1

Space-Time Adaptive Processing and MotionParameter Estimation in Multi-Static Passive Radar

Using Sparse Bayesian LearningQisong Wu, Member, IEEE, Yimin D. Zhang, Senior Member, IEEE,

Moeness G. Amin, Fellow, IEEE, and Braham Himed, Fellow, IEEE

Abstract—Conventional space-time adaptive processing suffersfrom the requirement of a large number of secondary samples. Inthis paper, a novel method is proposed to accurately estimate theclutter covariance matrix based on a small number of secondarysamples by exploiting the common clutter support across nearbyrange cells in the angle-Doppler domain. By taking advantageof the intrinsic sparsity of the clutter in the angle-Dopplerdomain, the recently developed sparse Bayesian learning tech-nique is employed for high-resolution clutter profile estimation.The proposed method does not require the independent andidentically distributed secondary sample assumption, and therequired number of secondary data samples can be significantlyreduced. In addition, we propose a sparse reconstruction basedapproach to acquire the two-dimensional motion parameters ofmoving targets by exploiting their group sparsity in the velocitydomain in the multi-static passive radar systems. Simulationresults verify the effectiveness of the proposed algorithm.

Index Terms—Group sparsity, motion parameter estimation,multi-static passive radar, space-time adaptive processing (STAP),sparse Bayesian learning (SBL)

I. INTRODUCTION

Passive radar (PR) systems, which utilize broadcast, nav-igation, and wireless communication signals as sources ofopportunity, have recently attracted great interest due to theirdistinct advantages over conventional active radar systems,primarily in terms of their low cost and covertness [1]. PRs areexpected to play an increasing role in future reconnaissancesystems [2].

Moving target indication (MTI) and tracking are commonlyapplied in radar applications aiming at target detection and

Copyright (c) 2015 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

The work was supported in part by a subcontract with Defense EngineeringCorporation for research sponsored by the Air Force Research Laboratory(AFRL) under Contract FA8650-12-D-1376 and by a subcontract with Dy-netics, Inc. for research sponsored by the AFRL under Contract FA8650-08-D-1303.

Q. Wu was with the Center for Advanced Communications, VillanovaUniversity, Villanova, PA 19085, USA. He is now with the Key Laboratory ofUnderwater Acoustic Signal Processing of Ministry of Education, SoutheastUniversity, Nanjing 210096, China (email: qisong.wu@seu.edu.cn).

Y. D. Zhang was with the Center for Advanced Communications, VillanovaUniversity, Villanova, PA 19085, USA. He is now with the Departmentof Electrical and Computer Engineering, College of Engineering, TempleUniversity, Philadelphia, PA 19122, USA (email: ydzhang@temple.edu).

M. G. Amin is with the Center for Advanced Communications, VillanovaUniversity, Villanova, PA 19085, USA.

B. Himed is with the RF Technology Branch, Air Force Research Lab(AFRL/RYMD), WPAFB, OH 45433, USA.

motion parameter estimation. The latter has been extensivelystudied, particularly for active radar platforms, and has ben-efited from time-frequency analysis [3–5], and motion andrange migration compensations [6–8]. However, for airborneradar receivers, those algorithms are effective only whenclutter suppression is achieved.

Space-time adaptive processing (STAP) is considered aneffective technique to MTI in cluttered environments. STAP, inessence, performs signal processing in the joint angle-Dopplerdomain to better separate moving targets from clutter [9, 10].Application of STAP techniques requires a large number ofsecondary data samples from other range gates to estimatethe clutter covariance matrix (CCM) [9]. More specifically,for a STAP system consisting of N antennas and L azimuthsamples, the well-known Reed-Mallett-Brennan’s (RMB) rulestates that 2NL independent and identically distributed (i.i.d.)secondary data samples are required to achieve an output SINRthat is within 3 dB loss from the Clairvoyant solution [11].Although a number of reduced-rank STAP algorithms havebeen developed to robustly estimate the CCM with a smallernumber of secondary data samples (e.g., [10, 12–14]), therequired number of secondary data samples is still at leasttwice the rank of the dominant clutter subspace for a satis-factory performance. By jointly exploiting the reduced-ranktechnique and the structure of the covariance matrix [15, 16],this number can be further reduced. However, this approachis still considered impractical in passive radar systems dueto the narrow signal bandwidth and the corresponding lowrange resolution. For example, a Digital Video Broadcasting-Terrestrial (DVB-T) channel occupies a 7.6 MHz bandwidth inthe 450–900 MHz band and its corresponding bistatic rangeresolution is approximately 20 m. The range resolution be-comes even coarser when signals with a narrower bandwidth,such as the Digital Audio Broadcasting (DAB) broadcast andGlobal System for Mobile Communications (GSM) signals,are used. With a much lower range resolution as comparedto conventional active radar systems, it is often difficult toacquire a sufficient number of i.i.d. secondary data samplesrequired for the estimation of a full-rank CCM [17, 18].

In order to relax the condition on the number of secondarysamples, a more recent approach is to exploit the latestadvances of sparsity reconstruction and compressive sensing(CS) techniques. By taking advantage of the intrinsic sparsityof the clutter in the angle-Doppler domain, the sparse clutterspectrum can be recovered by using only one or a few samples

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[19–23]. In essence, we formulate the sparse clutter profileestimation as a regularized optimization problem, and thenconstruct the CCM based on the estimated clutter profiles toachieve effective clutter suppression. In [19], clutter suppres-sion is achieved by assuming the knowledge of the supportof the clutter ridge in the angle-Doppler plane. By applyinga mask to the assumed clutter ridge, moving targets areestimated as the sparse solution in the angle-Doppler domainoutside this clutter ridge. In [20], the clutter at each secondaryrange cell is separately estimated in the Bayesian framework.The estimated clutter at each range cell is used to compute theCCM and perform STAP in the traditional sense. In [21, 22],two approaches are proposed for the clutter profile estimationby utilizing multiple data samples where the clutter profile isestimated separately in each range cell. The former averagesthe estimated clutter profiles to construct the CCM, whereas inthe latter method, the maximum value is chosen for each angle-Doppler entry from all range cells being evaluated. In [23],the STAP processor is assumed to be persymmetry in bothspatial and azimuthal domains, and a complex sparse clutterestimation problem is cast into a group sparsity formulationto be solved in the multi-task Bayesian CS framework. Wemaintain that none of the above methods utilizes the groupsparsity of the clutter across range cells for improved cluttersupport or profile estimation. Except [19] in which the supportof the clutter ridge in the angle-Doppler plane is assumed to beknown, the other aforementioned algorithms assume that themoving target signals in the range cell under test are excludedin the reconstructed clutter profiles. While this assumption mayhold true for weak target signals due to the non-linear behaviorof sparse reconstruction, it is nevertheless difficult to guaranteesuch an outcome in various situations.

A separate issue arising in the sparse reconstruction ofthe clutter profile is its angle-Doppler domain resolution. Inaddition to satisfying the sparsity requirement, most existingCS algorithms, such as greedy algorithms like the orthogonalmatching pursuit (OMP) [24] and regularized optimizationalgorithms like the LASSO [25], generally expect a lowcoherence measurement matrix. A much higher number ofmeasurement samples would be required when the measure-ment matrix is highly coherent [26–28]. On the other hand,sparse Bayesian learning (SBL) methods, which are generallydeveloped under the relevance vector machine (RVM) frame-work [29], form a different class of effective tools for sparsesignal reconstruction. These methods are known to achieveimproved performance in many applications, particularly whenthe measurement matrix is of a high coherence [30]. High-resolution sparse reconstruction problems are encountered inthe underlying STAP [22, 23] as well as a large class of radarapplications, such as in direction-of-arrival (DOA) estimation[31, 32], microwave imaging [33–35], and synthesis of lineararray [36, 37]. While the original SBL framework is developedto handle real-valued data, a number of methods have beenmade available to process complex-valued data in the contextof the SBL framework [34, 38]. In [34], the complex-valuedproblem is reformulated as fictitious real-valued tasks withinterrelationship between the real and imaginary components.The idea that common parameters are shared by the real

and imaginary components is taken into account in [38].In this paper, we use the SBL method developed in thecontext of complex Gaussian distribution [39] to estimate thecomplex-valued clutter profiles. This method directly modelsthe complex-valued sparse reconstruction problem in the con-text of complex Gaussian distributions, and the problem issolved using the standard RVM approach [29].

In this paper, we propose a novel approach to accuratelyestimate the CCM and implement STAP based on a smallnumber of secondary data samples. By exploiting the groupsparsity that nearby range cells share the same non-zeroclutter support in the angle-Doppler domain, we first use thesecondary data samples to estimate the common non-zeroclutter support that excludes the target signals. The estimatedclutter support is then used to estimate the clutter profile inthe range cell under test with the target signals excluded. Theproposed method does not require the i.i.d. secondary sampleassumption, and the number of secondary data samples canbe significantly reduced. The SBL algorithm is introduced toimprove clutter profile reconstruction, particularly when themeasurement matrix has a high coherence.

A number of methods have been developed to considerthe group sparse problems and multi-channel observations.In the block orthogonal matching pursuit (BOMP) [40] andgroup Lasso (GLasso) [41], the group sparsity is consideredby evaluating the l2-norm of the member entries in each group.The multi-task CS algorithm [42] is a sparse Bayesian learningmethod where the group sparsity is accounted using the sharedpriors. The M-FOCUSS algorithm [43] is another popularlyused group sparse reconstruction method that is developedunder the multiple measurement vector (MMV) model. Thereare different approaches that deal with multichannel observa-tions. Reference [44] considers the joint sparse signal modelto account for multiple aspect observations in a wide-anglesynthetic aperture radar (SAR) imaging. The sparsity similaritybetween different aspect angles is considered through thechange in the scattering magnitude at each pixel. A similartechnique is applied in [45] to achieve enhanced image frommultichannel complex-valued images while preserving thecross-channel information.

In this paper, considering the fact that the clutter profilesshare the same non-zero clutter support across nearby rangecells, we incorporate group sparsity into sparse Bayesianlearning. This incorporation is generally performed within themulti-task CS framework [42]. In this case, the group sparsityis encouraged by exploiting a common precision parameterfor each discretized clutter entry corresponding to all rangecells. We extend the multi-task CS algorithm, which is devel-oped under real-valued data model, to process complex-valuedvariables by using complex Gaussian distributions [39]. It isimportant to note that the observation model is characterizedby the MMV model which assumes identical measurementmatrices for observations from nearby range cells in eachbisatic pair. It is shown that, while only an approximatesolution of the aforementioned precision parameter can beobtained in the general multi-task model, we are able to derivean analytical solution of these parameters under the MMVmodel.

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Another important task of a radar STAP system is the targetdetection and motion parameter estimation of moving targets.Unlike conventional mono-static radars, which only estimatethe radial velocity as the result of single-direction Dopplerfrequency, the multi-static operation of a passive radar systemenables two-dimensional (2-D) motion parameter estimation[46, 47]. In addition, multiple moving targets may be presentin a region and are inseparable in range due to the low rangeresolution, rendering motion parameter estimation difficult.The task of target detection and motion parameter estimationbecomes possible only after effective suppression of the clutter.In this paper, we propose a novel sparse reconstruction basedapproach to acquire 2-D motion parameters by exploiting thegroup sparsity in the velocity domain. The proposed approachenables 2-D velocity estimation of each moving target andoutperforms other techniques that non-coherently combinemulti-static STAP outputs.

To summarize, the key contribution of the paper is two-fold: (1) We propose a novel approach for reliable andhigh-resolution CCM estimation based on a small numberof secondary samples by using the group SBL approach toexploit the group sparsity of clutter profiles across the nearbyrange cells. There is no requirement of i.i.d. property of thesecondary samples. (2) We obtain the 2-D motion parametersof multiple moving targets by utilizing the group sparsityin the velocity domain and coherently combining multi-staticmeasurements.

Notations: We use lower-case (upper-case) bold charactersto denote vectors (matrices). In particular, IN denotes the N×N identity matrix. (.)∗ represents complex conjugate, whereas(.)T and (.)H , respectively, denote the transpose and conjugatetranspose of a matrix or vector. diag(x) represents a diagonalmatrix that uses the elements of x as its diagonal elements ,and std(x) denotes the standard deviation of x. ‖ · ‖22 impliesthe Euclidean (l2) norm of a vector, whereas ‖ · ‖1 denotesthe l1 norm. Pr(·) expresses the probability density function(pdf), and CN (x|a, b) denotes the random variable x followsa complex Gaussian distribution with mean a and varianceb. In addition, ⊗ is the Kronecker product, and vec(·) is thevectorization operator.

II. SYSTEM MODEL

A. Radar Geometry and Signal Model

Consider a multi-static radar scene consisting of K station-ary transmitters and a moving receiver as depicted in Fig.1. The ith transmitter is assumed to be located at a knownposition pTi = [pTi,x, pTi,y, pTi,z]

T and uses a single antennato emit a signal at frequency fi, i = 1, ...,K. The radarreceiver, which utilizes an N -element uniform linear array(ULA) with inter-element spacing d, moves in a straight linewith a constant horizontal velocity vR.

For convenience of notation and without loss of generality,we assume that the receiver moves in the x-direction. Theinitial position of the receiver is expressed in the Cartesiancoordinate system as pR(0) = [pR,x(0), 0, HR]T , and thevelocity vector is vR = [vR, 0, 0]T . As such, the position of

Rx1Tx

P

t1

2Tx

2Ty

px

py1t

r

t2

rRx

1tR2tRrR

z

x

y1Tx mTx Rv

RH1tH

2tHr

Fig. 1. Multi-static passive radar geometry.

the receiver at time instant t is expressed as

pR(t) = [pR,x(0) + vRt, 0, HR]T . (1)

The signal vector received at the receive array is the super-position of the signals corresponding to the K transmitters,expressed as,

x(t) =

K∑k=1

xk(t)+n(t) =

K∑k=1

xks(t)+

K∑k=1

xkc(t)+n(t), (2)

where xk(t) is the combined received signal corresponding tothe kth illuminated source. It consists of two components: thetarget signal xks(t) and the clutter xkc(t). In addition, n(t)is the additive noise, which is characterized as i.i.d. complexGaussian with zero mean. The target signal and the clutter willbe described in the next two subsections.

Note that, in the above expression, we have assumed thatthe received signal is properly conditioned. That is, the direct-path signal from the transmitters to the receiver is perfectlysuppressed through, for example, spatial filtering utilizing theknown geometric relationship between each illuminator andthe receiver. In addition, we assume that the signal waveformtransmitted from each illuminator is perfectly reconstructedas the result of channel error correction coding. These arestandard assumptions commonly used for passive radar [46,48].

B. Target Signal

Assume that Q ground moving targets are located in closeproximity in bistatic ranges, rendering them unresolvable. Theelevation coordinate of all targets is assumed to be 0. Theqth ground moving target is located at pq = [xq, yq, 0]T witha velocity of vq = [vxq, vyq, 0]T . The desired signal xs(t)received at the receiver is expressed as

xs(t) =

K∑k=1

xks(t)

=

K∑k=1

Q∑q=1

√PTk

Gkqσkq

rTkqrqRsk[t− τTkq(t)− τqR(t)]

×e−j2πfk[τTkq(t)+τqR(t)]ak(φq),(3)

where PTkis the transmit signal power of the kth illuminator,

Gkq represents the antenna gain of the kth illuminator in the

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direction of the qth target, and σkq is a complex reflectioncoefficient associated with the radar cross section (RCS) ofthe qth target and depends on the specific illuminator kbecause the RCS varies with the aspect angle. In addition,sk(t) is the waveform transmitted from the kth illuminator,τTkq(t) = rTkq(t)/c and τqR(t) = rqR(t)/c are the timedelays respectively corresponding to the range between thekth illuminator and the qth moving target,

rTkq(t) = ‖pTk− pq(t)‖, (4)

and the range between the qth moving target and the receiver,

rqR(t) = ‖pq(t)− pR(t)‖, (5)

where c is the velocity of light. Furthermore,

ak(φq) = [1, ejκkd sin(φq), ..., ejκk(N−1)d sin(φq)]T ∈ CN (6)

is the steering vector of the receive array toward the directionof the target with a DOA φq , which is defined as the coneangle of the qth target with respect to the x-axis. In the aboveexpression, κk = 2π/λk is the wavenumber with λk = c/fkdenoting the wavelength of the signal transmitted from the kthilluminator. Note that, the DOA is assumed to be time-invariantin the sense that its change during the coherent processinginterval (CPI) is negligible.

For a given area, specified by delay τi,n for the ith bistaticpair, we perform matched filtering by using the reconstructedwaveform of the signal emitted from the ith illuminator as thereference signal. The matched filter output at the L azimuthtime instants tl = lT , l = 0, ..., L− 1, is given by

y(n)is (tl) =

∫ tl+T

tl

xs(t)s∗i (t− τi,n)ej2πfiτi,ndt

=

Q∑q=1

√PTiGiqσiq

rTiqrqRai(φq)

∫ tl+T

tl

si[t− τTiq(t)−τqR(t)]

×s∗i (t− τi,n)e−j2πfi[τTiq(t)+τqR(t)−τi,n]dt,

(7)where rTiq and rqR in the denominator are assumed to beconstant by neglecting the effect of variation of rTiq and rqRon the amplitude within the short CPI. The small phase changewithin the matched filtering window [48] permits choosing thedelay τi,n to align the received signal with the reference signalduring the center of the CPI, i.e., τi,n = (τTin + τnR) + (L−1)T/2. By ignoring the range migration issue due to the shortCPI, the above expression is simplified as

y(n)is (tl) =

Q∑q=1

√PTiGiqσiq

rTiqrqRρiai(φq)e

j2πνiq [tl−(L−1)T/2],

(8)where ρi = ρi(0), with

ρi(τ) =

∫ tl+T

tl

si(t)s∗i (t+ τ)dt (9)

denoting the signal auto-correlation function, which is inde-pendent of tl as most digital waveforms have a stable auto-correlation property [2]. In addition, νiq denotes the Dopplerfrequency of the qth moving target which is determined by the

change rate of the combined bistatic range, expressed as

νiq = − 1

λi· ddt

[rTiq(t) + rqR(t)]

= − 1

λi·[

[pTi− pq(0)]T · vq

‖pTi − pq(0)‖+

[pR(0)− pq(0)]T · vq‖pR(0)− pq(0)‖

+[pR(0)− pq(0)]T · vR‖pR(0)− pq(0)‖

], (10)

where pq(0) is the the initial position of the qth moving target.The total Doppler frequency νiq can be regarded as the com-bination of two components. The first one, which correspondsto the first two terms in Eq. (10), represents the contributionof moving target, whereas the second component, given asthe third term in Eq. (10), represents the contribution of themoving receiver. Note that the first component disappears forstationary clutter, which is discussed in the following sub-section.

Conventional mono-static STAP systems detect movingtargets and estimate the target Doppler frequencies projectedonto the observation direction, thus failing to acquire the 2-Dmotion parameters. In the multi-static operation, on the otherhand, 2-D motion parameters can be estimated due to differenttransmitting aspect angles. It is clear from Eq. (10) that, whilethe second and third terms are common to all transmitters,the first term varies with the illuminator positions due to thediverse observation aspect angles. These illuminator-dependentDoppler frequencies provide sufficient information for thereceiver to acquire 2-D motion parameters of moving targets.Motion parameter estimation is discussed in Section V.

Stack y(n)is (tl) over the L collected azimuth time samples

as

yis =[[y

(n)is (t0)]T , [y

(n)is (t1)]T , ..., [y

(n)is (tL−1)]T

]T=

Q∑q=1

√PTiGiqσiq

rTiqrqRρih(νiq, φq),

(11)where

h(νiq, φq) = b(νiq)⊗ a(φq) ∈ CNL (12)

is the spatio-temporal signature of the qth target, and

b(νiq) =[e−j2πνiq(L−1)T/2, e−j2πνiq(L−3)T/2, ...,

ej2πνiq(L−1)T/2]T ∈ CL (13)

is the temporal signature vector of the same target.

C. Clutter

The clutter received at the receiver is considered as a sum-mation of Nc statistically independent scatterers, expressed as,

xc(t) =

K∑k=1

xkc(t)

=

K∑k=1

Nc∑m=1

√PTk

Gkmσkm

rTkmrmR(t)sk[t− τTkm(t)− τmR(t)]

×e−j2πfk(τTkm+τmR(t))ak(φm),(14)

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where, similar to the target signal, rTkm is the range fromthe kth illuminator and the mth clutter scatterer with m ∈1, · · · , Nc, rmR(t) is the range from the same scatterer tothe receiver, τTkm and τmR(t) are respectively the delayscorresponding to rTkm and rmR(t). In addition, σkm is thereflectivity of the mth clutter scatterer with respect to the kthilluminator, and ak(φm) is its corresponding steering vectorobserved at the receive array. Similar to processing the targetsignals, the matched filter output of the clutter signals atazimuth time tl = lT , l = 0, ..., L − 1, corresponding to theith bistatic pair and the nth range bin, is expressed as

y(n)ic (tl) =

Nc∑m=1

√PTi

Gimσim

rTimrmR(tl)ρiai(φm)ej2πνim(tl−(L−1)T/2),

(15)where

νim = − 1

λi

d

dtrmR(t) = − 1

λivR cosφm (16)

is the Doppler frequency of the mth clutter scatterer and isdetermined by the scatterer-receiver path, where φm is thecone angle between the direction of the scatterer and the arraybaseline. It should be noted in Eq. (16) that the clutter Dopplerfrequency depends on the receiver velocity and the transmitfrequency, but is independent of range cells. This characteristicprovides a feasible way to acquire the clutter profiles in theangle-Doppler domain by exploiting the group sparsity acrossthe nearby range cells.

Stacking y(n)ic (tl) over the L collected azimuth time samples

yields

yic =[[y

(n)ic (t0)]T , [y

(n)ic (t1)]T , ..., [y

(n)ic (tL−1)]T

]T=

Nc∑m=1

√PTi

Gimσim

rTimrmR(tl)ρih(νim, φm),

(17)where

h(νim, φm) = b(νim)⊗ ai(φm) ∈ CNL (18)

is the spatio-temporal signature of the mth scatterer, and

b(νim) =[e−j2πνim(L−1)T/2, e−j2πνim(L−3)T/2, ...,

ej2πνim(L−1)T/2]T∈ CL

(19)is the temporal signature vector of the mth clutter scatterer.

III. GROUP SPARSE BAYESIAN LEARNING METHOD

Because of the group sparsity of the clutter profile in theangle-Doppler domain across nearby range cells, group sparsereconstruction approaches can be applied and are consideredeffective for clutter profile estimation. In this paper, we usethe group SBL method developed in the context of complexGaussian distribution [39] to reconstruct the sparse clutterprofile. This approach is briefly summarized below.

We discretize the angle-Doppler domain into a grid with NdDoppler samples and Ns azimuth samples. Denote wk as anM×1 vector that includes vectorized clutter entries defined inthe entire angle-Doppler domain, where M = NdNs. Considera general multi-task model where the complex measurement

vector of spatio-temporal samples in the kth bistatic pair isdescribed as

yk = Ψkwk + εk, k ∈ [1, · · · ,K], (20)

where the dimension of vector yk is N0 = NL, Ψk ∈ CN0×M

is the kth measurement matrix, and εk denotes an additivecomplex Gaussian noise. When K = 1, it is a canonicalsparse reconstruction problem for the coefficient vector. Thelikelihood of the observed data yk can be expressed as,

Pr(yk |wk,Ψk, β0 ) = CN (yk |Ψkwk, β0IN )

=1

(πβ0)N0exp

(− 1

β0‖yk −Ψkwk‖22

),

(21)

where β0 is the variance of the complex Gaussian noise.The prior distribution of the weight vector wm· =[wm1, · · · , wmK ], which is the mth row of W =[w1, · · · ,wK ] ∈ CM×K , is modeled as a zero-mean complexGaussian distributions with unknown precision (reciprocal ofvariance) αm, i.e.,

wm· ∼ CN (wm·|0, α−1m IK), (22)

where {αm}Mm=1 is a nonnegative parameter controlling thegroup sparsity of wm·. When αm = ∞, the mth row wm·becomes zero. During the learning procedure, most elementsof αm tend to be infinite, due to the mechanism of automaticrelevance determination [29]. Thus sparsity at the group levelis encouraged.

In the group SBL framework, the posterior distribution forwk can be evaluated analytically based on the Bayes’ rule as

Pr(wk|yk,Ψk,α, β0) = CN (wk|µk,Σk),

where

µk = β−10 ΣkΨ

Hk yk, k ∈ [1, · · · ,K], (23)

Σk =(β−1

0 ΨHk Ψk + A

)−1

, (24)

where A = diag(α1, · · · , αM ). Once the parameters α and β0

are estimated, the maximum a posteriori (MAP) estimate ofwk, denoted by wk, can be obtained from the posterior meanas,

wk = µk, k ∈ [1, · · · ,K]. (25)

With known α and β0, the mean and covariance of eachscattering coefficient are derived in (23) and (24). The as-sociated learning problem, in the context of the RVM, thusbecomes searching for the parameters α and β0. The empiricalBayesian estimate for α is determined by maximizing themarginal likelihood [29, 49], or equivalently, its logarithm

{α, β0} = argmaxα,β0

L(α, β0), (26)

6

where

L(α, β0) =

K∑k=1

log Pr(yk|α, β0)

≡K∑k=1

− log |Ck| − yHk C−1k yk, (27)

in which Ck = β0I + ΨkA−1ΨH

k . Following the fast greedyalgorithm based on the marginal likelihood maximizationcriterion [50], we acquire the approximate solution of αm inthe multi-task model as [42]

α(new)m

≈ KK∑k=1

1

a2k,m

(b2k,m − ak,m)

,

if

K∑k=1

1

a2k,m

(b2k,m − ak,m) > 0,

=∞,

if

K∑k=1

1

a2k,m

(b2k,m − ak,m) ≤ 0,

(28)where ak,m , ψHk,mC−1

k,−mψk,m, bk,m , ψHk,mC−1k,−myk,

ψk,m represents the mth column in Ψk, and Ck,−m is Ck

with the contribution of atom ψk,m removed.When Ψ1 = · · · = ΨK = Ψ, the model in Eq. (20) reduces

to a typical multiple measurement vector (MMV) model [43].In this case, unlike approximate solutions in the multi-taskmodel, we can derive an analytical solution of αm as,

α(new)m =

Ka2

m

K∑k=1

b2k,m −Kam, if

∑Kk=1 b

2k,m > Kam,

∞, if∑Kk=1 b

2k,m ≤ Kam,

(29)

where am , ψHmC−1−mψm, ψm represents the mth column

in the common measurement matrix Ψ, and C−m is C =β0I + ΨA−1ΨH with the contribution of atom ψm removed.It is observed that these two solutions of α(new)

m are consistent.Following the approach of MacKay [49], we differentiate

Eq. (26) with respect to β0 and set the result to zero, yielding

β(new)0 =

K∑k=1

‖yk −Ψkµk‖22

K∑k=1

[N0 −M +

M∑m=1

αmΣk,(mm)

] , (30)

where Σk,(mm) denotes the (m,m)th entry in the matrix Σk.Instead of going through all the elements in the M groups

in each iteration, the group SBL algorithm described aboveoperates in a constructive manner, i.e., sequentially adds(or deletes) candidate atom to the model until all relevantatoms, for which the associate weights are non-zero, havebeen included. Therefore, the computational complexity of thealgorithm isO(KMm2

0), with m0 denoting the sparsity, whichis on the same order as the multi-task CS [42]. The complexity

is linearly proportional to M and thus is comparable to thatof the well-known M-FOCUSS [43].

IV. CLUTTER PROFILE ESTIMATION AND SUPPRESSION

A. Group-Sparsity Representation of Clutter Profile acrossRange Cells

Because the non-zero support of clutter profiles is sharedacross nearby range cells, we first employ a small number ofsecondary samples to learn the common clutter support. Theexact clutter profile in the range cell under test is then obtainedthrough the SBL method by only considering the clutter entrieswithin the learned clutter support.

According to the signal and clutter model described inSection II, the output of the matched filter of the receivedsignal in the range cell under test for the ith bistatic pair isexpressed as

y(t)i = y

(t)ic + y

(t)is + y

(t)in = Φi(w

(t)ic + w

(t)is ) + y

(t)in , (31)

where the superscript (t) is used to emphasize the range undertest, w

(t)ic and w

(t)is are, respectively, M × 1 vectorized clutter

and signal entries in the angle-Doppler domain. In addition,y

(t)in denotes its noise component. On the other hand, the

matched filter output of received signals in the nlth secondarysample, where the described target signals are absent, isexpressed for the ith bistatic pair as,

y(nl)i = y

(nl)ic + y

(nl)in . (32)

To ensure the exclusion of the target in the secondarysamples, the range cells corresponding to these samples shouldbe sufficiently separated from the range cell under test by oneor multiple guard cells in each side [9]. Note that passive radarsystems exploit narrowband signals and thus causing a coarserange cell resolution, implying a low probability for a targetto move across multiple range cells within the CPI.

Rewrite the clutter term in Eq. (32) as the product of anoverdetermined dictionary matrix Φi and sparse vector w

(nl)ic ,

we havey

(nl)i = Φiw

(nl)ic + y

(nl)in , (33)

where w(nl)ic ∈ CNdNs .

In this paper, we consider a small number of secondarydata samples from range cells that are close to that undertest, and thus it is well justified that they share the samenon-zero clutter support, i.e., their scatterers are located inthe same positions in the angle-Doppler domain. However,the exact values of the scatterers generally differ. Note thatthe common support assumption is much more relaxed andpractical compared to the i.i.d. requirement. We only need asmall number of secondary samples in this case. The sparseclutter entries w

(nl)ic are separately computed for each bistaic

pair.

B. Clutter Support Estimation and Clutter Suppression

The group SBL algorithm described in Section III canbe used to recover the sparse clutter profiles described asan MMV model in Eq. (33) and subsequently acquire the

7

common clutter support Φi,cs, which is a submatrix of Φi

and acts as a collection of the columns whose correspondingcoefficients in w

(nl)ic are non-zero. One way to confine the

clutter estimate for the range cell under test is to project thereceived signal to the clutter profile through the weighted leastsquare (WLS) method, expressed as,

w(t)ic = (Φ

Hi,csU0Φi,cs)

−1ΦHi,csU0y, (34)

where ul is the lth diagonal entry of the diagonal weightedmatrix U0 and its value is chosen to be the reciprocal of noisevariance. A more reliable solution is to estimate the clutterprofile w

(t)ic by solving the following equation based on the

SBL algorithm, similar to that depicted in Section III,

y(t)i = Φi,csw

(t)ic + y

(t)in . (35)

The posterior mean, as depicted in Eq. (23), will be used asthe estimate of w

(t)ic . Note that, when compared with Eq. (33),

the measurement matrix Φi is replaced by Φi,cs correspondingonly to the clutter support and w

(t)ic is a corresponding subvec-

tor. The target signal term disappeared in Eq. (35), because itis out of the clutter support. As such, the estimated coefficientvector w

(t)ic only involves clutter components, whereas the

target signals are excluded.

Once the clutter coefficient vector w(t)ic is estimated, the

corresponding CCM is expressed as [9, 22]

R(i)z =

M∑m=1

|w(t)ic (νim, φm)|2h(νim, φm)hH(νim, φm)+βi,0INL,

(36)where βi,0 represents the noise power in the ith bistatic pair,which is adaptively estimated from the observation data asdescribed above in Eq. (30).

Based on this result, we acquire the weight vector for therange cell under test in the ith bistatic pair as [9]

ui =

[R

(i)z

]−1

h0(νp, φm)

hH0 (νp, φm)[R

(i)z

]−1

h0(νp, φm), (37)

where h0(νp, φm) is the spatio-temporal signature of thehypothesized target corresponding to Doppler frequency νpand spatial angle φm.

V. MOTION PARAMETER ESTIMATION IN THEMULTI-STATIC PASSIVE SYSTEM

In the conventional mono-static STAP, the Doppler fre-quency of a moving target only reflects its radial velocity,whereas its 2-D motion parameters cannot be estimated. Inthe multi-static passive radar system, on the other hand, the2-D motion parameters can be estimated due to observationsfrom multiple transmitter-receiver pairs with diverse bistaticaspect angles.

Eq. (10) is rewritten according to the geometry of the multi-

static passive system, as depicted in Fig. 1, as

νiq = ν(T )iq + ν

(R)iq , (38)

ν(T )iq = − 1

λi

[pTi− pq(0)]T · vq

‖pTi − pq(0)‖

= −cosφTivxq + sinϕTi

cos θTivyq

λi, (39)

ν(R)iq = − 1

λi

[[pR(0)− pq(0)]T · vq‖pR(0)− pq(0)‖

+[pR(0)− pq(0)]T · vR‖pR(0)− pq(0)‖

]= − 1

λi[cosφrvxq + sinϕr cos θrvyq + cosφrvR] ,

(40)

where φTi and φr are the respective cone angles of the ithtransmitter and the receiver, whereas θTi

and θr , ϕTiand ϕr

are respectively the depression angles and azimuth angles ofthe ith illuminator and the receiver. In Eq. (38), we divide theentire Doppler frequency into two parts, where ν(T )

iq is inducedby the relative motion of the qth moving target with respect tothe ith illuminator. Thus, it varies with the illuminator positionand, as such, enables the estimation of 2-D motion parameters.The other term, ν(R)

iq , is induced by the motions of the targetand the receiver platform and is invariant to the illuminators.

In the following two subsections, we respectively considerthe motion parameter estimation in the presence of a singleand multiple moving targets.

A. Parameter Estimation in Single Moving Target Case

We first consider a simple case where only one movingtarget exists in the spatial angle φm. In the conventional STAP,the estimated weight vector is first used to whiten the clutterprofile and perform matched filtering on the spatio-temporalsignature of the hypothesized target, and individual constantfalse alarm rate (CFAR) detection is then applied. We can thenacquire the corresponding Doppler frequency of the movingtarget in each bistatic pair. From Eq. (38), we obtain

ν = Dv + e, (41)

with

D =

cosφr + cosφT1

λ1

sinϕr cos θr + sinϕT1cos θT1

λ1...

...cosφr + cosφTK

λK

sinϕr cos θr + sinϕTKcos θTK

λK

,(42)

where ν = [ν1, · · · , νK ]T , v = [vx, vy]T , and e =[−cosφrvR/λ1, · · · ,−cosφrvR/λK ]T . Therefore, the 2-Dmotion parameters can be obtained as the least square (LS)solution to Eq. (41), expressed as

v = (DHD)−1DH(ν − e). (43)

Note that the LS algorithm described in Eq. (43) is effectiveonly when the signal power is sufficiently high to allow targetdetection in each bistatic pair. It may become difficult, how-ever, to detect the moving target and acquire its correspondingDoppler frequency for each bistatic pair when the signals are

8

noisy.

B. Parameter Estimation in Multiple Moving Target Case

Multiple moving targets may appear in the same spatialangle φm and are unresolvable in bistatic ranges due to acoarse range resolution. In this case, the application of theLS algorithm for motion parameter estimation in the presenceof multiple targets requires proper association of the Dopplerfrequencies with the corresponding targets [46]. The com-plexity of procedure increases exponentially as the number oftargets increases. On the other hand, as we discussed earlier forthe single-target case, the LS algorithm works only when theDoppler frequency corresponding to each bistatic pair can bereliably estimated. To perform multi-target motion parameterestimation that is robust for noisy signals, a novel jointparameter estimation method based on sparse reconstructiontechniques is proposed.

We first acquire the spatio-temporal weight vector of hy-pothesized target in the ith bistatic pair specified by theDoppler frequency νp and the spatial angle φm as,

upi(φm) =

[R

(i)z

]−1

h(νp, φm)

hH(νp, φm)[R

(i)z

]−1

h(νp, φm), (44)

where

h(νp, φm) = b(νp)⊗ a(φm), (45)

b(νp) =[e−j2πνp(L−1)T/2, e−j2πνp(L−3)T/2, ...,

ej2πνp(L−1)T/2]T, (46)

for p ∈ [1, · · · , P ] with P denoting the total number of thehypothesized Doppler frequency points corresponding to thesame spatial angle φm. The STAP output corresponding to theith bistatic pair can be expressed as,

gpi(φm) = uHpi(φm)y(t)i . (47)

In the traditional threshold detection technique, the existenceof a target is determined for each individual bistatic pair bychoosing a proper threshold. For weak signals, however, targetdetection based on an individual bistatic pair is not reliable.Consider the fact that, although motion-induced Doppler fre-quencies differ for each bistatic pair due to the different bistaticangles and carrier frequencies, they share the same motionparameters in the velocity domain and thus have the groupsparsity. In the proposed algorithm, we project the spatio-temporal signature to the 2-D velocity vector of v = [vx, vy]T

in the velocity space. We uniformly discretize 2-D velocityparameters [−vxmax, vxmax] and [−vymax, vymax] to form anNx × Ny grid. Given a certain spatial angle φm, the spatio-temporal steering vector of the moving target with motionparameters of v(a)

x and v(b)y in the ith bistatic pair is denoted

as

h(a,b)i (φm) = b(f

(a,b)is (φm))⊗ a(φm), (48)

where

f(a,b)is (φm) = − 1

λi

[(cosφTi + cosφm)v(a)

x

+ (sinϕTicos θTi

+ sinϕm cos θr)v(b)y

]+

cosφmvRλi

(49)

is the Doppler frequency for a ∈ [1, · · · , Nx] and b ∈[1, · · · , Ny]. We construct a dictionary matrix Θ(mi) ∈CP×NxNy for the mth spatial angle φm under test in the ithbistatic pair and express the entry of Θ(mi) in the pth row andthe lth column as,

θ(mi)(p, l) = uHpi(φm)h(a,b)i (φm), (50)

where l = (a − 1)Nx + b, p ∈ [1, · · · , P ], and l ∈[1, · · · , NxNy].

According to Eq. (31) and Eq. (47), we have,

gpi(φm) = uHpi(φm)y(t)is + npi, (51)

npi = uHpi(φm)y(t)ic + uHpi(φm)y

(t)in . (52)

The role of the STAP weight vector upi(φm) based on theestimated CCM in Eq. (36) is to suppress the clutter andenhance the target signals through matched filtering. The termscaused by the clutter signal y

(t)ic and the noise signal y

(t)in in

the output gpi(φm) can be defined as a new noise signal npiin Eq. (52) in the new model. As a result, the output signalvector gi = [g1i, · · · , gPi]T ∈ CP can be expressed as,

gi = Θ(mi)ω(mi) + ni, i ∈ [1, · · · ,K] (53)

where ω(mi) ∈ CNxNy and ni = [npi, · · · , nPi]T . The motionparameter estimate problem can thus be formulated as thesparse reconstruction problem of ω(mi) from gi. It shouldbe pointed out that the vectors {ω(mi)}Ki=1 share the samevelocity supports across all bistatic pairs because the truemotion parameters are shared by all illuminators in the velocitydomain, although their corresponding Doppler frequencies ineach bistatic pair are different. It is interesting to note thatEq. (53) is a group sparse reconstruction problem. Therefore,it can be solved using the aforementioned group SBL method,and the estimated motion parameters are the posterior meandepicted in Eq. (23). However, consider the fact that thenumber of moving targets in each range cell is usually smalland can be estimated, other CS methods, such as the BOMPalgorithm, may also yield good performance with a lowercomputational complexity, provided that the their velocitiesare not very close.

VI. SIMULATION RESULTS

Consider a multi-static passive radar system including K =4 stationary illuminators and 1 moving receiver. As depictedin Fig. 2, the illuminators are located several kilometers awayfrom the scene center with a height of 150 m, and theirrespective aspect angles are 0◦, 90◦,−150◦ and −45◦. Theseilluminators emit DVB-T signals with their respective carrierfrequencies from 800 MHz to 860 MHz with a 20 MHzfrequency interval. The initial position of the receiver is [0,

9

1Tx

2Tx

3Tx

4Tx

RxRv

Clutter scene

Moving target

x

yz

Fig. 2. Geometry of multi-static passive in the simulations.

6 ,8] km, and the receiver is equipped with a 20-elementULA whose inter-element spacing is half of the minimumwavelength. The velocity of the receiver is vR = [100, 0, 0]m/s. The azimuth sampling frequency is 600 Hz, yielding anunaliased observation of Doppler frequency between −300 Hzand 300 Hz. 30 azimuthal samples are used to perform STAP.As such, a total number of NL = 600 spatio-temporal samplesis adopted.

In the group SBL reconstruction processing, we empiricallyinitialize β0 = 10−4 × std(y)2, the threshold η = 10−6

for stopping the algorithm, and the maximum number ofiterations is 5000. As the clutter is usually much strongerthan the noise, the above values of parameters β0 and η aresuited in the underlying clutter profile reconstruction problem.In this case, a large value of threshold η will degrade thereconstruction performance because the algorithm would beterminated before a finer steady state performance could bereached. Compared to η, the impact of the noise variance β0

on the reconstruction performance is less direct. A discussionabout this impact is available at [51].

A. Clutter Profiles Estimation

1) Clutter Profiles with Group Sparsity: In Fig. 3, we showthe clutter profiles in the 1st bistatic pair with nt = 4 nearbysecondary samples in the angle-Doppler domain, where eachscatterer coefficient is drawn from a complex Gaussian distri-bution. The angle-Doppler profile of the clutter is discretizedinto a grid of Nd = 90 Doppler bins from −300 Hz to 300 Hzand Ns = 40 angle bins from −180◦ to 180◦, yielding 3600entries. It is shown in [52] that the value of NsNd shouldbe chosen between 4NL and 16NL. In our simulations, theclutter profiles in each angular bin are represented by twoadjacent non-zero random complex entries in each bistaticpair, and thus the total number of non-zero entries is 80, asshown in Fig. 3. It is observed that these clutter profiles in thenearby range cells share the same group sparsity. In addition, acomplex Gaussian noise is added and the clutter-to-noise ratio(CNR) is 40 dB.

2) Reconstruction of Clutter Profiles Based on Group SBL: In the following, the group SBL algorithm is compared withseveral state-of-the-art group sparse reconstruction algorithms,including BOMP [40], GLasso [41], and M-FOCUSS [43]. Toquantitatively evaluate the performances of those algorithms,

we introduce the normalized mean square error (NMSE),defined as‖ϑ− ϑgen‖2/‖ϑgen‖2, to be the performance metric,where ϑ is the estimate of the scatterer coefficient vector ϑgen.

In this simulation, one neighboring secondary sample ineach bistatic pair is used to estimate the common cluttersupport. Fig. 4(a) shows the original clutter profile supportin the 1st bistatic pair, whereas Figs. 4(b)–4(f) shows theestimated support from different algorithms, where the BOMPalgorithm assumes the knowledge of the true sparsity. TheBOMP fails to recover the clutter profile and leads to a largenumber of spurious clutter entries around the true positionsbecause of the high coherence in the measurement matrix.Fig. 5(a) shows the NMSE of the reconstructed clutter profileto quantitatively compare the performance of the differentmethods in reconstructing the clutter profile in the nearbyrange cells. It is observed that the M-FOCUSS algorithm offersa lower reconstruction error than that by GLasso, and the groupSBL method yields the least spurious entries with a minimumNMSE. These results clearly demonstrate the superiority ofthe group SBL method over other methods in the presence ofhighly coherent measurement matrix.

To demonstrate the advantage of group sparsity, we sep-arately perform the SBL algorithm and acquire individualclutter support in each range cell. In the latter case, the overallclutter support is obtained as the union of the individual cluttersupports corresponding to each range cell. This technique isreferred to as the single SBL (S-SBL). It is shown in Fig.4(f) that, without taking the advantage of the group sparsitybetween multiple range cells, the S-SBL yields a poor cluttersupport estimation.

Once the clutter support is estimated, the clutter profilesin the range cell under test can be acquired by using theWLS algorithm in Eq. (34) and the CS algorithms in Eq.(35). The corresponding NMSEs are respectively shown inFigs. 5(b) and 5(c). It is observed that the reconstruction errorobtained from the WLS algorithm, shown in Fig. 5(b), isgenerally higher than that obtained from the correspondingsparse reconstruction method as shown in Fig. 5(c). Due tosuperiority of the CS method over the WLS method, we usethe CS method for the processing in the sequel.

Fig. 6(a) shows the clutter profile of the range cell undertest in the 1st bistatic pair. By using the method proposedin Section IV, we estimate the clutter coefficients based onthe clutter supports which are respectively obtained fromthe BOMP, GLasso, M-FOCUSS, S-SBL, and group SBLalgorithms. The results are respectively depicted in Figs. 6(b)-6(f). To quantitatively evaluate the performance of clutterscatterer coefficients, the output signal-to-inference-plus-noiseratio (SINR) loss, which is defined as the difference betweenoutput SINR and the output signal-to-noise ratio (SNR) [9], isevaluated, and the results are depicted in Fig. 5(d). It is ob-served that the SINR loss of the group SBL algorithm closelyapproaches the Clairvoyant solution. This is attributed to thegroup SBL algorithm which, under the MMV model basedon nonparametric Bayesian model, can acquire an improvedclutter profile estimation, even though the measurement matrixis highly coherent.

To summarize, the proposed method avoids the restrictive

10

cos

Dop

pler

freq

uenc

y

-1 0 1-300

-200

-100

0

100

200

cos -1 0 1

cos -1 0 1

cos -1 0 1

300

Fig. 3. Clutter profiles with 4 nearby secondary samples.

requirement of a high number of the secondary data samples asin the conventional STAP and acquires an accurate estimationof the CCM for clutter suppression by utilizing only a smallnumber of secondary samples.

cos

Dop

pler

freq

uenc

y

-1 -0.5 0 0.5 1-300

-200

-100

0

100

200

300

(a)

cos

Dop

pler

freq

uenc

y

-1 -0.5 0 0.5 1-300

-200

-100

0

100

200

300

(b)

cos

Dop

pler

freq

uenc

y

-1 -0.5 0 0.5 1-300

-200

-100

0

100

200

300

(c)

cos

Dop

pler

freq

uenc

y

-1 -0.5 0 0.5 1-300

-200

-100

0

100

200

300

(d)

cos

Dop

pler

freq

uenc

y

-1 -0.5 0 0.5 1-300

-200

-100

0

100

200

(e)

cos

Dop

pler

freq

uenc

y

-1 -0.5 0 0.5 1-300

-200

-100

0

100

200

300

(f)

Fig. 4. Estimated clutter profile support in the 1st bistatic pair andestimated NMSE. (a) Original clutter support. (b) Reconstructedclutter support using BOMP. (c) Reconstructed clutter support usingGLasso. (d) Reconstructed clutter support using M-FOCUSS. (e)Reconstructed clutter support using group SBL. (f) Reconstructedclutter support using S-SBL.

3) Effect of Clutter Correlation on Reconstructed Clut-ter Profile: All simulations above are performed under theassumption that the clutter scattering coefficients in nearbyrange cells and range cell under test are independent. Inpractice, this assumption may not be met in the real-worldclutter environment. In this case, we investigate effect ofthe correlated clutter on the sparse clutter reconstruction in

1 2 3 4

10−3

10−2

10−1

100

101

Bistatic pairN

MS

E

BOMPGLassoM−FOCUSSS−SBLGroup SBL

(a)

1 2 3 410

−3

10−2

10−1

100

Bistatic pair

NM

SE

BOMPGLassoM−FOCUSSS−SBLGroup SBL

(b)

1 2 3 4

10−3

10−2

10−1

100

Bistatic pair

NM

SE

BOMPGLassoM−FOCUSSS−SBLGroup SBL

(c)

−50 0 50

−30

−20

−10

0

Doppler frequency (Hz)

Out

put S

INR

loss

(dB

)

ClairvoyantGLassoBOMPM−FOCUSSS−SBLGroup SBL

(d)

Fig. 5. Clutter reconstruction and output SINR loss Performances. (a)NMSE of the reconstructed clutter profiles in the nearby range cellsobtained from different methods. (b) NMSE of the estimated clutterprofile in the range cells under test using the WLS algorithm. Eachcurve corresponds to a different clutter support estimation method.(c) NMSE of the reconstructed clutter profiles in the range cellsunder test using the CS algorithm. Each curve corresponds to adifferent clutter support estimation method. (d) Output SINR losscorresponding to Fig. 5(c).

the proposed method. Fig. 7 shows the reconstructed averageNMSE versus correlation of the clutter scattering coefficients,which varies from 0 to 0.9. It is observed that the proposedmethod has outstanding reconstruction performance with avery small NMSE, no matter what the correlation is. As aresult, we can conclude that the proposed method is robust tothe clutter correlation.

B. Motion Parameter Estimation

1) Parameter Estimation in Single Moving Target Case:In the following simulation, a moving target located at spa-tial angle φm = 106.86◦, with x- and y-axis velocities of(15.0, 10.2) m/s, is added in the under test range cell. TheDoppler frequencies corresponding to the four bistatic pairsare respectively −131.76 Hz, −121.93 Hz, −45.70 Hz, and−108.37 Hz. These Doppler frequencies are shown in vertical

11

cos

Dop

pler

freq

uenc

y

-1 -0.5 0 0.5 1-300

-200

-100

0

100

200

300

(a)

cos

Dop

pler

freq

uenc

y

-1 -0.5 0 0.5 1-300

-200

-100

0

100

200

300

(b)

cos

Dop

pler

freq

uenc

y

-1 -0.5 0 0.5 1-300

-200

-100

0

100

200

300

(c)

cos

Dop

pler

freq

uenc

y

-1 -0.5 0 0.5 1-300

-200

-100

0

100

200

300

(d)

cos

Dop

pler

freq

uenc

y

-1 -0.5 0 0.5 1-300

-200

-100

0

100

200

300

(e)

cos

Dop

pler

freq

uenc

y

-1 -0.5 0 0.5 1-300

-200

-100

0

100

200

300

(f)

Fig. 6. Estimated clutter profile in the range cells under test in the1st bistatic pair. (a) Original clutter profile. (b) Estimated clutterprofile using BOMP. (c) Estimated clutter profile using GLasso. (d)Estimated clutter profile using M-FOCUSS. (e) Estimated clutterprofile using S-SBL. (f) Estimated clutter profile using the proposedmethod.

0 0.2 0.4 0.6 0.8 110-4

10-3

10-2

10-1

100

Correlation

NM

SE

Fig. 7. Effect of clutter correlation on NMSE.

dash lines in Fig. 8(a). The signal-to-clutter ratio (SCR) is −36dB. Fig. 8(a) shows the output amplitudes of the conventionalSTAP in which the weight vector is obtained using the groupSBL algorithm under the MMV model. In the traditionalway, the maximum peaks will be selected. The correspondingestimated frequencies are respectively −129.20 Hz, −128.38Hz, −44.51 Hz and −115.02 Hz for the four bistatic pairsthrough 32 times of interpolation operation. We computer themotion parameters by utilizing the WLS approach describedin Eq. (43) and acquire the result of vx = 15.30 m/s andvy = 10.12 m/s. The estimated Doppler frequencies in the2nd and 4th bistatic pairs have an offset of about 6.45 Hz and

6.65 Hz, respectively, yielding velocity estimation errors. TheCS-based method proposed in Section V effectively alleviatesthis effect by utilizing multiple transmitters configuration inthe passive radar system. The group SBL is used to estimatethe 2-D motion parameters, where the ranges of vx and vy areboth from −50.0 m/s to 50.0 m/s and the velocity step is 0.5m/s. We acquire the estimated motion parameter of vx = 15.0m/s and vy = 10.0 m/s.

-200 -150 -100 -50 0 500

0.01

0.02

0.03

Doppler frequency (Hz)

Am

plitu

de

1st pair2nd pair3rd pair4th pair

True object

(a)

-200 -150 -100 -50 0 500

0.01

0.02

0.03

Doppler frequency (Hz)

Am

plitu

de

1st pair2nd pair3rd pair4th pairT1T2

(b)

Fig. 8. Motion parameter estimation of moving targets. (a) Single target; (b)Multiple targets.

2) Parameter Estimation in the Multiple Moving TargetsCase: Next, we consider a more general case with multi-ple moving targets and a low SNR. We add an additionalmoving target with motion parameters vx = −5.6 m/s andvy = 18.0 m/s in the same spatial angle φm = 106.86◦.All other parameters remain the same as those in the singletarget case. Fig. 8(b) shows the output amplitudes in eachbistatic pair, where T1 and T2 represent the correspondingDoppler frequencies from the 1st and 2nd moving targets,respectively. The vertical lines respectively denote the trueDoppler frequencies of those two targets corresponding tothe four bistatic pairs. It is noted that the output amplitudecorresponding to the 2nd moving target in the 4th pair, whichis located around a true Doppler frequency of −89.33 Hz, isnearly zero. In this case, the Doppler frequency offset causedby the target motion parameter is only −6.2 Hz and, as such,the moving target signal is unresolvable from the clutter and isthus substantially suppressed by the designed spatio-temporalfilter. The individual Doppler frequencies are difficult to beaccurately estimated in this case and, as such, the LS algorithmcompletely fails to determine the motion parameters.

Before we present the results obtained from the group SBLalgorithm, we perform a conventional non-coherent imagefusion method to estimate motion parameters of multipletargets [46]. Fig. 8(b) shows the output amplitude distributionswith respect to the Doppler frequency in each bistatic pair.Based on the relationship in Eq. (49) between the Dopplerfrequencies and motion parameters, we represent those am-plitude distributions in the vx and vy domain, as shown inFigs. 9(a)−9(d). One Doppler frequency corresponds to a linein the velocity domain according to the linear relationshipin Eq. (49) and the strength of the line is represented bythe output amplitude in each Doppler frequency point. Theslope of the line is completely determined by the geometryof the illuminators and the receiver, and thus differs for eachbistatic pair. For comparison, we depict a heat image in Fig.9(e) as the result of an intuitional non-coherent method thatexploits those acquired images in each bistatic pair. However,

12

this method has poor velocity resolution. As can be observed,the combined strength formed from a relatively weak targetmay still be weaker than the sidelobes of a strong target dueto the difference in the target signal power. Accordingly, it isdifficult to estimate the correct motion parameters. Fig. 9(f)shows the result obtained from the proposed method. It is clearthat the proposed algorithm successfully estimates the motionparameters with [15.0, 10.5] m/s for the 1st moving target and[−5.5, 18.5] m/s for the 2nd moving target.

Vx (m/s)

Vy

(m/s

)

-20 0 20-30

-20

-10

0

10

20

30

0

0.01

0.02

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Fig. 9. Motion parameter estimation in the multiple moving targetcase. (a)–(d) Outputs amplitude distribution in the 4 bistatic pairs.(e) Motion parameter estimation based on non-coherent fusion. (f)Motion parameter estimation based on the proposed method.

VII. CONCLUSION

Conventional space-time adaptive processing (STAP) suffersfrom the high number requirement of independent identicallydistributed (i.i.d.) secondary data samples for a reliable esti-mation of statistical clutter covariance matrix (CCM) so as toachieve effective clutter suppression in the range cell undertest. One possible solution to relax this condition is to utilizethe range cell data under test to estimate its CCM. The sparseBayesian learning (SBL) technique provides a feasible meansfor accurate reconstruction of the clutter profile in the angle-Doppler domain, particularly when the measurement matrixhas a high coherence.

In this paper, a novel method was proposed to estimatethe CCM based on a small number of secondary samples.The common clutter support was estimated by exploitingthe fact that this support is shared across neighboring rangecells. This estimate is used in the range cell under test toacquire the clutter profile with the target signal excluded.This strategy does not require i.i.d. secondary samples, neither

does it include the target signals in the estimated CCM. Thegroup sparse problem is effectively solved using the groupSBL algorithm in the multiple measurement vector model toachieve accurate and robust performance in the presence ofhighly coherent measurement matrix that relates the spatio-temporal measurements and the angle-Doppler domain clutterprofile.

Another important contribution of this paper was the groupsparsity-based two-dimensional motion parameter estimationof a single or multiple ground moving targets. While thetraditional least-square solutions provide a simple approach toachieve this objective, they are infeasible when dealing withmultiple target case, or when the Doppler estimation from eachindividual bistatic pair is unreliable. The proposed technique,on the other hand, achieves improved motion parameter esti-mation in such challenging situations.

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