sample size can mit-igate this effect The simplest way could be to use the asymp-totic approximation using the well-known linkage factors[19]

120582119894=

119894minus1

119870119894

119873

sum

119895=1

119895 = 119894

119894

119894minus

119895

(25)

More refined methods are also possible see [16] Howeveras explained in [16] simple diagonal loading can improveAIC and MDL for small sample size and make these criteriarobust against errors For the WNT this loading is containedin the setting of the assumed noise level 1205902 Figure 8 showsan example of a comparison of MDL and AIC without anycorrectionsMDL andWNTwith asymptotic correction (25)and MDL and WNT with diagonal loading of 120583 = 1120590

2 The


threshold for WNT was set for a probability to overestimatethe target number of 120572 = 10 The scenario consists of foursources at 119906 = minus07 minus055 minus031 minus024 with SNR of 18 6 20204 dB and a uniform linear antenna with 14 elements and10 relative bandwidth leading to some eigenvalue leakageEmpirical probabilities were determined from 100 MonteCarlo trials Note that the asymptotic correction seems towork better for WNT than for MDL With diagonal loadingall decisions with both MDL and WNT were correct (equalto 4)

A more thorough study of the small sample size dimJSSestimation problem considering the ldquoeffective number ofidentifiable signalsrdquo has been performed in [20] and a newmodified information theoretic criterion has been derived

5 Parameter Estimation and Superresolution

The objective of radar processing is not to maximize the SNRbut to detect targets and determine their parameters Fordetection the SNR is a sufficient statistic (for the likelihoodratio test) that is if we maximize the SNR we maximize alsothe probability of detection Only for these detected targetswe have then a subsequent procedure to estimate the targetparameters direction range and possibly Doppler Standardradar processing can be traced back to maximum likelihoodestimation of a single target which leads to the matched filter[21] The properties of the matched filter can be judged bythe beam shape (for angle estimation) and by the ambiguityfunction (for range andDoppler estimation) If the ambiguityfunction has a narrow beam and sufficiently low sidelobesthe model of a single target is a good approximation as othertargets are attenuated by the sidelobes However if we haveclosely spaced targets or high sidelobes multiple target mod-els have to be used for parameter estimation A variety of suchestimation methods have been introduced which we termhere ldquosuperresolution methodsrdquo Historically these methodshave often been introduced to improve the limited resolutionof the matched filter

51 Superresolution The resolution limit for classical beam-forming is the 3 dB beamwidth An antenna array providesspatial samples of the impinging wavefronts and one maydefine a multitarget model for this case This opens thepossibility for enhanced resolutionThesemethods have beendiscussed since decades and textbooks on this topic are avail-able for example [22] We formulate here the angle param-eter estimation problem (spatial domain) but correspondingversions can be applied in the time domain as well In thespatial domain we are faced with the typical problems ofirregular sampling and subarray processing

From the many proposed methods we mention hereonly some classical methods to show the connections andrelationships We have spectral methods which generate aspiky estimate of the angular spectral density like

Caponrsquos method

119878119862(u) = (a(u)119867Rminus1MLa (u))

minus1

with RML =1

119870

119870

sum

119896=1

z119896z119867119896

(26)

and MUSIC method (Multiple Signal Classification)

119878MUSIC (u) = (a(u)119867Pperpa (u))minus1

(27)

with Pperp = I minusXX119867 and X spanning the dominant subspaceAn LMI-version instead of MUSIC would also be possibleThe target directions are then found by the119872highestmaximaof these spectra (119872 1- or 2-dimensional maximizations)

An alternative group ofmethods are parametricmethodswhich deliver only a set of ldquooptimalrdquo parameter estimateswhich explain in a sense the data for the inserted model by119872 or 2119872 dimensional optimization [21]

Deterministic ML method (complex amplitudes areassumed deterministic)

119865det (120579) = tr (PperpARML) with PperpA = I minus A(A119867A)minus1

A119867

A = (a (u1) a (u

119872))

(28)

Stochastic ML method (complex amplitudes are complexGaussian)

119865sto (120579) = logdet (R (120579)) + tr (R(120579)minus1RML) (29)

whereR(120579) denotes the completely parameterized covariancematrix A formulation with the unknown directions as theonly parameters can be given as

119865sto (120579) = det A (120579)B (120579)A119867 (120579) + 1205902 (120579) I with

1205902(120579) =

1

119873 minus119872tr PperpARML

B (120579) = (A119867A)minus1

A119867 (RML minus 1205902(120579) I)A(A119867A)

minus1

for A = A (120579)

(30)

The deterministic ML method has some intuitive interpreta-tions

(1) 119865det(120579)= (1119870)sum119870

119896=1 z119867

119896PperpAz119896 = (1119870)sum

119870

119896=1PperpAz119896

2

=

(1119870)sum119870

119896=1z119896minus A(A119867A)minus1A119867z

119896⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=b

2

which means

that the mean squared residual error after signalextraction is minimized

(2) 119865det(120579) = 119862 minus sum119870

119896=1z119867119896A(A119867A)minus1A119867z

119896 which can

be interpreted as maximizing a set of decoupled sumbeams (a119867(u

1)zk a119867(u119872)zk)

(3) 119865det(120579) = 119862minusa119867nullRanulla119867

nullanull with anull = PperpAa(u)where we have partitioned the matrix of steeringvectors into A = (a A) This property is valid due tothe projection decomposition lemma which says thatfor any partitioning A = (FG) we can write PperpA =

PperpGminusPperp

GF(F119867PperpGF)

minus1F119867PperpG If we keep the directions inAfixed this relation says thatwe have tomaximize thescan pattern over uwhile the sources in the directionsof A are deterministically nulled (see (7)) One can


minus09 minus08 minus07 minus06 minus05 minus04 minus03minus10

0

10

20

30

40

50

60

70

80

u

1 12 12

(dB)

(a) Simulated data

minus05

0

05

minus05

0

050

5

10

15

20

u

(dB)

(b) Measured real data

Figure 9 MUSIC spectra with a planar array of 7 elements

now perform the multidimensional maximization byalternating 1-dimensionalmaximizations and keepingthe remaining directions fixed This is the basis of thealternating projection (AP) method or IMP (Incre-mental MultiParameter) method [22 page 105]

A typical feature of the MUSIC method is illustrated inFigure 9 This figure shows the excellent resolution in sim-ulation while for real data the pattern looks almost the sameas with Caponrsquos method

A result with real data with the deterministic MLmethodis shown in Figure 10Minimizationwas performed here witha stochastic approximation method This example shows inparticular that the deterministicML-method is able to resolvehighly correlated targets which arise due to the reflection onthe sea surface for low angle tracking The behavior of themonopulse estimates reflect the variation of the phase differ-ences of direct and reflected path between 0∘ and 180∘ For0∘ phase difference the monopulse points into the centre for180∘ it points outside the 2-target configuration

The problems of superresolution methods are describedin [21 23] A main problem is the numerical effort of findingthe 119872 maxima (one 119872-dimensional optimization or 119872 1-dimensional optimizations for a linear antenna) To mitigatethis problem a stochastic approximation algorithm or theIMP method has been proposed for the deterministic MLmethod The IMP method is an iteration of maximizationsof an adaptively formed beam pattern Therefore the gener-alized monopulse method can be used for this purpose seeSection 71 and [24]

Another problem is the exact knowledge of the signalmodel for all possible directions (the vector function a(u))The codomain of this function is sometimes called the arraymanifold This is mainly a problem of antenna accuracy orcalibration While the transmission of a plane wave in themain beam direction can be quite accurately modeled (usingcalibration) this can be difficult in the sidelobe region

For an array with digital subarrays superresolution hasto be performed only with these subarray outputs The array

manifold has then to be taken at the subarray outputs as in(2) This manifold (the subarray patterns) is well modeled inthe main beam region but often too imprecise in the sideloberegion to obtain a resolution better than the conventional Inthat case it is advantageous to use a simplified array manifoldmodel based only on the subarray gains and centers calledthe Direct Uniform Manifold model (DUM) This simpli-fied model has been successfully applied to MUSIC (calledSpotlightMUSIC [25]) and to the deterministic MLmethodUsing the DUM model requires little calibration effort andgives improved performance [25]

More refined parametricmethodswith higher asymptoticresolution property have been suggested (eg COMETCovarianceMatching Estimation Technique [26]) Howeverapplication of such methods to real data often revealed noimprovement (as is the case with MUSIC in Figure 9) Thereason is that these methods are much more sensitive to thesignal model than the accuracy of the system provides Asensitivity with an very sharp ideal minimum of the objectivefunction may lead to a measured data objective functionwhere the minimum has completely disappeared

52 Target NumberDetermination Superresolution is a com-bined target number and target parameter estimation prob-lem As a starting point all the methods of Section 43 canbe used If we use the detML method we can exploit thatthe objective function can be interpreted as the residual errorbetween model (interpretation 2) and data The WNT teststatistic (23) is just an estimate of this quantity The detMLresidual can therefore be used for this test instead of the sumof the eigenvalues

These methods may lead to a possibly overestimatedtarget number To determine the power allocated to eachtarget a refined ML power estimate using the estimateddirections A(120579) can be used B(120579) = (A119867A)minus1A119867(RML minus

1205902(120579)I)A(A119867A)minus1 with 120590

2(120579) = (1(119873 minus 119872)) trPperpARML as

in (30) This estimate can even reveal correlations betweenthe targets This has been successfully demonstrated with


257 308 358 408 459 509 560 610 661 711 762 812 862

10

06

02

minus02

minus06

Range (km)

Elev

atio

n (B

W) Monopulse estimate

with glint effect dueto multipath

Superresolutionwith deterministic ML

True elevationfrom opticaltracker

Figure 10 Superresolution of multipath propagation over sea with deterministic ML method (real data from vertical linear array with 32elements scenario illustrated above)

SNR (dB)

101

203

304

406

507

608

710

811

Test down (dB) Test up (dB)

minus05 0 05

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

u

Figure 11 Combined target number and direction estimation for 2 targets with 7-element planar array

the low angle tracking data of Figure 10 In case that sometarget power is too low the target number can be reducedand the angle estimates can be updated This is an iterativeprocedure of target number estimation and confirmation orreduction This way all target modeling can be accuratelymatched to the data

The deterministic ML method (28) together with thewhite noise test (24) is particularly suited for this kind ofiterative model fitting It has been implemented in an exper-imental system with a 7-element planar array at FraunhoferFHR and was first reported in [21 page 81] An example ofthe resulting output plot is shown in Figure 11 The estimateddirections in the 119906 V-plane are shown by small dishes havinga color according to the estimated target SNR correspondingto the color bar The circle indicates the 3 dB contour of thesum beam One can see that the two targets are at about 05

beamwidth separation The directions were estimated by thestochastic approximation algorithm used in Figure 10 Thetest statistic for increasing the target number is shown bythe right most bar The thresholds for increasing the numberare indicated by lines The dashed line is the actually validthreshold (shown is the threshold for 2 targets) The targetnumber can be reduced if the power falls below a thresholdshown in two yellow bars in themiddleThewhole estimationand testing procedure can also be performed adaptively withchanging target situations We applied it to two blinkingtargets alternating between the states ldquotarget 1 onrdquo ldquobothtargets onrdquo ldquotarget 2 onrdquo ldquoboth targets onrdquo and so forthClearly these test works only if the estimation procedure hasconverged This is indicated by the traffic light in the rightup corner We used a fixed empirically determined iterationnumber to switch the test procedure on (=green traffic light)


K range cells

M pulses

N channels

Figure 12 Symmetric auxiliary sensorecho processor of Klemm [4] as an example for forming space-time subarrays

All thresholds and iteration numbers have to be selectedcarefully Otherwise situations may arise where this adaptiveprocedure switches between two target models for examplebetween 2 and 3 targets

The problem of resolution of two closely spaced targetsbecomes a particular problem in the so called thresholdregion which denotes configurations where the SNR or theseparation of the targets lead to an angular variance departingsignificantly from the Cramer-Rao bound (CRB) The designof the tests and this threshold region must be compatible togive consistent joint estimation-detection resolution resultThese problems have been studied in [27 28] One way toachieve consistency and improving resolution proposed in[27] is to detect and remove outliers in the data which arebasically responsible for the threshold effect A general dis-cussion about the achievable resolution and the best realisticrepresentation of a target cluster can be found in [28]

6 Extension to Space-Time Arrays

As mentioned in Section 24 there is mathematically no dif-ference between spatial and temporal samples as long as thedistributional assumptions are the same The adaptive meth-ods and superresolution methods presented in the previoussections can therefore be applied analogously in the time orspace-time domain

In particular subarraying in time domain is an importanttool to reduce the numerical complexity for space-timeadaptive processing (STAP) which is the general approach foradaptive clutter suppression for airborne radar [4] With theformalism of transforming space-time 2D-beamforming of adata matrix into a usual beamforming operation of vectorsintroduced in (4) the presented adaptive beamforming andsuperresolution methods can be easily transformed intocorresponding subarrayed space-time methods

Figure 12 shows an example of an efficient space-timesubarraying scheme used for STAP clutter cancellation forairborne radar

7 Embedding of Array Processing intoFull Radar Data Processing

A key problem that has to be recognized is that the task of aradar is not tomaximize the SNR but to give the best relevant

information about the targets after all processingThis meansthat for implementing refined methods of interference sup-pression or superresolutionwe have also to consider the effecton the subsequent processing To get optimum performanceall subsequent processing should exploit the properties of therefined array signal processing methods applied before Inparticular it has been shown that for the tasks of detectionangle estimation and tracking significant improvements canbe achieved by considering special features

71 Adaptive Monopulse Monopulse is an established tech-nique for rapid and precise angle estimation with arrayantennas It is based on two beams formed in parallel a sumbeam and a difference beam The difference beam is zero atthe position of the maximum of the sum beam The ratioof both beams gives an error value that indicates the offsetof a target from the sum beam pointing direction In fact itcan be shown that this monopulse estimator is an approxima-tion of the Maximum-Likelihood angle estimator [24] Themonopulse estimator has been generalized in [24] to arrayswith arbitrary subarrays and arbitrary sum and differencebeams

When adaptive beams are used the shape of the sumbeam will be distorted due to the interference that is to besuppressed The difference beam must adaptively suppressthe interference as well which leads to another distortionThen the ratio of both beams will no more indicate the targetdirection The generalized monopulse procedure of [24]provides correction values to compensate these distortions

The generalizedmonopulse formula for estimating angles( V)119879 with a planar array and sum and difference beamsformed into direction (119906

0 V0)119879 is

(

V) = (

1199060

V0

) minus (119888119909119909

119888119909119910

119888119910119909

119888119910119910

)(119877119909minus 120583

119909

119877119910minus 120583

119910

) (31)

whereC = (119888119909119909 119888119909119910

119888119910119909 119888119910119910) is a slope correctionmatrix and 120583 = (

120583119909120583119910)

is a bias correction 119877119909= Red119867

119909zw119867z is the monopulse

ratio formed with the measured difference and sum beamoutputs119863

119909= d119867

119909z and 119878 = w119867z respectively with difference

and sum beam weight vectors d119909 w (analogous for elevation

estimation with d119910) The monopulse ratio is a function of the

unknown target directions (119906 V) Let the vector ofmonopulseratios be denoted by R(119906 V) = (119877

119909(119906 V) 119877

119910(119906 V))119879 The cor-

rection quantities are determined such that the expectation


of the error is unbiased and a linear function with slope 1 isapproximated More precisely for the following function ofthe unknown target direction

M (119906 V) = C sdot (119864 R (119906 V) minus 120583) (32)

we require

M (1199060 V0) = 0

120597M120597119906

(1199060 V0) = (

1

0)

120597M120597V

(1199060 V0) = (

0

1) or

C(120597R120597119906

120597R120597V

) (1199060 V0) = I

(33)

These conditions can only approximately be fulfilled for suf-ficiently high SNR Then one obtains for the bias correctionfor a pointing direction a

0= a(119906

0 V0) [24]

120583120572= Re

d119867120572a0

w119867a0

for 120572 = 119909 119910 (34)

For the elements of the inverse slope correction matrix(119888120572ℎ) 119886=119909119910

ℎ=119906V= Cminus1 one obtains

119888120572ℎ

=

Re d119867120572aℎ0a1198670w + d119867

120572a0a119867ℎ0w

1003816100381610038161003816w119867a01003816100381610038161003816

2minus 120583

1198862Re

w119867aℎ0

w119867a0

(35)

with 120572 = 119909 or 119910 and ℎ = 119906 or V and aℎ0

denotes thederivative (120597a120597ℎ)|

(1199060 V0) In general these are fixed antenna

determined quantities For example for omnidirectionalantenna elements and phase steering at the elements we havea0= 119866

119890(1 1)

119879 where119866119890is the antenna element gain and

a1198791199060

= 119866119890(1198952120587119891119888)(119909

1 119909

119873)

It is important to note that this formula is independentof any scaling of the difference and sum weights Constantfactors in the difference and sum weight will be cancelledby the corresponding slope correction Figure 13 shows the-oretically calculated bias and variances for this correctedgeneralized monopulse using the formulas of [24] for thearray of Figure 2The biases are shown by arrows for differentpossible single target positions with the standard deviationellipses at the tip A jammer is located in the asterisk symboldirection with JNR = 27 dB The hypothetical target has aSNR of 6 dB The 3 dB contour of the unadapted sum beamis shown by a dashed circle The 3 dB contour of the adaptedbeamwill be of course different One can see that in the beampointing direction (0 0) the bias is zero and the variance issmall The errors increase for target directions on the skirt ofthe main beam and close to the jammer

The large bias may not be satisfying However one mayrepeat the monopulse procedure by repeating the monopulseestimate with a look direction steered at subarray level intothe new estimated direction This is an all-offline procedurewith the given subarray data No new transmit pulse isneeded We have called this the multistep monopulse proce-dure [24]Multistepmonopulse reduces the bias considerably

minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u

Figure 13 Bias and standard deviation ellipses for different targetpositions

minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u

Figure 14 Bias for 2-step monopulse for different target positionsand jammer scenario of Figure 13

with only one additional iteration as shown in Figure 14 Thevariances appearing in Figure 13 are virtually not changedwith the multistep monopulse procedure and are omitted forbetter visibility

72 Adaptive Detection For detection with adaptive beamsthe normal test procedure is not adequate because we have atest statistic depending on twodifferent kinds of randomdatathe training data for the adaptive weight and the data undertest Various kinds of tests have been developed accounting


for this fact The first and basic test statistics were the GLRT[29] the AMF detector [30] and the ACE detector [31]These have the form

119879GLRT (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (1 + (1119870) z119867Qminus1

SMIz) (36)

119879AMF (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (37)

119879ACE (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 sdot z119867Qminus1

SMIz (38)

The quantities z a0 Q are here all generated at the subarray

outputs a0denotes the plane wave model for a direction u

0

Basic properties of these tests are

(i) 119879GLRT =119879AMF

1 + (1119870) z119867Qminus1

SMIz 119879ACE =

119879AMF

z119867Qminus1

SMIz

(39)

(ii)TheAMF detector represents an estimate of the signal-to-noise ratio because it can be written as

119879AMF =

10038161003816100381610038161003816w119867z10038161003816100381610038161003816

2

w119867QSMIwwith w = Qminus1

SMIa0 (40)

This provides a meaningful physical interpretation A com-plete statistical description of these tests has been givenin very compact form in [32 33] These results are validas well for planar arrays with irregular subarrays and alsomismatched weighting vector

Actually all these detectors use the adaptive weight ofthe SMI algorithm which has unsatisfactory performance asmentioned in Section 42 The unsatisfactory finite sampleperformance is just the motivation for introducing weightestimators like LSMI LMI or CAPS Clutter insufficientadaptive suppression and surprise interference are the moti-vation for requiring low sidelobes Recently several morecomplicated adaptive detectors have been introduced withthe aim of achieving additional robustness properties [34ndash38] However another and quite simple way would be togeneralize the tests of (36) (37) (38) to arbitrary weightvectors with the aim of inserting well known robust weightsas derived in Section 41 This has been done in [39] Firstwe observe that the formulation of (40) can be used for anyweight vector Second one can observe that the ACE andGLRT have the form of a sidelobe blanking device Inparticular it has already been shown in [35] that diagonalloading provides significant better detection performance

A guard channel is implemented in radar systems toeliminate impulsive interference (hostile or fromother neigh-boring radars) using the sidelobe blanking (SLB) device Theguard channel receives data from an omnidirectional antennaelement which is amplified such that its power level is abovethe sidelobe level of the highly directional radar antenna butbelow the power of the radar main beam [1 page 99] If the

received signal power in the guard channel is above the powerof the main channel this must be a signal coming via thesidelobes Such signals will be blanked If the guard channelpower is below the main channel power it is considered as adetection

With phased arrays it is not necessary to provide anexternal omnidirectional guard channel Such a channelcan be generated from the antenna itself all the requiredinformation is in the antenna We may use the noncoherentsum of the subarrays as guard channelThis is the same as theaverage omnidirectional power Some shaping of the guardpattern can be achieved by using a weighting for the nonco-herent sum

119866 =

119871

sum

119894=1

119892119894

10038161003816100381610038161198941003816100381610038161003816

2

(41)

If all subarrays are equal a uniform weighting g = (1 1)119879

may be suitable for unequal irregular subarrays as in Figure 2the different contributions of the subarrays can be weightedThe directivity pattern of such guard channel is given by119878119866(u) = sum

119871

119894=1119892119894|119886119894(u)|2 More generally we may use a com-

bination of noncoherent and coherent sums of the subarrayswith weights contained in the matricesD K respectively

119866 = z119867KDK119867z 119878119866(u) = a119867 (u)KDK119867a (u) (42)

Examples of such kind of guard channels are shown inFigure 15 for the generic array of Figure 2 with minus35 dB Taylorweighing for low sidelobes The nice feature of these guardchannels is (i) that they automatically scan together withthe antenna look direction and (ii) that they can easily bemade adaptive This is required if we want to use the SLBdevice in the presence of CW plus impulsive interference ACW jammer would make the SLB blank all range cells thatis would just switch off the radar To generate an adaptiveguard channel we only have to replace in (42) the datavector of the cell under test (CUT) by the pre-whitened datazpre-119908 = Rminus12z Then the test statistic can be written as119879 = 119879AMF(z)119866adapt(z) where 119866adapt(z) = z119867Rminus1z forACE and 119866adapt(z) = 1 + (1119870)z119867Rminus1z for GLRT Hence119866adapt is just the incoherent sumof the pre-whitened subarrayoutputs in other words 119879ACE can be interpreted as an AMFdetector with an adaptive guard channel and 119879GLRT the samewith guard channel on a pedestal Figure 16 shows examplesof some adapted guard channels generated with the genericarray of Figure 2 andminus35 dBTaylorweightingTheunadaptedpatterns are shown by dashed lines

This is the adaptive generalization of the usual sidelobeblanking device (SLB) and the AMF ACE andGLRT tests canbe used as extension of the SLB detector to the adaptive case[32] called the 2D adaptive sidelobe blanking (ASB) detectorThe AMF is then the test for the presence of a potential targetand the generalized ACE or GRLT are used confirming thistarget or adaptive sidelobe blanking

A problemwith these modified tests is to define a suitablethreshold for detection For arbitraryweight vector it is nearlyimpossible to determine this analytically In [39] the detection


minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)

minus35 dB Taylor


u

ni = 32

(a) Uniform subarray weighting

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

minus35 dB Taylor ni = 32

(b) Power equalized weighting

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u


(c) Power equalized + difference weighting

Figure 15 Guard channel and sum beam patterns for generic array of Figure 2

margin has been introduced as an empirical tool for judging agood balance between the AMF and ASB threshold for givenjammer scenarios The detection margin is defined as thedifference between the expectation of the AMF statistic andthe guard channel where the expectation is taken only overthe interference complex amplitudes for a known interferencescenario In addition one can also calculate the standarddeviation of these patternsTheperformance against jammersclose to the main lobe is the critical feature The detectionmargin provides the mean levels together with standarddeviations of the patterns An example of the detectionmargin is shown in Figure 17 (same antenna and weightingas in Figures 15 and 16)

Comparing the variances of the ACE and GLRT guardchannels in [39] revealed that the GLRT guard performssignificantly better in terms of fluctuations The GLRT guardchannel may therefore be preferred for its better sidelobeperformance and higher statistical stability

73 Adaptive Tracking A key feature of ABF is that overallperformance is dramatically influenced by the proximity ofthe main beam to an interference source The task of targettracking in the proximity of a jammer is of high operationalrelevance In fact the information on the jammer directioncan be made available by a jammer mapping mode whichdetermines the direction of the interferences by a backgroundprocedure using already available data Jammers are typicallystrong emitters and thus easy to detect In particular theSpotlightMUSICmethod [25]workingwith subarray outputsis suited for jammer mapping with a multifunction radar

Let us assume here for simplicity that the jammer direc-tion is known This is highly important information for thetracking algorithmof amultifunction radar where the trackerdetermines the pointing direction of the beam We will usefor angle estimation the adaptive monopulse procedure ofSection 71 ABF will form beams with a notch in the jammerdirection Therefore one cannot expect target echoes from


minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

34


(a) Weighting for equal subarray power

34

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u


(b) Difference type guard with weighting for equal subarray power

Figure 16 Adapted guard patterns for jammer at 119906 = minus027 (minus157∘) with JNR of 34 dB for generic array

minus30

minus20

minus10

0

10

20

30

40

50

(dB)

AMFACE detection marginsGuard=uniform LSMIload=3

ACE-detect marginAMF pattern120578ACEConf bound(AMF)Conf bound(AMF)Adapted guard

Conf bound(adG)Conf bound(adG)Ant patt clear120578ACEGuard clear120578AMF 120578ACE


u

Figure 17 Detection margin (gray shading) between AMF andnonweighted adapted guard pattern with confidence bounds forACE with LSMI (64 snapshots and 31205902 diagonal loading) Two CWinterferences of 40 dB INR are present at 119906 = minus08 minus04

directions close to the jammer and therefore it does not makesense to steer the beam into the jammer notch Furthermorein the case of amissingmeasurement of a tracked target insidethe jammer notch the lack of a successful detection supportsthe conclusion that this negative contact is a direct result ofjammer nulling by ABF This is so-called negative informa-tion [40] In this situation we can use the direction of thejammer as a pseudomeasurement to update andmaintain the

track file The width of the jammer notch defines the uncer-tainty of this pseudo measurement Moreover if one knowsthe jammer direction one can use the theoretically calculatedvariances for the adaptivemonopulse estimate of [24] as a pri-ori information in the tracking filterThe adaptivemonopulsecan have very eccentric uncertainty ellipses as shown inFigure 13 which is highly relevant for the tracker The largebias appearing in Figure 13 which is not known by thetracker can be reduced by applying the multistep monopulseprocedure [24]

All these techniques have been implemented in a trackingalgorithm and refined by a number of stabilization measuresin [41]The following special measures for ABF tracking havebeen implemented and are graphically visualized in Figure 18

(i) Look direction stabilization the monopulse estimatemay deliver measurements outside of the 3 dB con-tour of the sum beam Such estimates are also heavilybiased especially for look directions close to the jam-mer despite the use of the multistep monopulse pro-cedure Estimates of that kind are therefore correctedby projecting them onto the boundary circle of sumbeam contour

(ii) Detection threshold only those measurements areconsidered in the update step of the tracking algo-rithm whose sum beam power is above a certaindetection threshold (typically 13 dB) This guaranteesuseful and valuable monopulse estimates It is wellknown that the variance of the monopulse estimatedecreases monotonically with this threshold increas-ing

(iii) Adjustment of antenna look direction look directionsin the jammer notch should generally be avoided dueto the expected lack of good measurements In casethat the proposed look direction lies in the jammernotch we select an adjusted direction on the skirt ofthe jammer notch


Elev

atio

n

rarr

Azimuth u rarr

Projection of monopulse estimate

BW120572 middot BW

Antenna look direction

Projection

(a)

Adjustment of antenna look direction

BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)

Figure 18 Illustration of different stabilization measures to improve track stability and track continuity

(iv) Variable measurement covariance a variable covari-ance matrix of the adaptive monopulse estimationaccording to [24] is considered only for a mainlobejammer situation For jammers in the sidelobes thereis little effect on the angle estimates and we can usethe fixed covariance matrix of the nonjammed case

(v) QuadSearch and Pseudomeasurements if the pre-dicted target direction lies inside the jammer notchand if despite all adjustments of the antenna lookdirection the target is not detected a specific searchpattern is initiated (named QuadSearch) which useslook directions on the skirt of the jammer notch toobtain acceptable monopulse estimates If this proce-dure does not lead to a detection we know that thetarget is hidden in the jammer notch and we cannotsee it We use then the direction of the jammer asa pseudobearing measurement to maintain the trackfile The pseudomeasurement noise is determined bythe width of the jammer notch

(vi) LocSearch in case of a permanent lack of detections(eg for three consecutive scans) while the trackposition lies outside the jammer notch a specificsearch pattern is initiated (named LocSearch) that issimilar to the QuadSearch The new look directionslie on the circle of certain radius around the predictedtarget direction

(vii) Modeling of target dynamics the selection of asuitable dynamics model plays a major role for thequality of tracking results In this context the so-called interacting multiple model (IMM) is a well-known method to reliably track even those objectswhose dynamic behavior remains constant only dur-ing certain periods

(viii) Gating in the vicinity of the jammer the predictedtarget direction (as an approximation of the truevalue) is used to compute the variable angle measure-ment covariance Strictly speaking this is only validexactly in the particular look direction Moreover thetracking algorithm regards all incoming sensor data

30 5050

70

70

90

90

110

110

130

130

10

SensorTarget

Standoff jammerTarget in jammer notch

East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)

Figure 19 Target tracking scenario with standoff jammer geo-graphic plot of platform trajectories

as unbiased measurements To avoid track instabil-ities an acceptance region is defined for each mea-surement depending on the predicted target state andthe assumed measurement accuracy Sensor reportslying outside this gate are considered as invalid

In order to evaluate our stabilizationmeasures we considereda realistic air-to-air target tracking scenario [41] Figure 19provides an overview of the different platform trajectoriesIn this scenario the sensor (on a forward looking radarplatform flying with a constant speed of 265ms) employsthe antenna array of Figure 2 (sum beamwidth BW = 34∘field of view 120∘ scan interval 1 s) and approaches the target(at velocity 300ms) which thereupon veers away after ashort time During this time the target is hidden twice inthe jammer notch of the standoff jammer (SOJ)mdashfirst for3 s and then again for 4 s The SOJ is on patrol (at 235ms)and follows a predefined race track at constant altitudeFigure 20 shows exemplary the evaluation of the azimuthmeasurements and estimates over time in a window wherethe target first passes through the jammer notchThedifferenterror bars of a single measurement illustrate the approxi-mation error of the variable measurement covariance 120590SIM

119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu

Jammer notchTarget in notchAzimuth (truth)

PseudomeasurementNo detectionGating120590SIM

u119896

120590FILTu119896

Measurement (update)

Figure 20 Exemplary azimuth measurements and models for aspecific time window for tracking scenario of Figure 19

denotes the true azimuth standard deviation (std) which isgenerated in the antenna simulation120590FILT

119906119896corresponds to the

std which is used in the tracking algorithm More preciselythe tracking program computes the adaptive angle measure-ment covariance only in the vicinity of the jammer with adiameter of this zone of 85∘ Outside of this region thetracking algorithm uses a constant std of 0004 for bothcomponents of the angle measurement The constant std forthe other parameters are 75m and 75ms for range andrange-rate measurements The signal-to-noise and jammer-to-noise ratioswere set to 26 dB and 27 dB at a reference rangeof 70 km From Figure 20 the benefits of using pseudobearingmeasurements become apparent

From these investigations it turned out that tracking onlywith adaptive beamforming and adaptive monopulse nearlyalways leads to track loss in the vicinity of the jammer Withadditional stabilization measures that did not require theknowledge of the jammer direction (projection ofmonopulseestimate detection threshold LocSearch gating) still trackinstabilities occurred culminating finally in track loss Anadvanced tracking version which used pseudomeasurementsmitigated this problem to some degree Finally the additionalconsideration of the variable measurement covariance witha better estimate of the highly variable shape of the angleuncertainty ellipse resulted in significantly fewer measure-ments that were excluded due to gating In this case all the sta-bilization measures could not only improve track continuitybut also track accuracy and thus track stability [41]This tellsus that it is absolutely necessary to use all information ofthe adaptive process for the tracker to achieve the goal ofdetection and tracking in the vicinity of the interference

8 Conclusions and Final Remarks

In this paper we have pointed out the links between arraysignal processing and antenna design hardware constraintsand target detection and parameter estimation and trackingMore specifically we have discussed the following features

(i) Interference suppression by deterministic and adap-tive pattern shaping both approaches can be rea-sonably combined Applying ABF after deterministicsidelobe reduction allows reducing the requirementson the low sidelobe level Special techniques areavailable tomake ABF preserve the low sidelobe level

(ii) General principles and relationships between ABFalgorithms and superresolution methods have beendiscussed like dependency on the sample numberrobustness the benefits of subspace methods prob-lems of determining the signalinterference subspaceand interference suppressionresolution limit

(iii) Array signal processing methods like adaptive beam-forming and superresolution methods can be appliedto subarrays generated from a large fully filled arrayThis means applying these methods to the sparsesuperarray formed by the subarray centers We havepointed out problems and solutions for this specialarray problem

(iv) ABF can be combined with superresolution in acanonical way by applying the pre-whiten and matchprinciple to the data and the signal model vector

(v) All array signal processing methods can be extendedto space-time processing (arrays) by defining a corre-sponding space-time plane wave model

(vi) Superresolution is a joint detection-estimation prob-lem One has to determine a multitarget model whichcontains the number directions and powers of the tar-getsThese parameters are strongly coupled A practi-cal joint estimation and detection procedure has beenpresented

(vii) The problems for implementation in real system havebeen discussed in particular the effects of limitedknowledge of the array manifold effect of channelerrors eigenvalue leakage unequal noise power inarray channels and dynamic range of AD-converters

(viii) For achieving best performance an adaptation of theprocessing subsequent to ABF is necessary Directionestimation can be accommodated by using ABF-monopulse the detector can be accommodated byadaptive detection with ASLB and the tracking algo-rithms can be extended to adaptive tracking and trackmanagement with jammer mapping

With a single array signal processing method alone nosignificant improvement will be obtained The methods haveto be reasonably embedded in the whole system and allfunctionalities have to be mutually tuned and balancedThis is a task for future research The presented approachesconstitute only a first ad hoc step and more thorough studiesare required Note that in most cases tuning the functionali-ties is mainly a software problem So there is the possibilityto upgrade existing systems softly and step-wise


Disclosure

The main part of this work was performed while the authorwas with the Fraunhofer Institute forHigh Frequency Physicsand Radar Techniques (FHR) in Wachtberg Germany

References

[1] M I SkolnikRadarHandbookMcGrawHill 2nd edition 1990[2] M A Richards J A Scheer and W A Holden Principles of

Modern Radar SciTech Publishing 2010[3] W D Wirth Radar Techniques Using Array Antennas IEE

Publishers 2001[4] R Klemm Principles of Space-Time Adaptive Processing IET

Publishers London UK 3rd edition 2006[5] J H G Ender ldquoOn compressive sensing applied to radarrdquo Signal

Processing vol 90 no 5 pp 1402ndash1414 2010[6] U Nickel ldquoSubarray configurations for digital beamforming

with low sidelobes and adaptive interference suppressionrdquo inProceedings of the IEEE International Radar Conference pp 714ndash719 Alexandria Egypt May 1995

[7] U Nickel ldquoProperties of digital beamforming with subarraysrdquoin Proceedings of the International Conference on Radar (CIErsquo06) pp 6ndash19 Shanghai China October 2006

[8] W Burger ldquoSidelobe forming for ground clutter and jammersuppression for airborne active array radarrdquo in Proceedings ofthe IEEE International Symposium on Phased Array Systems andTechnology Boston Mass USA 2003

[9] J Liu A B Gershman Z-Q Luo and K M Wong ldquoAdaptivebeamforming with sidelobe control a second-order cone pro-gramming approachrdquo IEEE Signal Processing Letters vol 10 no11 pp 331ndash334 2003

[10] S A Vorobyov A B Gershman Z-Q Luo and N Ma ldquoAdap-tive beamforming with joint robustness against mismatchedsignal steering vector and interference nonstationarityrdquo IEEESignal Processing Letters vol 11 no 2 pp 108ndash111 2004

[11] G A Fabrizio A B Gershman and M D Turley ldquoRobustadaptive beamforming for HF surface wave over-the-horizonradarrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 40 no 2 pp 510ndash525 2004

[12] Y Hua A B Gershman and Q Cheng High Resolution andRobust Signal Processing Marcel Dekker 2004

[13] U Nickel ldquoAdaptive Beamforming for Phased Array Radarsrdquoin Proceedings of the IEEE International Radar Symposium (IRSrsquo98) pp 897ndash906 DGON and VDEITG September 1998

[14] C H Gierull ldquoFast and effective method for low-rank inter-ference suppression in presence of channel errorsrdquo ElectronicsLetters vol 34 no 6 pp 518ndash520 1998

[15] G M Herbert ldquoNew projection based algorithm for lowsidelobe pattern synthesis in adaptive arraysrdquo in Proceedingsof the Radar Edinburgh International Conference pp 396ndash400October 1997

[16] U Nickel ldquoDetermination of the dimension of the signalsubspace for small sample sizerdquo in Proceedings of the IASTEDinternational conference on Signal Processing and Communica-tion Systems pp 119ndash122 IASTEDActa Press 1998

[17] S Kritchman and B Nadler ldquoNon-parametric detection ofthe number of signals hypothesis testing and random matrixtheoryrdquo IEEE Transactions on Signal Processing vol 57 no 10pp 3930ndash3941 2009

[18] U Nickel ldquoOn the influence of channel errors on array signalprocessing methodsrdquo International Journal of Electronics andCommunications vol 47 no 4 pp 209ndash219 1993

[19] R J Muirhead Aspects of Multivariate Analysis Theory JohnWiley amp Sons New York NY USA 1982

[20] R R Nadakuditi and A Edelman ldquoSample eigenvalue baseddetection of high-dimensional signals in white noise usingrelatively few samplesrdquo IEEE Transactions on Signal Processingvol 56 no 7 pp 2625ndash2638 2008

[21] U Nickel ldquoRadar target parameter estimation with antennaarraysrdquo in Radar Array Processing S Haykin J Litva and T JShepherd Eds pp 47ndash98 Springer 1993

[22] S Haykin Advances in Spectrum Analysis and Array ProcessingVol II Prentice Hall 1991

[23] U Nickel ldquoAspects of implementing super-resolution methodsinto phased array radarrdquo International Journal of Electronics andCommunications vol 53 no 6 pp 315ndash323 1999

[24] U Nickel ldquoOverview of generalized monopulse estimationrdquoIEEE Aerospace and Electronic Systems Magazine vol 21 no 6pp 27ndash55 2006

[25] U Nickel ldquoSpotlight MUSIC super-resolution with subarrayswith low calibration effortrdquo IEE Proceedings vol 149 no 4 pp166ndash173 2002

[26] B Ottersten P Stoica and R Roy ldquoCovariance matchingestimation techniques for array signal processing applicationsrdquoDigital Signal Processing vol 8 no 3 pp 185ndash210 1998

[27] Y I Abramovich N K Spencer and A Y Gorokhov ldquoGLRT-based threshold detection-estimation performance improve-ment and application to uniform circular antenna arraysrdquo IEEETransactions on Signal Processing vol 55 no 1 pp 20ndash31 2007

[28] Y I Abramovich and B A Johnson ldquoDetection-estimation ofvery close emitters performance breakdown ambiguity andgeneral statistical analysis of maximum-likelihood estimationrdquoIEEE Transactions on Signal Processing vol 58 no 7 pp 3647ndash3660 2010

[29] E J Kelly ldquoPerformance of an adaptive detection algorithmrejection of unwanted signalsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 25 no 2 pp 122ndash133 1992

[30] F C Robey D R Fuhrmann E J Kelly and R Nitzberg ldquoACFAR adaptive matched filter detectorrdquo IEEE Transactions onAerospace and Electronic Systems vol 28 no 1 pp 208ndash2161992

[31] S Kraut and L L Scharf ldquoAdaptive subspace detectorsrdquo IEEETransactions on Signal Processing vol 49 no 1 pp 1ndash16 2001

[32] C D Richmond ldquoPerformance of the adaptive sidelobe blankerdetection algorithm in homogeneous environmentsrdquo IEEETransactions on Signal Processing vol 48 no 5 pp 1235ndash12472000

[33] C D Richmond ldquoPerformance of a class of adaptive detectionalgorithms in nonhomogeneous environmentsrdquo IEEE Transac-tions on Signal Processing vol 48 no 5 pp 1248ndash1262 2000

[34] T F Ayoub and A M Haimovich ldquoModified GLRT signaldetection algorithmrdquo IEEE Transactions on Aerospace andElectronic Systems vol 36 no 3 pp 810ndash818 2000

[35] Y I Abramovich N K Spencer and A Y Gorokhov ldquoModifiedGLRT and AMF framework for adaptive detectorsrdquo IEEETransactions on Aerospace and Electronic Systems vol 43 no3 pp 1017ndash1051 2007

[36] O Besson J-Y Tourneret and S Bidon ldquoKnowledge-aidedBayesian detection in heterogeneous environmentsrdquo IEEE Sig-nal Processing Letters vol 14 no 5 pp 355ndash358 2007


[37] A De Maio S De Nicola A Farina and S Iommelli ldquoAdaptivedetection of a signal with angle uncertaintyrdquo IET Radar Sonarand Navigation vol 4 no 4 pp 537ndash547 2010

[38] A De Maio and E Conte ldquoAdaptive detection in gaussianinterference with unknown covariance after reduction byinvariancerdquo IEEE Transactions on Signal Processing vol 58 no6 pp 2925ndash2934 2010

[39] U Nickel ldquoDesign of generalised 2D adaptive sidelobe blankingdetectors using the detectionmarginrdquo Signal Processing vol 90no 5 pp 1357ndash1372 2010

[40] W R Blanding W Koch and U Nickel ldquoAdaptive phased-array tracking in ECM using negative informationrdquo IEEETransactions on Aerospace and Electronic Systems vol 45 no1 pp 152ndash166 2009

[41] M Feldmann and U Nickel ldquoTarget parameter estimation andtracking with adaptive beamformingrdquo in Proceedings of theInternational Radar Symposium (IRS rsquo11) pp 585ndash590 LeipzigGermany September 2011

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Submit your manuscripts athttpwwwhindawicom

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



we point out how direction estimation must be modified ifadaptive beams are used and how the radar detector thetracking algorithm and the track management should beadapted for ABF

2 Design Factors for Arrays

Array processing starts with the array antenna This is hard-ware and is a selected construction that cannot be alteredIt must therefore be carefully designed to fulfill all require-ments Digital array processing requires digital array outputsThe number and quality of these receivers (eg linearity andnumber of ADC bits) determine the quality and the cost ofthe whole system It may be desirable to design a fully digitalarray with AD-converters at each antenna element Howeverweight and cost will often lead to a system with reducednumber of digital receivers On the other hand becauseof the 11198774-decay of the received power radar needs antennaswith high gain and high directional discrimination whichmeans arrays with many elements There are different solu-tions to solve this contradiction

(i) Thinned arrays the angular discrimination of anarray with a number of elements can be improvedby increasing the separation of the elements and thusincreasing the aperture of the antenna Note that thethinned array has the same gain as the correspondingfully filled array

(ii) Subarrays element outputs are summed up in ananalog manner into subarrays which are then AD-converted and processed digitally The size and theshape of the subarrays are an important design cri-terion The notion of an array with subarrays is verygeneral and includes the case of steerable directionalarray elements

21 Impact of the Dimensionality of the Array Antenna ele-ments may be arranged on a line (1-dimensional array) on aplane (a ring or a 2-dimensional planar array) on a curvedsurface (conformal array) or within a volume (3D-arrayalso called Crowrsquos Nest antenna [3 Section 461]) A 1-dimensional array can only measure one independent angle2D and 3D arrays can measure the full polar coordinates inR3

Antenna element design and the need for fixing elementsmechanically lead to element patterns which are never omni-directional The elements have to be designed with patternsthat allow a unique identification of the direction Typicallya planar array can only observe a hemispherical half spaceTo achieve full spherical coverage several planar arrays canbe combined (multifacetted array) or a conformal or volumearray may be used

211 Arrays with Equal Patterns For linear planar and vol-ume arrays elements with nearly equal patterns can berealized These have the advantage that the knowledge of theelement pattern is for many array processing methods notnecessary An equal complex value can be interpreted as amodified target complex amplitude which is often a nuisance

parameter More important is that if the element patternsare really absolutely equal any cross-polar components of thesignal are in all channels equal and fulfill the array model inthe sameway as the copolar components that is they produceno error effect

212 Arrays with Unequal Array Patterns This occurs typi-cally by tilting the antenna elements as is done for conformalarrays For a planar array this tilt may be used to realize anarray with polarization diversity Single polarized elementsare then mounted with orthogonal alignment at differentpositions Such an array can provide some degree of dualpolarization receptionwith single channel receivers (contraryto more costly fully polarimetric arrays with receivers forboth polarizations for each channel)

Common to arrays with unequal patterns is that we haveto know the element patterns for applying array processingmethods The full element pattern function is also called thearray manifold In particular the cross-polar (or short 119909-pol)component has a different influence for each element Thismeans that if this component is not known and if the 119909-polcomponent is not sufficiently attenuated it can be a signifi-cant source of error

22 Thinned Arrays To save the cost of receiving modulessparse arrays are considered that is with fewer elements thanthe full populated 1205822 grid Because such a ldquothinned arrayrdquospans the same aperture it has the same beamwidth Hencethe angular accuracy and resolution are the same as the fullyfilled array Due to the gaps ambiguities or at least high side-lobesmay arise In early publications like [1] it was advocatedto simply take out elements of the fully filled array It wasearly recognized that this kind of thinning does not implyldquosufficiently randomrdquo positions Random positions on a 12058216grid as used in [3 Chapter 17] can provide quite acceptablepatterns Note that the array gain of a thinned array with 119873elements is always 119873 and the average sidelobe level is 1119873Today we know from the theory of compressed sensing thata selection of sufficiently random positions can produce aunique reconstruction of a not too large number of impingingwave fields with high probability [5]

23 Arrays with Subarrays If a high antenna gain with lowsidelobes is desired one has to go back to the fully filledarray For large arrays with thousands of elements the largenumber of digital channel constitutes a significant cost factorand a challenge for the resulting data rate Therefore oftensubarrays are formed and all digital (adaptive) beamformingand sophisticated array processing methods are applied tothe subarray outputs Subarraying is a very general conceptAt the elements we may have phase shifters such that allsubarrays are steered into a given direction and wemay applysome attenuation (tapering) to influence the sidelobe levelThe sum of the subarrays then gives the sum beam outputThe subarrays can be viewed as a superarray with elementshaving different patterns steered into the selected directionThe subarrays should have unequal size and shape to avoidgrating effects for subsequent array processing because thesubarray centers constitute a sparse array (for details see [6])


Analogue

Digital

sumADC withbandpass

matched filtering

(1)

(2)



m m m(n)




0 V0) =


119894(u0) = 119890




119894119886119894(u0) which can be




119910 = m119867z (1)


119891 (u) = m119867T119867a (u) = m119867a (u) (2)

where a(u) = (119886119894(u))








minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0(d

B)



sin (120579)




0 5 10 15 20 25 30 35 4001

02

03

04

05

06

07

08

09

1

n






119870 we have


1 z

119870) of the form

119878 = m119867

119904Zm

119905 (3)

where m119904 m



119878 = (m119905otimesm

119904)119867 vec Z (4)





1198921198941198901198952120587119891r119879



119894= cos](120587119909


120572) or 119892119894= 119890




minw int

Ω

10038161003816100381610038161003816w119867a (u)10038161003816100381610038161003816

2

119901 (u) 119889u


minw wCw

st w119867a0= 1

with C = int

Ω

a (u) a(u)119867119901 (u) 119889u

(5)


1

05

0

0

minus05


minus20

minus40

minus60

minus80

minus100

u

1

05

0

0

minus05


u

1

05

0

0

minus05


minus20

minus40

minus60

minus80

minus100

u

1

05

0

0

minus05


u





w =Cminus1a

0

a1198670Cminus1a

0

(6)


| 1199062+ V2 le 1


1 u


119872

119896=1120575(u minus


w =Pa

0

a1198670Pa

0


A119867

A = (a (u1) a (u

119872))

(7)









0

maxw

10038161003816100381610038161003816w119867a

0

10038161003816100381610038161003816

2

119864 1003816100381610038161003816w119867n

1003816100381610038161003816

2

= maxw

10038161003816100381610038161003816w119867a

0

10038161003816100381610038161003816

2

w119867Qw (8)


w = 120583Qminus1a0

with Q = 119864 nn119867 (9)



0Qminus1z = a119867

0L119867Lz = (La

0)119867(Lz)


0







119894= 119896


119894




minw w119867Q w

st w119867C = k (or w119867c119894= 119896

119894 119894 = 1 sdot sdot sdot 119903)

(10)


w =

119903

sum

119894=1

120582119894Qminus1c


minus1

k (11)




0)minus1Qminus1a




0=










optimization

minw w119867Qw +

119903

sum

119894=1

119887119894

10038161003816100381610038161003816w119867c

119894minus 119896

119894

10038161003816100381610038161003816

2

or



(12)

with B = diag1198871 119887


optimization is


CBk (13)




minus50

minus40

minus30

minus20

minus10

0

u

20 20 20

(dB)


minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

20 2020

SNIR

(dB)


u

(b) SNIR







1 z




QSMI =1

119870

119870

sum

119896=1

z119896z119867119896 (14)


QLSMI =1

119870

119870

sum

119896=1

z119896z119867119896+ 120575 sdot I (15)






minus50

minus40

minus30

minus20

minus10

0

u

20 20 20

(dB)


minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

20 20 20

SNIR

(dB)


u

(b) SNIR


minus30

minus25

minus20

minus15

minus10

minus5

0

SNIR

(dB)

2020

20

LSMI0LSMI4

EVP10

Optimum



with dim (JSS) = 3


1(119905) 119887

119872(119905) in

directionsu1 u


119896) =

sum119872

119898=1a(u

119898)119887119898(119905119896) + n(119905

119896) or short z

119896= Ab

119896+ n

119896 then

119864 zz119867 = Q = ABA119867 + I (16)




minus1

A119867

997888rarr

Brarrinfin


A119867 = PperpA(17)


119883= I minus XX119867








wCAPS =1



perpCX

perp)minus1

times X119867perpC(

1


SMIm minusm)

(19)





wCAPS = m + P[Xm] (

1


SMIm minusm) (20)





119879 (119898) =(1 (119873 minus 119898))sum

119873

119894=119898+1120582119894

(prod119873

119894=119898+1120582119894)1(119873minus119898)

(21)





(22)


119871 (119898) =2119870

1205902

119873

sum

119894=119898+1

120582119894 (23)

no

12

34

56

24

68

1012

0

20

40

60

80

100

dimSS






if 119871 (119898) le 1205942

2119870(119873minus119898)120572 = 119898 STOP

end

(24)

The symbol 1205942119903120572



120582119894=

119894minus1

119870119894

119873

sum

119895=1

119895 = 119894

119894

119894minus

119895

(25)


2 The








Caponrsquos method

119878119862(u) = (a(u)119867Rminus1MLa (u))

minus1

with RML =1

119870

119870

sum

119896=1

z119896z119867119896

(26)



(27)





A119867

A = (a (u1) a (u

119872))

(28)




119865sto (120579) = det A (120579)B (120579)A119867 (120579) + 1205902 (120579) I with

1205902(120579) =

1


B (120579) = (A119867A)minus1

A119867 (RML minus 1205902(120579) I)A(A119867A)

minus1

for A = A (120579)

(30)


(1) 119865det(120579)= (1119870)sum119870

119896=1 z119867

119896PperpAz119896 = (1119870)sum

119870

119896=1PperpAz119896

2

=

(1119870)sum119870


119896⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=b

2

which means


(2) 119865det(120579) = 119862 minus sum119870

119896=1z119867119896A(A119867A)minus1A119867z

119896 which can


1)zk a119867(u119872)zk)



PperpGminusPperp

GF(F119867PperpGF)




0

10

20

30

40

50

60

70

80

u

1 12 12

(dB)

(a) Simulated data

minus05

0

05

minus05

0

050

5

10

15

20

u

(dB)













1205902(120579)I)A(A119867A)minus1 with 120590




257 308 358 408 459 509 560 610 661 711 762 812 862

10

06

02

minus02

minus06

Range (km)

Elev

atio

n (B






SNR (dB)

101

203

304

406

507

608

710

811


minus05 0 05

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

u






K range cells

M pulses

N channels














0 V0)119879 is

(

V) = (

1199060

V0

) minus (119888119909119909

119888119909119910

119888119910119909

119888119910119910

)(119877119909minus 120583

119909

119877119910minus 120583

119910

) (31)

whereC = (119888119909119909 119888119909119910


120583119909120583119910)




119909= d119867





119909(119906 V) 119877

119910(119906 V))119879 The cor-




M (119906 V) = C sdot (119864 R (119906 V) minus 120583) (32)

we require

M (1199060 V0) = 0

120597M120597119906

(1199060 V0) = (

1

0)

120597M120597V

(1199060 V0) = (

0

1) or

C(120597R120597119906

120597R120597V

) (1199060 V0) = I

(33)


0= a(119906

0 V0) [24]

120583120572= Re

d119867120572a0

w119867a0

for 120572 = 119909 119910 (34)



119888120572ℎ

=

Re d119867120572aℎ0a1198670w + d119867

120572a0a119867ℎ0w

1003816100381610038161003816w119867a01003816100381610038161003816

2minus 120583

1198862Re

w119867aℎ0

w119867a0

(35)





119890(1 1)


a1198791199060

= 119866119890(1198952120587119891119888)(119909

1 119909

119873)



minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u


minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u






119879GLRT (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (1 + (1119870) z119867Qminus1

SMIz) (36)

119879AMF (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (37)

119879ACE (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1


SMIz (38)



0


(i) 119879GLRT =119879AMF

1 + (1119870) z119867Qminus1

SMIz 119879ACE =

119879AMF

z119867Qminus1

SMIz

(39)


119879AMF =

10038161003816100381610038161003816w119867z10038161003816100381610038161003816

2


SMIa0 (40)






119866 =

119871

sum

119894=1

119892119894

10038161003816100381610038161198941003816100381610038161003816

2

(41)



119871



119866 = z119867KDK119867z 119878119866(u) = a119867 (u)KDK119867a (u) (42)





minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)

minus35 dB Taylor


u

ni = 32


minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u



minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u









minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

34



34

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u




minus30

minus20

minus10

0

10

20

30

40

50

(dB)





u









Elev

atio

n

rarr

Azimuth u rarr


BW120572 middot BW


Projection

(a)


BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)







30 5050

70

70

90

90

110

110

130

130

10

SensorTarget


East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)




119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Analogue

Digital

sumADC withbandpass

matched filtering

(1)

(2)



m m m(n)




0 V0) =


119894(u0) = 119890




119894119886119894(u0) which can be




119910 = m119867z (1)


119891 (u) = m119867T119867a (u) = m119867a (u) (2)

where a(u) = (119886119894(u))








minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0(d

B)



sin (120579)




0 5 10 15 20 25 30 35 4001

02

03

04

05

06

07

08

09

1

n






119870 we have


1 z

119870) of the form

119878 = m119867

119904Zm

119905 (3)

where m119904 m



119878 = (m119905otimesm

119904)119867 vec Z (4)





1198921198941198901198952120587119891r119879



119894= cos](120587119909


120572) or 119892119894= 119890




minw int

Ω

10038161003816100381610038161003816w119867a (u)10038161003816100381610038161003816

2

119901 (u) 119889u


minw wCw

st w119867a0= 1

with C = int

Ω

a (u) a(u)119867119901 (u) 119889u

(5)


1

05

0

0

minus05


minus20

minus40

minus60

minus80

minus100

u

1

05

0

0

minus05


u

1

05

0

0

minus05


minus20

minus40

minus60

minus80

minus100

u

1

05

0

0

minus05


u





w =Cminus1a

0

a1198670Cminus1a

0

(6)


| 1199062+ V2 le 1


1 u


119872

119896=1120575(u minus


w =Pa

0

a1198670Pa

0


A119867

A = (a (u1) a (u

119872))

(7)









0

maxw

10038161003816100381610038161003816w119867a

0

10038161003816100381610038161003816

2

119864 1003816100381610038161003816w119867n

1003816100381610038161003816

2

= maxw

10038161003816100381610038161003816w119867a

0

10038161003816100381610038161003816

2

w119867Qw (8)


w = 120583Qminus1a0

with Q = 119864 nn119867 (9)



0Qminus1z = a119867

0L119867Lz = (La

0)119867(Lz)


0







119894= 119896


119894




minw w119867Q w

st w119867C = k (or w119867c119894= 119896

119894 119894 = 1 sdot sdot sdot 119903)

(10)


w =

119903

sum

119894=1

120582119894Qminus1c


minus1

k (11)




0)minus1Qminus1a




0=










optimization

minw w119867Qw +

119903

sum

119894=1

119887119894

10038161003816100381610038161003816w119867c

119894minus 119896

119894

10038161003816100381610038161003816

2

or



(12)

with B = diag1198871 119887


optimization is


CBk (13)




minus50

minus40

minus30

minus20

minus10

0

u

20 20 20

(dB)


minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

20 2020

SNIR

(dB)


u

(b) SNIR







1 z




QSMI =1

119870

119870

sum

119896=1

z119896z119867119896 (14)


QLSMI =1

119870

119870

sum

119896=1

z119896z119867119896+ 120575 sdot I (15)






minus50

minus40

minus30

minus20

minus10

0

u

20 20 20

(dB)


minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

20 20 20

SNIR

(dB)


u

(b) SNIR


minus30

minus25

minus20

minus15

minus10

minus5

0

SNIR

(dB)

2020

20

LSMI0LSMI4

EVP10

Optimum



with dim (JSS) = 3


1(119905) 119887

119872(119905) in

directionsu1 u


119896) =

sum119872

119898=1a(u

119898)119887119898(119905119896) + n(119905

119896) or short z

119896= Ab

119896+ n

119896 then

119864 zz119867 = Q = ABA119867 + I (16)




minus1

A119867

997888rarr

Brarrinfin


A119867 = PperpA(17)


119883= I minus XX119867








wCAPS =1



perpCX

perp)minus1

times X119867perpC(

1


SMIm minusm)

(19)





wCAPS = m + P[Xm] (

1


SMIm minusm) (20)





119879 (119898) =(1 (119873 minus 119898))sum

119873

119894=119898+1120582119894

(prod119873

119894=119898+1120582119894)1(119873minus119898)

(21)





(22)


119871 (119898) =2119870

1205902

119873

sum

119894=119898+1

120582119894 (23)

no

12

34

56

24

68

1012

0

20

40

60

80

100

dimSS






if 119871 (119898) le 1205942

2119870(119873minus119898)120572 = 119898 STOP

end

(24)

The symbol 1205942119903120572



120582119894=

119894minus1

119870119894

119873

sum

119895=1

119895 = 119894

119894

119894minus

119895

(25)


2 The








Caponrsquos method

119878119862(u) = (a(u)119867Rminus1MLa (u))

minus1

with RML =1

119870

119870

sum

119896=1

z119896z119867119896

(26)



(27)





A119867

A = (a (u1) a (u

119872))

(28)




119865sto (120579) = det A (120579)B (120579)A119867 (120579) + 1205902 (120579) I with

1205902(120579) =

1


B (120579) = (A119867A)minus1

A119867 (RML minus 1205902(120579) I)A(A119867A)

minus1

for A = A (120579)

(30)


(1) 119865det(120579)= (1119870)sum119870

119896=1 z119867

119896PperpAz119896 = (1119870)sum

119870

119896=1PperpAz119896

2

=

(1119870)sum119870


119896⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=b

2

which means


(2) 119865det(120579) = 119862 minus sum119870

119896=1z119867119896A(A119867A)minus1A119867z

119896 which can


1)zk a119867(u119872)zk)



PperpGminusPperp

GF(F119867PperpGF)




0

10

20

30

40

50

60

70

80

u

1 12 12

(dB)

(a) Simulated data

minus05

0

05

minus05

0

050

5

10

15

20

u

(dB)













1205902(120579)I)A(A119867A)minus1 with 120590




257 308 358 408 459 509 560 610 661 711 762 812 862

10

06

02

minus02

minus06

Range (km)

Elev

atio

n (B






SNR (dB)

101

203

304

406

507

608

710

811


minus05 0 05

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

u






K range cells

M pulses

N channels














0 V0)119879 is

(

V) = (

1199060

V0

) minus (119888119909119909

119888119909119910

119888119910119909

119888119910119910

)(119877119909minus 120583

119909

119877119910minus 120583

119910

) (31)

whereC = (119888119909119909 119888119909119910


120583119909120583119910)




119909= d119867





119909(119906 V) 119877

119910(119906 V))119879 The cor-




M (119906 V) = C sdot (119864 R (119906 V) minus 120583) (32)

we require

M (1199060 V0) = 0

120597M120597119906

(1199060 V0) = (

1

0)

120597M120597V

(1199060 V0) = (

0

1) or

C(120597R120597119906

120597R120597V

) (1199060 V0) = I

(33)


0= a(119906

0 V0) [24]

120583120572= Re

d119867120572a0

w119867a0

for 120572 = 119909 119910 (34)



119888120572ℎ

=

Re d119867120572aℎ0a1198670w + d119867

120572a0a119867ℎ0w

1003816100381610038161003816w119867a01003816100381610038161003816

2minus 120583

1198862Re

w119867aℎ0

w119867a0

(35)





119890(1 1)


a1198791199060

= 119866119890(1198952120587119891119888)(119909

1 119909

119873)



minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u


minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u






119879GLRT (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (1 + (1119870) z119867Qminus1

SMIz) (36)

119879AMF (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (37)

119879ACE (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1


SMIz (38)



0


(i) 119879GLRT =119879AMF

1 + (1119870) z119867Qminus1

SMIz 119879ACE =

119879AMF

z119867Qminus1

SMIz

(39)


119879AMF =

10038161003816100381610038161003816w119867z10038161003816100381610038161003816

2


SMIa0 (40)






119866 =

119871

sum

119894=1

119892119894

10038161003816100381610038161198941003816100381610038161003816

2

(41)



119871



119866 = z119867KDK119867z 119878119866(u) = a119867 (u)KDK119867a (u) (42)





minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)

minus35 dB Taylor


u

ni = 32


minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u



minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u









minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

34



34

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u




minus30

minus20

minus10

0

10

20

30

40

50

(dB)





u









Elev

atio

n

rarr

Azimuth u rarr


BW120572 middot BW


Projection

(a)


BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)







30 5050

70

70

90

90

110

110

130

130

10

SensorTarget


East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)




119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0(d

B)



sin (120579)




0 5 10 15 20 25 30 35 4001

02

03

04

05

06

07

08

09

1

n






119870 we have


1 z

119870) of the form

119878 = m119867

119904Zm

119905 (3)

where m119904 m



119878 = (m119905otimesm

119904)119867 vec Z (4)





1198921198941198901198952120587119891r119879



119894= cos](120587119909


120572) or 119892119894= 119890




minw int

Ω

10038161003816100381610038161003816w119867a (u)10038161003816100381610038161003816

2

119901 (u) 119889u


minw wCw

st w119867a0= 1

with C = int

Ω

a (u) a(u)119867119901 (u) 119889u

(5)


1

05

0

0

minus05


minus20

minus40

minus60

minus80

minus100

u

1

05

0

0

minus05


u

1

05

0

0

minus05


minus20

minus40

minus60

minus80

minus100

u

1

05

0

0

minus05


u





w =Cminus1a

0

a1198670Cminus1a

0

(6)


| 1199062+ V2 le 1


1 u


119872

119896=1120575(u minus


w =Pa

0

a1198670Pa

0


A119867

A = (a (u1) a (u

119872))

(7)









0

maxw

10038161003816100381610038161003816w119867a

0

10038161003816100381610038161003816

2

119864 1003816100381610038161003816w119867n

1003816100381610038161003816

2

= maxw

10038161003816100381610038161003816w119867a

0

10038161003816100381610038161003816

2

w119867Qw (8)


w = 120583Qminus1a0

with Q = 119864 nn119867 (9)



0Qminus1z = a119867

0L119867Lz = (La

0)119867(Lz)


0







119894= 119896


119894




minw w119867Q w

st w119867C = k (or w119867c119894= 119896

119894 119894 = 1 sdot sdot sdot 119903)

(10)


w =

119903

sum

119894=1

120582119894Qminus1c


minus1

k (11)




0)minus1Qminus1a




0=










optimization

minw w119867Qw +

119903

sum

119894=1

119887119894

10038161003816100381610038161003816w119867c

119894minus 119896

119894

10038161003816100381610038161003816

2

or



(12)

with B = diag1198871 119887


optimization is


CBk (13)




minus50

minus40

minus30

minus20

minus10

0

u

20 20 20

(dB)


minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

20 2020

SNIR

(dB)


u

(b) SNIR







1 z




QSMI =1

119870

119870

sum

119896=1

z119896z119867119896 (14)


QLSMI =1

119870

119870

sum

119896=1

z119896z119867119896+ 120575 sdot I (15)






minus50

minus40

minus30

minus20

minus10

0

u

20 20 20

(dB)


minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

20 20 20

SNIR

(dB)


u

(b) SNIR


minus30

minus25

minus20

minus15

minus10

minus5

0

SNIR

(dB)

2020

20

LSMI0LSMI4

EVP10

Optimum



with dim (JSS) = 3


1(119905) 119887

119872(119905) in

directionsu1 u


119896) =

sum119872

119898=1a(u

119898)119887119898(119905119896) + n(119905

119896) or short z

119896= Ab

119896+ n

119896 then

119864 zz119867 = Q = ABA119867 + I (16)




minus1

A119867

997888rarr

Brarrinfin


A119867 = PperpA(17)


119883= I minus XX119867








wCAPS =1



perpCX

perp)minus1

times X119867perpC(

1


SMIm minusm)

(19)





wCAPS = m + P[Xm] (

1


SMIm minusm) (20)





119879 (119898) =(1 (119873 minus 119898))sum

119873

119894=119898+1120582119894

(prod119873

119894=119898+1120582119894)1(119873minus119898)

(21)





(22)


119871 (119898) =2119870

1205902

119873

sum

119894=119898+1

120582119894 (23)

no

12

34

56

24

68

1012

0

20

40

60

80

100

dimSS






if 119871 (119898) le 1205942

2119870(119873minus119898)120572 = 119898 STOP

end

(24)

The symbol 1205942119903120572



120582119894=

119894minus1

119870119894

119873

sum

119895=1

119895 = 119894

119894

119894minus

119895

(25)


2 The








Caponrsquos method

119878119862(u) = (a(u)119867Rminus1MLa (u))

minus1

with RML =1

119870

119870

sum

119896=1

z119896z119867119896

(26)



(27)





A119867

A = (a (u1) a (u

119872))

(28)




119865sto (120579) = det A (120579)B (120579)A119867 (120579) + 1205902 (120579) I with

1205902(120579) =

1


B (120579) = (A119867A)minus1

A119867 (RML minus 1205902(120579) I)A(A119867A)

minus1

for A = A (120579)

(30)


(1) 119865det(120579)= (1119870)sum119870

119896=1 z119867

119896PperpAz119896 = (1119870)sum

119870

119896=1PperpAz119896

2

=

(1119870)sum119870


119896⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=b

2

which means


(2) 119865det(120579) = 119862 minus sum119870

119896=1z119867119896A(A119867A)minus1A119867z

119896 which can


1)zk a119867(u119872)zk)



PperpGminusPperp

GF(F119867PperpGF)




0

10

20

30

40

50

60

70

80

u

1 12 12

(dB)

(a) Simulated data

minus05

0

05

minus05

0

050

5

10

15

20

u

(dB)













1205902(120579)I)A(A119867A)minus1 with 120590




257 308 358 408 459 509 560 610 661 711 762 812 862

10

06

02

minus02

minus06

Range (km)

Elev

atio

n (B






SNR (dB)

101

203

304

406

507

608

710

811


minus05 0 05

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

u






K range cells

M pulses

N channels














0 V0)119879 is

(

V) = (

1199060

V0

) minus (119888119909119909

119888119909119910

119888119910119909

119888119910119910

)(119877119909minus 120583

119909

119877119910minus 120583

119910

) (31)

whereC = (119888119909119909 119888119909119910


120583119909120583119910)




119909= d119867





119909(119906 V) 119877

119910(119906 V))119879 The cor-




M (119906 V) = C sdot (119864 R (119906 V) minus 120583) (32)

we require

M (1199060 V0) = 0

120597M120597119906

(1199060 V0) = (

1

0)

120597M120597V

(1199060 V0) = (

0

1) or

C(120597R120597119906

120597R120597V

) (1199060 V0) = I

(33)


0= a(119906

0 V0) [24]

120583120572= Re

d119867120572a0

w119867a0

for 120572 = 119909 119910 (34)



119888120572ℎ

=

Re d119867120572aℎ0a1198670w + d119867

120572a0a119867ℎ0w

1003816100381610038161003816w119867a01003816100381610038161003816

2minus 120583

1198862Re

w119867aℎ0

w119867a0

(35)





119890(1 1)


a1198791199060

= 119866119890(1198952120587119891119888)(119909

1 119909

119873)



minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u


minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u






119879GLRT (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (1 + (1119870) z119867Qminus1

SMIz) (36)

119879AMF (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (37)

119879ACE (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1


SMIz (38)



0


(i) 119879GLRT =119879AMF

1 + (1119870) z119867Qminus1

SMIz 119879ACE =

119879AMF

z119867Qminus1

SMIz

(39)


119879AMF =

10038161003816100381610038161003816w119867z10038161003816100381610038161003816

2


SMIa0 (40)






119866 =

119871

sum

119894=1

119892119894

10038161003816100381610038161198941003816100381610038161003816

2

(41)



119871



119866 = z119867KDK119867z 119878119866(u) = a119867 (u)KDK119867a (u) (42)





minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)

minus35 dB Taylor


u

ni = 32


minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u



minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u









minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

34



34

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u




minus30

minus20

minus10

0

10

20

30

40

50

(dB)





u









Elev

atio

n

rarr

Azimuth u rarr


BW120572 middot BW


Projection

(a)


BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)







30 5050

70

70

90

90

110

110

130

130

10

SensorTarget


East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)




119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

05

0

0

minus05


minus20

minus40

minus60

minus80

minus100

u

1

05

0

0

minus05


u

1

05

0

0

minus05


minus20

minus40

minus60

minus80

minus100

u

1

05

0

0

minus05


u





w =Cminus1a

0

a1198670Cminus1a

0

(6)


| 1199062+ V2 le 1


1 u


119872

119896=1120575(u minus


w =Pa

0

a1198670Pa

0


A119867

A = (a (u1) a (u

119872))

(7)









0

maxw

10038161003816100381610038161003816w119867a

0

10038161003816100381610038161003816

2

119864 1003816100381610038161003816w119867n

1003816100381610038161003816

2

= maxw

10038161003816100381610038161003816w119867a

0

10038161003816100381610038161003816

2

w119867Qw (8)


w = 120583Qminus1a0

with Q = 119864 nn119867 (9)



0Qminus1z = a119867

0L119867Lz = (La

0)119867(Lz)


0







119894= 119896


119894




minw w119867Q w

st w119867C = k (or w119867c119894= 119896

119894 119894 = 1 sdot sdot sdot 119903)

(10)


w =

119903

sum

119894=1

120582119894Qminus1c


minus1

k (11)




0)minus1Qminus1a




0=










optimization

minw w119867Qw +

119903

sum

119894=1

119887119894

10038161003816100381610038161003816w119867c

119894minus 119896

119894

10038161003816100381610038161003816

2

or



(12)

with B = diag1198871 119887


optimization is


CBk (13)




minus50

minus40

minus30

minus20

minus10

0

u

20 20 20

(dB)


minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

20 2020

SNIR

(dB)


u

(b) SNIR







1 z




QSMI =1

119870

119870

sum

119896=1

z119896z119867119896 (14)


QLSMI =1

119870

119870

sum

119896=1

z119896z119867119896+ 120575 sdot I (15)






minus50

minus40

minus30

minus20

minus10

0

u

20 20 20

(dB)


minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

20 20 20

SNIR

(dB)


u

(b) SNIR


minus30

minus25

minus20

minus15

minus10

minus5

0

SNIR

(dB)

2020

20

LSMI0LSMI4

EVP10

Optimum



with dim (JSS) = 3


1(119905) 119887

119872(119905) in

directionsu1 u


119896) =

sum119872

119898=1a(u

119898)119887119898(119905119896) + n(119905

119896) or short z

119896= Ab

119896+ n

119896 then

119864 zz119867 = Q = ABA119867 + I (16)




minus1

A119867

997888rarr

Brarrinfin


A119867 = PperpA(17)


119883= I minus XX119867








wCAPS =1



perpCX

perp)minus1

times X119867perpC(

1


SMIm minusm)

(19)





wCAPS = m + P[Xm] (

1


SMIm minusm) (20)





119879 (119898) =(1 (119873 minus 119898))sum

119873

119894=119898+1120582119894

(prod119873

119894=119898+1120582119894)1(119873minus119898)

(21)





(22)


119871 (119898) =2119870

1205902

119873

sum

119894=119898+1

120582119894 (23)

no

12

34

56

24

68

1012

0

20

40

60

80

100

dimSS






if 119871 (119898) le 1205942

2119870(119873minus119898)120572 = 119898 STOP

end

(24)

The symbol 1205942119903120572



120582119894=

119894minus1

119870119894

119873

sum

119895=1

119895 = 119894

119894

119894minus

119895

(25)


2 The








Caponrsquos method

119878119862(u) = (a(u)119867Rminus1MLa (u))

minus1

with RML =1

119870

119870

sum

119896=1

z119896z119867119896

(26)



(27)





A119867

A = (a (u1) a (u

119872))

(28)




119865sto (120579) = det A (120579)B (120579)A119867 (120579) + 1205902 (120579) I with

1205902(120579) =

1


B (120579) = (A119867A)minus1

A119867 (RML minus 1205902(120579) I)A(A119867A)

minus1

for A = A (120579)

(30)


(1) 119865det(120579)= (1119870)sum119870

119896=1 z119867

119896PperpAz119896 = (1119870)sum

119870

119896=1PperpAz119896

2

=

(1119870)sum119870


119896⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=b

2

which means


(2) 119865det(120579) = 119862 minus sum119870

119896=1z119867119896A(A119867A)minus1A119867z

119896 which can


1)zk a119867(u119872)zk)



PperpGminusPperp

GF(F119867PperpGF)




0

10

20

30

40

50

60

70

80

u

1 12 12

(dB)

(a) Simulated data

minus05

0

05

minus05

0

050

5

10

15

20

u

(dB)













1205902(120579)I)A(A119867A)minus1 with 120590




257 308 358 408 459 509 560 610 661 711 762 812 862

10

06

02

minus02

minus06

Range (km)

Elev

atio

n (B






SNR (dB)

101

203

304

406

507

608

710

811


minus05 0 05

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

u






K range cells

M pulses

N channels














0 V0)119879 is

(

V) = (

1199060

V0

) minus (119888119909119909

119888119909119910

119888119910119909

119888119910119910

)(119877119909minus 120583

119909

119877119910minus 120583

119910

) (31)

whereC = (119888119909119909 119888119909119910


120583119909120583119910)




119909= d119867





119909(119906 V) 119877

119910(119906 V))119879 The cor-




M (119906 V) = C sdot (119864 R (119906 V) minus 120583) (32)

we require

M (1199060 V0) = 0

120597M120597119906

(1199060 V0) = (

1

0)

120597M120597V

(1199060 V0) = (

0

1) or

C(120597R120597119906

120597R120597V

) (1199060 V0) = I

(33)


0= a(119906

0 V0) [24]

120583120572= Re

d119867120572a0

w119867a0

for 120572 = 119909 119910 (34)



119888120572ℎ

=

Re d119867120572aℎ0a1198670w + d119867

120572a0a119867ℎ0w

1003816100381610038161003816w119867a01003816100381610038161003816

2minus 120583

1198862Re

w119867aℎ0

w119867a0

(35)





119890(1 1)


a1198791199060

= 119866119890(1198952120587119891119888)(119909

1 119909

119873)



minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u


minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u






119879GLRT (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (1 + (1119870) z119867Qminus1

SMIz) (36)

119879AMF (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (37)

119879ACE (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1


SMIz (38)



0


(i) 119879GLRT =119879AMF

1 + (1119870) z119867Qminus1

SMIz 119879ACE =

119879AMF

z119867Qminus1

SMIz

(39)


119879AMF =

10038161003816100381610038161003816w119867z10038161003816100381610038161003816

2


SMIa0 (40)






119866 =

119871

sum

119894=1

119892119894

10038161003816100381610038161198941003816100381610038161003816

2

(41)



119871



119866 = z119867KDK119867z 119878119866(u) = a119867 (u)KDK119867a (u) (42)





minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)

minus35 dB Taylor


u

ni = 32


minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u



minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u









minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

34



34

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u




minus30

minus20

minus10

0

10

20

30

40

50

(dB)





u









Elev

atio

n

rarr

Azimuth u rarr


BW120572 middot BW


Projection

(a)


BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)







30 5050

70

70

90

90

110

110

130

130

10

SensorTarget


East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)




119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













0

maxw

10038161003816100381610038161003816w119867a

0

10038161003816100381610038161003816

2

119864 1003816100381610038161003816w119867n

1003816100381610038161003816

2

= maxw

10038161003816100381610038161003816w119867a

0

10038161003816100381610038161003816

2

w119867Qw (8)


w = 120583Qminus1a0

with Q = 119864 nn119867 (9)



0Qminus1z = a119867

0L119867Lz = (La

0)119867(Lz)


0







119894= 119896


119894




minw w119867Q w

st w119867C = k (or w119867c119894= 119896

119894 119894 = 1 sdot sdot sdot 119903)

(10)


w =

119903

sum

119894=1

120582119894Qminus1c


minus1

k (11)




0)minus1Qminus1a




0=










optimization

minw w119867Qw +

119903

sum

119894=1

119887119894

10038161003816100381610038161003816w119867c

119894minus 119896

119894

10038161003816100381610038161003816

2

or



(12)

with B = diag1198871 119887


optimization is


CBk (13)




minus50

minus40

minus30

minus20

minus10

0

u

20 20 20

(dB)


minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

20 2020

SNIR

(dB)


u

(b) SNIR







1 z




QSMI =1

119870

119870

sum

119896=1

z119896z119867119896 (14)


QLSMI =1

119870

119870

sum

119896=1

z119896z119867119896+ 120575 sdot I (15)






minus50

minus40

minus30

minus20

minus10

0

u

20 20 20

(dB)


minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

20 20 20

SNIR

(dB)


u

(b) SNIR


minus30

minus25

minus20

minus15

minus10

minus5

0

SNIR

(dB)

2020

20

LSMI0LSMI4

EVP10

Optimum



with dim (JSS) = 3


1(119905) 119887

119872(119905) in

directionsu1 u


119896) =

sum119872

119898=1a(u

119898)119887119898(119905119896) + n(119905

119896) or short z

119896= Ab

119896+ n

119896 then

119864 zz119867 = Q = ABA119867 + I (16)




minus1

A119867

997888rarr

Brarrinfin


A119867 = PperpA(17)


119883= I minus XX119867








wCAPS =1



perpCX

perp)minus1

times X119867perpC(

1


SMIm minusm)

(19)





wCAPS = m + P[Xm] (

1


SMIm minusm) (20)





119879 (119898) =(1 (119873 minus 119898))sum

119873

119894=119898+1120582119894

(prod119873

119894=119898+1120582119894)1(119873minus119898)

(21)





(22)


119871 (119898) =2119870

1205902

119873

sum

119894=119898+1

120582119894 (23)

no

12

34

56

24

68

1012

0

20

40

60

80

100

dimSS






if 119871 (119898) le 1205942

2119870(119873minus119898)120572 = 119898 STOP

end

(24)

The symbol 1205942119903120572



120582119894=

119894minus1

119870119894

119873

sum

119895=1

119895 = 119894

119894

119894minus

119895

(25)


2 The








Caponrsquos method

119878119862(u) = (a(u)119867Rminus1MLa (u))

minus1

with RML =1

119870

119870

sum

119896=1

z119896z119867119896

(26)



(27)





A119867

A = (a (u1) a (u

119872))

(28)




119865sto (120579) = det A (120579)B (120579)A119867 (120579) + 1205902 (120579) I with

1205902(120579) =

1


B (120579) = (A119867A)minus1

A119867 (RML minus 1205902(120579) I)A(A119867A)

minus1

for A = A (120579)

(30)


(1) 119865det(120579)= (1119870)sum119870

119896=1 z119867

119896PperpAz119896 = (1119870)sum

119870

119896=1PperpAz119896

2

=

(1119870)sum119870


119896⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=b

2

which means


(2) 119865det(120579) = 119862 minus sum119870

119896=1z119867119896A(A119867A)minus1A119867z

119896 which can


1)zk a119867(u119872)zk)



PperpGminusPperp

GF(F119867PperpGF)




0

10

20

30

40

50

60

70

80

u

1 12 12

(dB)

(a) Simulated data

minus05

0

05

minus05

0

050

5

10

15

20

u

(dB)













1205902(120579)I)A(A119867A)minus1 with 120590




257 308 358 408 459 509 560 610 661 711 762 812 862

10

06

02

minus02

minus06

Range (km)

Elev

atio

n (B






SNR (dB)

101

203

304

406

507

608

710

811


minus05 0 05

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

u






K range cells

M pulses

N channels














0 V0)119879 is

(

V) = (

1199060

V0

) minus (119888119909119909

119888119909119910

119888119910119909

119888119910119910

)(119877119909minus 120583

119909

119877119910minus 120583

119910

) (31)

whereC = (119888119909119909 119888119909119910


120583119909120583119910)




119909= d119867





119909(119906 V) 119877

119910(119906 V))119879 The cor-




M (119906 V) = C sdot (119864 R (119906 V) minus 120583) (32)

we require

M (1199060 V0) = 0

120597M120597119906

(1199060 V0) = (

1

0)

120597M120597V

(1199060 V0) = (

0

1) or

C(120597R120597119906

120597R120597V

) (1199060 V0) = I

(33)


0= a(119906

0 V0) [24]

120583120572= Re

d119867120572a0

w119867a0

for 120572 = 119909 119910 (34)



119888120572ℎ

=

Re d119867120572aℎ0a1198670w + d119867

120572a0a119867ℎ0w

1003816100381610038161003816w119867a01003816100381610038161003816

2minus 120583

1198862Re

w119867aℎ0

w119867a0

(35)





119890(1 1)


a1198791199060

= 119866119890(1198952120587119891119888)(119909

1 119909

119873)



minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u


minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u






119879GLRT (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (1 + (1119870) z119867Qminus1

SMIz) (36)

119879AMF (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (37)

119879ACE (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1


SMIz (38)



0


(i) 119879GLRT =119879AMF

1 + (1119870) z119867Qminus1

SMIz 119879ACE =

119879AMF

z119867Qminus1

SMIz

(39)


119879AMF =

10038161003816100381610038161003816w119867z10038161003816100381610038161003816

2


SMIa0 (40)






119866 =

119871

sum

119894=1

119892119894

10038161003816100381610038161198941003816100381610038161003816

2

(41)



119871



119866 = z119867KDK119867z 119878119866(u) = a119867 (u)KDK119867a (u) (42)





minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)

minus35 dB Taylor


u

ni = 32


minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u



minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u









minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

34



34

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u




minus30

minus20

minus10

0

10

20

30

40

50

(dB)





u









Elev

atio

n

rarr

Azimuth u rarr


BW120572 middot BW


Projection

(a)


BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)







30 5050

70

70

90

90

110

110

130

130

10

SensorTarget


East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)




119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus50

minus40

minus30

minus20

minus10

0

u

20 20 20

(dB)


minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

20 2020

SNIR

(dB)


u

(b) SNIR







1 z




QSMI =1

119870

119870

sum

119896=1

z119896z119867119896 (14)


QLSMI =1

119870

119870

sum

119896=1

z119896z119867119896+ 120575 sdot I (15)






minus50

minus40

minus30

minus20

minus10

0

u

20 20 20

(dB)


minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

20 20 20

SNIR

(dB)


u

(b) SNIR


minus30

minus25

minus20

minus15

minus10

minus5

0

SNIR

(dB)

2020

20

LSMI0LSMI4

EVP10

Optimum



with dim (JSS) = 3


1(119905) 119887

119872(119905) in

directionsu1 u


119896) =

sum119872

119898=1a(u

119898)119887119898(119905119896) + n(119905

119896) or short z

119896= Ab

119896+ n

119896 then

119864 zz119867 = Q = ABA119867 + I (16)




minus1

A119867

997888rarr

Brarrinfin


A119867 = PperpA(17)


119883= I minus XX119867








wCAPS =1



perpCX

perp)minus1

times X119867perpC(

1


SMIm minusm)

(19)





wCAPS = m + P[Xm] (

1


SMIm minusm) (20)





119879 (119898) =(1 (119873 minus 119898))sum

119873

119894=119898+1120582119894

(prod119873

119894=119898+1120582119894)1(119873minus119898)

(21)





(22)


119871 (119898) =2119870

1205902

119873

sum

119894=119898+1

120582119894 (23)

no

12

34

56

24

68

1012

0

20

40

60

80

100

dimSS






if 119871 (119898) le 1205942

2119870(119873minus119898)120572 = 119898 STOP

end

(24)

The symbol 1205942119903120572



120582119894=

119894minus1

119870119894

119873

sum

119895=1

119895 = 119894

119894

119894minus

119895

(25)


2 The








Caponrsquos method

119878119862(u) = (a(u)119867Rminus1MLa (u))

minus1

with RML =1

119870

119870

sum

119896=1

z119896z119867119896

(26)



(27)





A119867

A = (a (u1) a (u

119872))

(28)




119865sto (120579) = det A (120579)B (120579)A119867 (120579) + 1205902 (120579) I with

1205902(120579) =

1


B (120579) = (A119867A)minus1

A119867 (RML minus 1205902(120579) I)A(A119867A)

minus1

for A = A (120579)

(30)


(1) 119865det(120579)= (1119870)sum119870

119896=1 z119867

119896PperpAz119896 = (1119870)sum

119870

119896=1PperpAz119896

2

=

(1119870)sum119870


119896⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=b

2

which means


(2) 119865det(120579) = 119862 minus sum119870

119896=1z119867119896A(A119867A)minus1A119867z

119896 which can


1)zk a119867(u119872)zk)



PperpGminusPperp

GF(F119867PperpGF)




0

10

20

30

40

50

60

70

80

u

1 12 12

(dB)

(a) Simulated data

minus05

0

05

minus05

0

050

5

10

15

20

u

(dB)













1205902(120579)I)A(A119867A)minus1 with 120590




257 308 358 408 459 509 560 610 661 711 762 812 862

10

06

02

minus02

minus06

Range (km)

Elev

atio

n (B






SNR (dB)

101

203

304

406

507

608

710

811


minus05 0 05

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

u






K range cells

M pulses

N channels














0 V0)119879 is

(

V) = (

1199060

V0

) minus (119888119909119909

119888119909119910

119888119910119909

119888119910119910

)(119877119909minus 120583

119909

119877119910minus 120583

119910

) (31)

whereC = (119888119909119909 119888119909119910


120583119909120583119910)




119909= d119867





119909(119906 V) 119877

119910(119906 V))119879 The cor-




M (119906 V) = C sdot (119864 R (119906 V) minus 120583) (32)

we require

M (1199060 V0) = 0

120597M120597119906

(1199060 V0) = (

1

0)

120597M120597V

(1199060 V0) = (

0

1) or

C(120597R120597119906

120597R120597V

) (1199060 V0) = I

(33)


0= a(119906

0 V0) [24]

120583120572= Re

d119867120572a0

w119867a0

for 120572 = 119909 119910 (34)



119888120572ℎ

=

Re d119867120572aℎ0a1198670w + d119867

120572a0a119867ℎ0w

1003816100381610038161003816w119867a01003816100381610038161003816

2minus 120583

1198862Re

w119867aℎ0

w119867a0

(35)





119890(1 1)


a1198791199060

= 119866119890(1198952120587119891119888)(119909

1 119909

119873)



minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u


minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u






119879GLRT (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (1 + (1119870) z119867Qminus1

SMIz) (36)

119879AMF (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (37)

119879ACE (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1


SMIz (38)



0


(i) 119879GLRT =119879AMF

1 + (1119870) z119867Qminus1

SMIz 119879ACE =

119879AMF

z119867Qminus1

SMIz

(39)


119879AMF =

10038161003816100381610038161003816w119867z10038161003816100381610038161003816

2


SMIa0 (40)






119866 =

119871

sum

119894=1

119892119894

10038161003816100381610038161198941003816100381610038161003816

2

(41)



119871



119866 = z119867KDK119867z 119878119866(u) = a119867 (u)KDK119867a (u) (42)





minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)

minus35 dB Taylor


u

ni = 32


minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u



minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u









minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

34



34

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u




minus30

minus20

minus10

0

10

20

30

40

50

(dB)





u









Elev

atio

n

rarr

Azimuth u rarr


BW120572 middot BW


Projection

(a)


BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)







30 5050

70

70

90

90

110

110

130

130

10

SensorTarget


East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)




119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus50

minus40

minus30

minus20

minus10

0

u

20 20 20

(dB)


minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

20 20 20

SNIR

(dB)


u

(b) SNIR


minus30

minus25

minus20

minus15

minus10

minus5

0

SNIR

(dB)

2020

20

LSMI0LSMI4

EVP10

Optimum



with dim (JSS) = 3


1(119905) 119887

119872(119905) in

directionsu1 u


119896) =

sum119872

119898=1a(u

119898)119887119898(119905119896) + n(119905

119896) or short z

119896= Ab

119896+ n

119896 then

119864 zz119867 = Q = ABA119867 + I (16)




minus1

A119867

997888rarr

Brarrinfin


A119867 = PperpA(17)


119883= I minus XX119867








wCAPS =1



perpCX

perp)minus1

times X119867perpC(

1


SMIm minusm)

(19)





wCAPS = m + P[Xm] (

1


SMIm minusm) (20)





119879 (119898) =(1 (119873 minus 119898))sum

119873

119894=119898+1120582119894

(prod119873

119894=119898+1120582119894)1(119873minus119898)

(21)





(22)


119871 (119898) =2119870

1205902

119873

sum

119894=119898+1

120582119894 (23)

no

12

34

56

24

68

1012

0

20

40

60

80

100

dimSS






if 119871 (119898) le 1205942

2119870(119873minus119898)120572 = 119898 STOP

end

(24)

The symbol 1205942119903120572



120582119894=

119894minus1

119870119894

119873

sum

119895=1

119895 = 119894

119894

119894minus

119895

(25)


2 The








Caponrsquos method

119878119862(u) = (a(u)119867Rminus1MLa (u))

minus1

with RML =1

119870

119870

sum

119896=1

z119896z119867119896

(26)



(27)





A119867

A = (a (u1) a (u

119872))

(28)




119865sto (120579) = det A (120579)B (120579)A119867 (120579) + 1205902 (120579) I with

1205902(120579) =

1


B (120579) = (A119867A)minus1

A119867 (RML minus 1205902(120579) I)A(A119867A)

minus1

for A = A (120579)

(30)


(1) 119865det(120579)= (1119870)sum119870

119896=1 z119867

119896PperpAz119896 = (1119870)sum

119870

119896=1PperpAz119896

2

=

(1119870)sum119870


119896⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=b

2

which means


(2) 119865det(120579) = 119862 minus sum119870

119896=1z119867119896A(A119867A)minus1A119867z

119896 which can


1)zk a119867(u119872)zk)



PperpGminusPperp

GF(F119867PperpGF)




0

10

20

30

40

50

60

70

80

u

1 12 12

(dB)

(a) Simulated data

minus05

0

05

minus05

0

050

5

10

15

20

u

(dB)













1205902(120579)I)A(A119867A)minus1 with 120590




257 308 358 408 459 509 560 610 661 711 762 812 862

10

06

02

minus02

minus06

Range (km)

Elev

atio

n (B






SNR (dB)

101

203

304

406

507

608

710

811


minus05 0 05

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

u






K range cells

M pulses

N channels














0 V0)119879 is

(

V) = (

1199060

V0

) minus (119888119909119909

119888119909119910

119888119910119909

119888119910119910

)(119877119909minus 120583

119909

119877119910minus 120583

119910

) (31)

whereC = (119888119909119909 119888119909119910


120583119909120583119910)




119909= d119867





119909(119906 V) 119877

119910(119906 V))119879 The cor-




M (119906 V) = C sdot (119864 R (119906 V) minus 120583) (32)

we require

M (1199060 V0) = 0

120597M120597119906

(1199060 V0) = (

1

0)

120597M120597V

(1199060 V0) = (

0

1) or

C(120597R120597119906

120597R120597V

) (1199060 V0) = I

(33)


0= a(119906

0 V0) [24]

120583120572= Re

d119867120572a0

w119867a0

for 120572 = 119909 119910 (34)



119888120572ℎ

=

Re d119867120572aℎ0a1198670w + d119867

120572a0a119867ℎ0w

1003816100381610038161003816w119867a01003816100381610038161003816

2minus 120583

1198862Re

w119867aℎ0

w119867a0

(35)





119890(1 1)


a1198791199060

= 119866119890(1198952120587119891119888)(119909

1 119909

119873)



minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u


minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u






119879GLRT (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (1 + (1119870) z119867Qminus1

SMIz) (36)

119879AMF (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (37)

119879ACE (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1


SMIz (38)



0


(i) 119879GLRT =119879AMF

1 + (1119870) z119867Qminus1

SMIz 119879ACE =

119879AMF

z119867Qminus1

SMIz

(39)


119879AMF =

10038161003816100381610038161003816w119867z10038161003816100381610038161003816

2


SMIa0 (40)






119866 =

119871

sum

119894=1

119892119894

10038161003816100381610038161198941003816100381610038161003816

2

(41)



119871



119866 = z119867KDK119867z 119878119866(u) = a119867 (u)KDK119867a (u) (42)





minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)

minus35 dB Taylor


u

ni = 32


minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u



minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u









minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

34



34

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u




minus30

minus20

minus10

0

10

20

30

40

50

(dB)





u









Elev

atio

n

rarr

Azimuth u rarr


BW120572 middot BW


Projection

(a)


BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)







30 5050

70

70

90

90

110

110

130

130

10

SensorTarget


East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)




119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











wCAPS =1



perpCX

perp)minus1

times X119867perpC(

1


SMIm minusm)

(19)





wCAPS = m + P[Xm] (

1


SMIm minusm) (20)





119879 (119898) =(1 (119873 minus 119898))sum

119873

119894=119898+1120582119894

(prod119873

119894=119898+1120582119894)1(119873minus119898)

(21)





(22)


119871 (119898) =2119870

1205902

119873

sum

119894=119898+1

120582119894 (23)

no

12

34

56

24

68

1012

0

20

40

60

80

100

dimSS






if 119871 (119898) le 1205942

2119870(119873minus119898)120572 = 119898 STOP

end

(24)

The symbol 1205942119903120572



120582119894=

119894minus1

119870119894

119873

sum

119895=1

119895 = 119894

119894

119894minus

119895

(25)


2 The








Caponrsquos method

119878119862(u) = (a(u)119867Rminus1MLa (u))

minus1

with RML =1

119870

119870

sum

119896=1

z119896z119867119896

(26)



(27)





A119867

A = (a (u1) a (u

119872))

(28)




119865sto (120579) = det A (120579)B (120579)A119867 (120579) + 1205902 (120579) I with

1205902(120579) =

1


B (120579) = (A119867A)minus1

A119867 (RML minus 1205902(120579) I)A(A119867A)

minus1

for A = A (120579)

(30)


(1) 119865det(120579)= (1119870)sum119870

119896=1 z119867

119896PperpAz119896 = (1119870)sum

119870

119896=1PperpAz119896

2

=

(1119870)sum119870


119896⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=b

2

which means


(2) 119865det(120579) = 119862 minus sum119870

119896=1z119867119896A(A119867A)minus1A119867z

119896 which can


1)zk a119867(u119872)zk)



PperpGminusPperp

GF(F119867PperpGF)




0

10

20

30

40

50

60

70

80

u

1 12 12

(dB)

(a) Simulated data

minus05

0

05

minus05

0

050

5

10

15

20

u

(dB)













1205902(120579)I)A(A119867A)minus1 with 120590




257 308 358 408 459 509 560 610 661 711 762 812 862

10

06

02

minus02

minus06

Range (km)

Elev

atio

n (B






SNR (dB)

101

203

304

406

507

608

710

811


minus05 0 05

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

u






K range cells

M pulses

N channels














0 V0)119879 is

(

V) = (

1199060

V0

) minus (119888119909119909

119888119909119910

119888119910119909

119888119910119910

)(119877119909minus 120583

119909

119877119910minus 120583

119910

) (31)

whereC = (119888119909119909 119888119909119910


120583119909120583119910)




119909= d119867





119909(119906 V) 119877

119910(119906 V))119879 The cor-




M (119906 V) = C sdot (119864 R (119906 V) minus 120583) (32)

we require

M (1199060 V0) = 0

120597M120597119906

(1199060 V0) = (

1

0)

120597M120597V

(1199060 V0) = (

0

1) or

C(120597R120597119906

120597R120597V

) (1199060 V0) = I

(33)


0= a(119906

0 V0) [24]

120583120572= Re

d119867120572a0

w119867a0

for 120572 = 119909 119910 (34)



119888120572ℎ

=

Re d119867120572aℎ0a1198670w + d119867

120572a0a119867ℎ0w

1003816100381610038161003816w119867a01003816100381610038161003816

2minus 120583

1198862Re

w119867aℎ0

w119867a0

(35)





119890(1 1)


a1198791199060

= 119866119890(1198952120587119891119888)(119909

1 119909

119873)



minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u


minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u






119879GLRT (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (1 + (1119870) z119867Qminus1

SMIz) (36)

119879AMF (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (37)

119879ACE (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1


SMIz (38)



0


(i) 119879GLRT =119879AMF

1 + (1119870) z119867Qminus1

SMIz 119879ACE =

119879AMF

z119867Qminus1

SMIz

(39)


119879AMF =

10038161003816100381610038161003816w119867z10038161003816100381610038161003816

2


SMIa0 (40)






119866 =

119871

sum

119894=1

119892119894

10038161003816100381610038161198941003816100381610038161003816

2

(41)



119871



119866 = z119867KDK119867z 119878119866(u) = a119867 (u)KDK119867a (u) (42)





minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)

minus35 dB Taylor


u

ni = 32


minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u



minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u









minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

34



34

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u




minus30

minus20

minus10

0

10

20

30

40

50

(dB)





u









Elev

atio

n

rarr

Azimuth u rarr


BW120572 middot BW


Projection

(a)


BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)







30 5050

70

70

90

90

110

110

130

130

10

SensorTarget


East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)




119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















Caponrsquos method

119878119862(u) = (a(u)119867Rminus1MLa (u))

minus1

with RML =1

119870

119870

sum

119896=1

z119896z119867119896

(26)



(27)





A119867

A = (a (u1) a (u

119872))

(28)




119865sto (120579) = det A (120579)B (120579)A119867 (120579) + 1205902 (120579) I with

1205902(120579) =

1


B (120579) = (A119867A)minus1

A119867 (RML minus 1205902(120579) I)A(A119867A)

minus1

for A = A (120579)

(30)


(1) 119865det(120579)= (1119870)sum119870

119896=1 z119867

119896PperpAz119896 = (1119870)sum

119870

119896=1PperpAz119896

2

=

(1119870)sum119870


119896⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=b

2

which means


(2) 119865det(120579) = 119862 minus sum119870

119896=1z119867119896A(A119867A)minus1A119867z

119896 which can


1)zk a119867(u119872)zk)



PperpGminusPperp

GF(F119867PperpGF)




0

10

20

30

40

50

60

70

80

u

1 12 12

(dB)

(a) Simulated data

minus05

0

05

minus05

0

050

5

10

15

20

u

(dB)













1205902(120579)I)A(A119867A)minus1 with 120590




257 308 358 408 459 509 560 610 661 711 762 812 862

10

06

02

minus02

minus06

Range (km)

Elev

atio

n (B






SNR (dB)

101

203

304

406

507

608

710

811


minus05 0 05

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

u






K range cells

M pulses

N channels














0 V0)119879 is

(

V) = (

1199060

V0

) minus (119888119909119909

119888119909119910

119888119910119909

119888119910119910

)(119877119909minus 120583

119909

119877119910minus 120583

119910

) (31)

whereC = (119888119909119909 119888119909119910


120583119909120583119910)




119909= d119867





119909(119906 V) 119877

119910(119906 V))119879 The cor-




M (119906 V) = C sdot (119864 R (119906 V) minus 120583) (32)

we require

M (1199060 V0) = 0

120597M120597119906

(1199060 V0) = (

1

0)

120597M120597V

(1199060 V0) = (

0

1) or

C(120597R120597119906

120597R120597V

) (1199060 V0) = I

(33)


0= a(119906

0 V0) [24]

120583120572= Re

d119867120572a0

w119867a0

for 120572 = 119909 119910 (34)



119888120572ℎ

=

Re d119867120572aℎ0a1198670w + d119867

120572a0a119867ℎ0w

1003816100381610038161003816w119867a01003816100381610038161003816

2minus 120583

1198862Re

w119867aℎ0

w119867a0

(35)





119890(1 1)


a1198791199060

= 119866119890(1198952120587119891119888)(119909

1 119909

119873)



minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u


minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u






119879GLRT (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (1 + (1119870) z119867Qminus1

SMIz) (36)

119879AMF (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (37)

119879ACE (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1


SMIz (38)



0


(i) 119879GLRT =119879AMF

1 + (1119870) z119867Qminus1

SMIz 119879ACE =

119879AMF

z119867Qminus1

SMIz

(39)


119879AMF =

10038161003816100381610038161003816w119867z10038161003816100381610038161003816

2


SMIa0 (40)






119866 =

119871

sum

119894=1

119892119894

10038161003816100381610038161198941003816100381610038161003816

2

(41)



119871



119866 = z119867KDK119867z 119878119866(u) = a119867 (u)KDK119867a (u) (42)





minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)

minus35 dB Taylor


u

ni = 32


minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u



minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u









minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

34



34

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u




minus30

minus20

minus10

0

10

20

30

40

50

(dB)





u









Elev

atio

n

rarr

Azimuth u rarr


BW120572 middot BW


Projection

(a)


BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)







30 5050

70

70

90

90

110

110

130

130

10

SensorTarget


East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)




119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

10

20

30

40

50

60

70

80

u

1 12 12

(dB)

(a) Simulated data

minus05

0

05

minus05

0

050

5

10

15

20

u

(dB)













1205902(120579)I)A(A119867A)minus1 with 120590




257 308 358 408 459 509 560 610 661 711 762 812 862

10

06

02

minus02

minus06

Range (km)

Elev

atio

n (B






SNR (dB)

101

203

304

406

507

608

710

811


minus05 0 05

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

u






K range cells

M pulses

N channels














0 V0)119879 is

(

V) = (

1199060

V0

) minus (119888119909119909

119888119909119910

119888119910119909

119888119910119910

)(119877119909minus 120583

119909

119877119910minus 120583

119910

) (31)

whereC = (119888119909119909 119888119909119910


120583119909120583119910)




119909= d119867





119909(119906 V) 119877

119910(119906 V))119879 The cor-




M (119906 V) = C sdot (119864 R (119906 V) minus 120583) (32)

we require

M (1199060 V0) = 0

120597M120597119906

(1199060 V0) = (

1

0)

120597M120597V

(1199060 V0) = (

0

1) or

C(120597R120597119906

120597R120597V

) (1199060 V0) = I

(33)


0= a(119906

0 V0) [24]

120583120572= Re

d119867120572a0

w119867a0

for 120572 = 119909 119910 (34)



119888120572ℎ

=

Re d119867120572aℎ0a1198670w + d119867

120572a0a119867ℎ0w

1003816100381610038161003816w119867a01003816100381610038161003816

2minus 120583

1198862Re

w119867aℎ0

w119867a0

(35)





119890(1 1)


a1198791199060

= 119866119890(1198952120587119891119888)(119909

1 119909

119873)



minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u


minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u






119879GLRT (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (1 + (1119870) z119867Qminus1

SMIz) (36)

119879AMF (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (37)

119879ACE (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1


SMIz (38)



0


(i) 119879GLRT =119879AMF

1 + (1119870) z119867Qminus1

SMIz 119879ACE =

119879AMF

z119867Qminus1

SMIz

(39)


119879AMF =

10038161003816100381610038161003816w119867z10038161003816100381610038161003816

2


SMIa0 (40)






119866 =

119871

sum

119894=1

119892119894

10038161003816100381610038161198941003816100381610038161003816

2

(41)



119871



119866 = z119867KDK119867z 119878119866(u) = a119867 (u)KDK119867a (u) (42)





minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)

minus35 dB Taylor


u

ni = 32


minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u



minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u









minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

34



34

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u




minus30

minus20

minus10

0

10

20

30

40

50

(dB)





u









Elev

atio

n

rarr

Azimuth u rarr


BW120572 middot BW


Projection

(a)


BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)







30 5050

70

70

90

90

110

110

130

130

10

SensorTarget


East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)




119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










257 308 358 408 459 509 560 610 661 711 762 812 862

10

06

02

minus02

minus06

Range (km)

Elev

atio

n (B






SNR (dB)

101

203

304

406

507

608

710

811


minus05 0 05

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

05

u






K range cells

M pulses

N channels














0 V0)119879 is

(

V) = (

1199060

V0

) minus (119888119909119909

119888119909119910

119888119910119909

119888119910119910

)(119877119909minus 120583

119909

119877119910minus 120583

119910

) (31)

whereC = (119888119909119909 119888119909119910


120583119909120583119910)




119909= d119867





119909(119906 V) 119877

119910(119906 V))119879 The cor-




M (119906 V) = C sdot (119864 R (119906 V) minus 120583) (32)

we require

M (1199060 V0) = 0

120597M120597119906

(1199060 V0) = (

1

0)

120597M120597V

(1199060 V0) = (

0

1) or

C(120597R120597119906

120597R120597V

) (1199060 V0) = I

(33)


0= a(119906

0 V0) [24]

120583120572= Re

d119867120572a0

w119867a0

for 120572 = 119909 119910 (34)



119888120572ℎ

=

Re d119867120572aℎ0a1198670w + d119867

120572a0a119867ℎ0w

1003816100381610038161003816w119867a01003816100381610038161003816

2minus 120583

1198862Re

w119867aℎ0

w119867a0

(35)





119890(1 1)


a1198791199060

= 119866119890(1198952120587119891119888)(119909

1 119909

119873)



minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u


minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u






119879GLRT (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (1 + (1119870) z119867Qminus1

SMIz) (36)

119879AMF (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (37)

119879ACE (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1


SMIz (38)



0


(i) 119879GLRT =119879AMF

1 + (1119870) z119867Qminus1

SMIz 119879ACE =

119879AMF

z119867Qminus1

SMIz

(39)


119879AMF =

10038161003816100381610038161003816w119867z10038161003816100381610038161003816

2


SMIa0 (40)






119866 =

119871

sum

119894=1

119892119894

10038161003816100381610038161198941003816100381610038161003816

2

(41)



119871



119866 = z119867KDK119867z 119878119866(u) = a119867 (u)KDK119867a (u) (42)





minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)

minus35 dB Taylor


u

ni = 32


minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u



minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u









minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

34



34

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u




minus30

minus20

minus10

0

10

20

30

40

50

(dB)





u









Elev

atio

n

rarr

Azimuth u rarr


BW120572 middot BW


Projection

(a)


BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)







30 5050

70

70

90

90

110

110

130

130

10

SensorTarget


East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)




119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










K range cells

M pulses

N channels














0 V0)119879 is

(

V) = (

1199060

V0

) minus (119888119909119909

119888119909119910

119888119910119909

119888119910119910

)(119877119909minus 120583

119909

119877119910minus 120583

119910

) (31)

whereC = (119888119909119909 119888119909119910


120583119909120583119910)




119909= d119867





119909(119906 V) 119877

119910(119906 V))119879 The cor-




M (119906 V) = C sdot (119864 R (119906 V) minus 120583) (32)

we require

M (1199060 V0) = 0

120597M120597119906

(1199060 V0) = (

1

0)

120597M120597V

(1199060 V0) = (

0

1) or

C(120597R120597119906

120597R120597V

) (1199060 V0) = I

(33)


0= a(119906

0 V0) [24]

120583120572= Re

d119867120572a0

w119867a0

for 120572 = 119909 119910 (34)



119888120572ℎ

=

Re d119867120572aℎ0a1198670w + d119867

120572a0a119867ℎ0w

1003816100381610038161003816w119867a01003816100381610038161003816

2minus 120583

1198862Re

w119867aℎ0

w119867a0

(35)





119890(1 1)


a1198791199060

= 119866119890(1198952120587119891119888)(119909

1 119909

119873)



minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u


minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u






119879GLRT (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (1 + (1119870) z119867Qminus1

SMIz) (36)

119879AMF (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (37)

119879ACE (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1


SMIz (38)



0


(i) 119879GLRT =119879AMF

1 + (1119870) z119867Qminus1

SMIz 119879ACE =

119879AMF

z119867Qminus1

SMIz

(39)


119879AMF =

10038161003816100381610038161003816w119867z10038161003816100381610038161003816

2


SMIa0 (40)






119866 =

119871

sum

119894=1

119892119894

10038161003816100381610038161198941003816100381610038161003816

2

(41)



119871



119866 = z119867KDK119867z 119878119866(u) = a119867 (u)KDK119867a (u) (42)





minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)

minus35 dB Taylor


u

ni = 32


minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u



minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u









minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

34



34

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u




minus30

minus20

minus10

0

10

20

30

40

50

(dB)





u









Elev

atio

n

rarr

Azimuth u rarr


BW120572 middot BW


Projection

(a)


BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)







30 5050

70

70

90

90

110

110

130

130

10

SensorTarget


East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)




119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











M (119906 V) = C sdot (119864 R (119906 V) minus 120583) (32)

we require

M (1199060 V0) = 0

120597M120597119906

(1199060 V0) = (

1

0)

120597M120597V

(1199060 V0) = (

0

1) or

C(120597R120597119906

120597R120597V

) (1199060 V0) = I

(33)


0= a(119906

0 V0) [24]

120583120572= Re

d119867120572a0

w119867a0

for 120572 = 119909 119910 (34)



119888120572ℎ

=

Re d119867120572aℎ0a1198670w + d119867

120572a0a119867ℎ0w

1003816100381610038161003816w119867a01003816100381610038161003816

2minus 120583

1198862Re

w119867aℎ0

w119867a0

(35)





119890(1 1)


a1198791199060

= 119866119890(1198952120587119891119888)(119909

1 119909

119873)



minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u


minus005 0

0

005

minus005

minus004

minus003

minus002

minus001

001

002

003

004

005

u






119879GLRT (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (1 + (1119870) z119867Qminus1

SMIz) (36)

119879AMF (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (37)

119879ACE (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1


SMIz (38)



0


(i) 119879GLRT =119879AMF

1 + (1119870) z119867Qminus1

SMIz 119879ACE =

119879AMF

z119867Qminus1

SMIz

(39)


119879AMF =

10038161003816100381610038161003816w119867z10038161003816100381610038161003816

2


SMIa0 (40)






119866 =

119871

sum

119894=1

119892119894

10038161003816100381610038161198941003816100381610038161003816

2

(41)



119871



119866 = z119867KDK119867z 119878119866(u) = a119867 (u)KDK119867a (u) (42)





minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)

minus35 dB Taylor


u

ni = 32


minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u



minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u









minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

34



34

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u




minus30

minus20

minus10

0

10

20

30

40

50

(dB)





u









Elev

atio

n

rarr

Azimuth u rarr


BW120572 middot BW


Projection

(a)


BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)







30 5050

70

70

90

90

110

110

130

130

10

SensorTarget


East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)




119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119879GLRT (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (1 + (1119870) z119867Qminus1

SMIz) (36)

119879AMF (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1

SMIa0 (37)

119879ACE (z) =10038161003816100381610038161003816a1198670Qminus1

SMIz10038161003816100381610038161003816

2

a1198670Qminus1


SMIz (38)



0


(i) 119879GLRT =119879AMF

1 + (1119870) z119867Qminus1

SMIz 119879ACE =

119879AMF

z119867Qminus1

SMIz

(39)


119879AMF =

10038161003816100381610038161003816w119867z10038161003816100381610038161003816

2


SMIa0 (40)






119866 =

119871

sum

119894=1

119892119894

10038161003816100381610038161198941003816100381610038161003816

2

(41)



119871



119866 = z119867KDK119867z 119878119866(u) = a119867 (u)KDK119867a (u) (42)





minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)

minus35 dB Taylor


u

ni = 32


minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u



minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u









minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

34



34

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u




minus30

minus20

minus10

0

10

20

30

40

50

(dB)





u









Elev

atio

n

rarr

Azimuth u rarr


BW120572 middot BW


Projection

(a)


BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)







30 5050

70

70

90

90

110

110

130

130

10

SensorTarget


East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)




119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)

minus35 dB Taylor


u

ni = 32


minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u



minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u









minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

34



34

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u




minus30

minus20

minus10

0

10

20

30

40

50

(dB)





u









Elev

atio

n

rarr

Azimuth u rarr


BW120572 middot BW


Projection

(a)


BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)







30 5050

70

70

90

90

110

110

130

130

10

SensorTarget


East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)




119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u

34



34

minus60

minus50

minus40

minus30

minus20

minus10

0

(dB)


u




minus30

minus20

minus10

0

10

20

30

40

50

(dB)





u









Elev

atio

n

rarr

Azimuth u rarr


BW120572 middot BW


Projection

(a)


BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)







30 5050

70

70

90

90

110

110

130

130

10

SensorTarget


East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)




119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Elev

atio

n

rarr

Azimuth u rarr


BW120572 middot BW


Projection

(a)


BWJ

Jammer

Jammer notch

Predicted

Direction

Adjusted

Direction

Elev

atio

n

rarr

Azimuth u rarr

(b)

BWJ

(middot)Diff look

Directions

Jammer

Jammer notch

(1) (2)

(3)(4)

Elev

atio

n

rarr

Azimuth u rarr

QuadSearch

(c)







30 5050

70

70

90

90

110

110

130

130

10

SensorTarget


East (km)

Nor

th (k

m)

0 120 2404

6

8

10

12

Time (s)

Alti

tude

(km

)




119906119896


90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










90 11095 100 105 115minus024

minus02

minus016

minus012

minus008

Time (s)

Azi

mut

hu



u119896

120590FILTu119896



















Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Disclosure


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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