In this work, we consider a two-channel multiple-input multiple-output (MIMO) passive detection problem, in which there is a surveillance array and a reference array. The reference array is known to carry a linear combination of broadband noise and a subspace signal of known dimension but unknown basis. The question is whether the surveillance channel carries a linear combination of broadband noise and a subspace signal of unknown basis, which is correlated with the subspace signal in the reference channel. We consider a second-order detection problem where these subspace signals are structured by an unknown, but common, p-dimensional random vector of symbols transmitted from sources of opportunity, and then received through unknown M × p matrices at each of the M-element arrays. The noises in each channel have arbitrary spatial correlation. We derive the generalized likelihood ratio test (GLRT) statistic and show it is a monotone function of canonical correlations between the reference and surveillance channels

Santamaría Caballero, Luis Ignacio

Scharf, Louis L.

Wang, Haonan

Wang, Yuan

English

UCrea

CANONICAL CORRELATIONS FOR TARGET DETECTION IN A PASSIVE RADARNETWORKYuan Wang1, Louis L. Scharf 2, Ignacio Santamarı́a3, and Haonan Wang41Department of Mathematics and Statistics, Washington State University, Pullman, WA, USA2Department of Mathematics, 4Department of Statistics, Colorado State University, Fort Collins, CO, USA3Department of Communications Engineering, University of Cantabria, Santander, Spainywang@math.wsu.edu, scharf@engr.colostate.edu, nacho@gtas.dicom.unican.es, wanghn@stat.colostate.eduABSTRACTIn this work, we consider a two-channel multiple-inputmultiple-output (MIMO) passive detection problem, inwhich there is a surveillance array and a referencearray. The reference array is known to carry a lin-ear combination of broadband noise and a subspacesignal of known dimension but unknown basis. Thequestion is whether the surveillance channel carries alinear combination of broadband noise and a subspacesignal of unknown basis, which is correlated with thesubspace signal in the reference channel. We consider asecond-order detection problem where these subspacesignals are structured by an unknown, but common,p-dimensional random vector of symbols transmittedfrom sources of opportunity, and then received throughunknown M × p matrices at each of the M -elementarrays. The noises in each channel have arbitrary spatialcorrelation. We derive the generalized likelihood ratiotest (GLRT) statistic and show it is a monotone func-tion of canonical correlations between the referenceand surveillance channels.Index Terms—Passive detection, MIMO channels, pas-sive radar, generalized likelihood ratio, canonical coor-dinates.I. INTRODUCTIONThis paper is motivated by a passive radar applica-tion, where the problem is to determine if there arecomplex demodulations and synchronizations in sev-eral surveillance antennas (or antenna arrays) that bringsignals in the surveillance antennas into coherencewith signals in the reference antennas. This coherenceis manifested in the synchronous sharing of transmitsymbols from several opportunistic transmitters (e.g.digital television, digital audio broadcast, or mobilecommunication systems), and as a consequence thereis correlation between signals observed at the MIMOsurveillance array and the MIMO reference array. Sothe problem is to detect correlated subspace signals ofdimension-p in two MIMO channels. In passive radarthe signal paths for the reference and the surveillancechannels are typically separated by digital beamform-ing using directional antennas.The conventional approach for passive detection usesthe cross-correlation (CC) between the data receivedin the reference and surveillance channels as thetest statistic [1]. However, the noise in the referencesignal renders the CC detection scheme suboptimal,especially in MIMO scenarios for which the inherentlow-dimensional subspace structure of the transmittedsignal can be exploited [2]. Passive MIMO target detec-tion with a noisy reference channel has recently beenconsidered in [3], where the transmitted waveform isconsidered as a deterministic unknown. The authors of[3] derived the generalized likelihood ratio test (GLRT)for this deterministic target model under spatially whitenoise of known variance. The work in [4] derivesthe GLRT in passive radar problem that models thereceived signal as a deterministic waveform scaled byan unknown single-input single-output (SISO) channeland under white noise of either known or unknownvariance. A passive detector that exploits the low-rankstructure of the received signal has been proposed in[5]. Instead of computing the cross-correlation betweenthe surveillance and reference channel measurements,the ad-hoc detector proposed in [5] cross-correlatesthe dominant left singular vectors of the matricescontaining the observations acquired at both channels.Detection of a dimension-one subspace signal with asingle array of sensors under white noise of unknownlevel has been addressed in [6], and extensions todiagonal noise covariance matrices and dimension-psubspace signals can be found in [7]. Other variantsof this problem, motivated by cognitive radio andmulti-static radar applications, and notably based onaveraging with respect to Haar measure on the spaceof dimension-p subspaces, have been considered in[8] and the references therein. These detection prob-lems are solved with a single array of sensors atthe surveillance channel (for radar applications) or atthe secondary user (for cognitive radio applications).The model considered in this paper is solved withthe assistance of an additional multi-antenna referencechannel which acquires a noisy and distorted versionof the transmitted signal.In this paper, we address the MIMO passive de-tection problem in a multivariate normal model whenthe surveillance and reference channels are equippedwith M antennas, the transmitted signal is an un-known rank-p signal, and the noises at surveillanceand reference channels are uncorrelated between them,but each having different spatial correlation models.It turns out that this is a problem in factor analysis[9], where there are constraints on the factor loadingsand the factors. The problem may be viewed as a one-channel factor analysis problem with constraints on thefactor loadings under the null hypothesis, or as a two-channel factor analysis problem, with constraints onthe factor loadings under the null, and with commonfactors under the alternative. We derive the generalizedlikelihood ratio test (GLRT) as a ratio of determinantsof covariance matrices. In the general case when thenoise covariaces are unstructured, the test statistic is amonotone function of canonical correlations betweenthe reference and surveillance channels.Notation: The superscripts (·)T and (·)H denotetranspose and Hermitian, respectively. The determinant,trace and Frobenius norm of a matrix A will bedenoted, respectively, as det(A), tr(A) and ||A||F . IMis the identity matrix of dimensions M×M , and 0 de-notes either a column vector with M zeros, or the zeromatrix of appropriate dimensions (the difference shouldbe clear from the context). We use A1/2 (A−1/2)to denote the square root matrix of the Hermitianmatrix A (A−1); diagM (A) is a block-diagonal matrixformed by M × M blocks on the diagonal of A.The expectation operator will be denoted by E[·], andx ∼ CNM (0,R) indicates that x is an M -dimensionalcomplex circular Gaussian random vector of zero meanand covariance R.II. PROBLEM FORMULATIONOur two-channel measurement model is[xs[n]xr[n]]=[θHsHr]s[n] +[vs[n]vr[n]]; n = 1, 2, . . . , N(1)where xs[n] ∈ CM and xr[n] ∈ CM are the surveil-lance and reference measurements; s[n] ∈ Cp containsthe signal transmitted by p opportunistic illuminators,Hs ∈ CM×p and Hr ∈ CM×p represent the M × pchannels from the transmitter(s) to the surveillanceand reference multiantenna receivers, respectively. Theparameter θ ∈ {0, 1} determines whether or not thereis a signal Hss[n] in the surveillance channel.We treat the symbol sequence as a sequence ofcircular, Gaussian random vectors with unknown co-variance E[s[n]s[m]H ] = Cδ[n − m]. The factorloadings Hs and Hr are unknown, to be identifiedin a maximum likelihood procedure. Without loss ofgenerality, the symbol covariance may be absorbed intothese factor loadings and thus we assume C = Ip. Thevectors vs[n] and vr[n] model the additive noise. Fornotational convenience, the signal, noise, and channelvectors can be stacked as x[n] = [xs[n]T ,xr[n]T ]T ,v[n] = [vs[n]T ,vr[n]T ]T and H = [HTs ,HTr ]T ,respectively. The additive noise is assumed to betemporally white, zero-mean Gaussian distributed, anduncorrelated between the surveillance and referencechannels. The noise covariance matrix can then bewritten asE[v[n]v[m]H ] =[Σss 00 Σrr]δ[m− n] (2)where Σss and Σrr are arbitrary psd matrices:The passive detection problem is to test the hypoth-esis that the surveillance channel contains no signal,versus the alternative that it does:H0 : θ = 0H1 : θ = 1(3)Denote by R0 and R1 the set of measurement covari-ance matrices under the null hypothesis and alternativehypothesis, respectively. We haveR0 ={[0 00 HrHHr]+[Σss 00 Σrr]}(4)R1 ={[HsHHs HsHHrHrHHs HrHHr]+[Σss 00 Σrr]}. (5)This detection problem essentially amounts to testingbetween two different structures for the compositecovariance matrix under the null hypothesis and alter-native hypothesis. It can be written asH0 : x[n] ∼ CN 2M (0,R), R ∈ R0H1 : x[n] ∼ CN 2M (0,R), R ∈ R1.(6)There are two possible interpretations of this model:(1) it is a one-channel factor model with special con-straint on the loadings under H0; or (2) it is a twochannel factor model with constraint under H0 andcommon factors in the two channels.III. THE GENERALIZED LIKELIHOODRATIOLet us now consider N consecutive array snapshotsunder a model with generic covariance matrix R[x[1] . . . x[N ]]= X ∈ C2M×N , (7)which are i.i.d. realizations of x[n] ∼ CN 2M (0,R). Asthere are unknown parameters under both hypotheses,the Neyman-Pearson detector is not implementable forthis composite test. Therefore, we adopt a generalizedlikelihood ratio test (GLRT), which usually resultsin simple detectors with good performance [9]. Thelikelihood may be written asf(X; R) =1π2MN det(R)Nexp{−N tr(SR−1)},(8)where S = 1N XXH is the sample covariance matrix,partitioned asS =[Sss SsrSHsr Srr]. (9)Here Sss is the sample covariance matrix of thesurveillance channel and the other blocks are definedsimilarly. The generalized likelihood ratio (GLR) isΛ =f(X; R̂1)f(X; R̂0),where R̂0 and R̂1 are, respectively, the maximumlikelihood (ML) estimates of the covariance matrix formodel j under H0 and H1. They maximize the log-likelihood functionL(R) = logdet(SR−1)− tr(SR−1), (10)The GLRT for noise model j reduces tolog(Λ) = log(det(R̂0)det(R̂1))−tr(S(R̂−11 − R̂−10))H1≷H0η,(11)with η a suitable threshold.The ML estimate of the covariance matrix under thenull is given byR̂0 = diagM (S) =[Sss 00 Srr]. (12)Under the alternative, the ML estimate has been derivedfor p = 1 in [2].To present the result for general p, letC = Sss−1/2SsrSrr−H/2 be the sample coherence ma-trix between the surveillance and reference channels,and let C = FKGH be its singular value decomposi-tion (SVD), where the matrix K = diag (k1, · · · , kM )contains the sample canonical correlations 1 ≥ k1 ≥· · · ≥ kM ≥ 0 along its diagonal. The ML estimate isR̂1 =[Sss S1/2ss CpS1/2rrS1/2rr CHp S1/2ss Srr](13)where Cp = FKpGH and Kp =diag (k1, · · · , kp, 0, . . . , 0) is a rank-p truncation ofK. As a result, the GLRT isΛ = det(I−K2p)−1 =p∏i=11(1− k2i )H1≷H0η, (14)where ki is the i-th sample canonical correlation be-tween the surveillance and reference channels, and ηis a suitable threshold. This formula was also derivedin [10] for a different problem. Equation (14) has aninteresting interpretation: 1 − Λ4−1 is the coherencestatistic, 0 ≤ 1−∏pi=1(1− k2i ) ≤ 1.Remark 1: Connection to Generalized Coherence.If the covariance matrix under H0 were assumed onlyblock diagonal and under H1 it were assumed anarbitrary PSD matrix, the GLRT statistic would be thefollowing generalized Hadamard ratio:H =det(S)det(Sss) det(Srr)=M∏i=1(1− k2i ) (15)Notice also that 1 − H is the Generalized Coherence(GC) originally defined in [11], and widely applied tomulti-channel detection problems. So the net of priorknowledge of rank p is to replace M by p in thecoherence statistic.Remark 2: The detection problem and its GLRTare invariant to the transformation group G = {G |G(X) = TXQ}, where the 2M × 2M matrix T isa diagonal matrix of nonsingular M × M matricesT1,T2, and Q is an N × N unitary matrix. As aspecial case of this general invariance, the coherencedetector is CFAR against unknown noise power inthe surveillance chanel and unknown signal-plus-noisepower in the reference channel. This is quite obviouslya desirable property.IV. SIMULATION STUDIESIn this section we evaluate the performance of theGLR detectors by means of Monte Carlo simulations.The input signal-to-noise-ratio (SNR) for both thesurveillance and reference channels is defined asSNRi = 10 log10tr(HHi Hi)tr(Σii), i = {s, r}.For given values of SNRs and SNRr, the probability ofdetection, Pd, and probability of false alarm, Pfa, areestimated by averaging 104 independent simulations,where at each simulation a different realization of theunknowns (Hr,Hs,Σss,Σrr) is generated.Fig. 1 depicts the ROC for the coherence detectorfor p = 1 (dimension-one subspace signal), M = 5antennas and N = 100 snapshots. The curves aroundthe chance line are ROCs for detectors, not discussedhere, which are mis-matched to the assumed noisemodel.Pfa0 0.2 0.4 0.6 0.8 1Pd00.20.40.60.81GLRT Model 1GLRT Model 2GLRT Model 3GLRT Model 4Fig. 1. ROC for coherence (or canonical correlation)detector, with M = 5 antennas, p = 1, N = 100snapshots and SNRs = SNRr = −4 dB.V. CONCLUSIONIn this paper we have addressed a problem motivatedby passive radar. The problem is to detect a commonsubspace signal in two MIMO channels. It turns outthat the problem is a problem in factor analysis, wherethere are constraints on the factor loadings and thefactors. The problem may be viewed as a one-channelfactor analysis problem with constraints on the factorloadings under the null hypothesis, or as a two-channelfactor analysis problem, with constraints on the factorloadings under the null, and with common factorsunder the alternative. The GLRT compares the product∏p1(1− k2i ) to a threshold, where the ki’s are squaredcanonical coordinates of the two channel sample co-variance matrix. The product may be replaced by1−∏p1(1−k2i ), which is coherence, so that the detectoris a coherence detector.VI. REFERENCES[1] J. Liu, H. Li, and B. Himed, “On the performanceof the cross-correlation detector for passive radarapplications,” Signal Process., vol. 113, pp. 32–37, 2015.[2] I. Santamaria, L. L. Scharf, D. Cochran, andJ. Via, “Passive detection of rank-one signalswith a multiantenna reference signal,” in Euro-pean Signal Processing Conference (EUSIPCO),Budapest, Hungary, Sep. 2016.[3] D. E. Hack, L. K. Patton, B. Himed, and M. A.Saville, “Detection in passive MIMO radar net-works,” IEEE Trans. Signal Process., vol. 62,no. 11, pp. 2999–3012, Jun. 2014.[4] G. Cui, J. Liu, H. Li, and B. Himed, “Targetdetection for passive radar with noisy referencechannel,” in Proceedings of 2014 IEEE RadarConference, Cincinnati (OH), USA, May 2014.[5] S. Gogineni, P. Setlur, M. Rangaswamy, and R. R.Nadakuditi, “Random matrix theory inspired pas-sive bistatic radar detection of low-rank signals,”in IEEE Int. Conf. Acoust., Speech and SignalProc. (ICASSP), Brisbane, Australia, Apr. 2015.[6] O. Besson, S. Kraut, and L. Scharf, “Detection ofan unknown rank-one component in white noise,”IEEE Trans. Signal Process., vol. 54, no. 7, pp.2835–2839, Jul. 2006.[7] D. Ramirez, G. Vazquez-Vilar, R. Lopez-Valcarce, J. Via, and I. Santamaria, “Detectionof rank-P signals in cognitive radio networkswith uncalibrated multiple antennas,” IEEE Trans.Signal Process., vol. 59, no. 8, pp. 3764–3774,Aug. 2011.[8] S. Sirianunpiboon, S. D. Howard, andD. Cochran, “Multiple-channel detection ofsignals having known rank,” in IEEE Int. Conf.Acoust., Speech and Signal Proc. (ICASSP),Vancouver, Canada, May 2013, pp. 6536–6540.[9] K. V. Mardia, J. T. Kent, and J. M. Bibby,Multivariate Analysis. Academic Press, 1979.[10] P. Stoica, K. M. Wong, and Q. Wu, “On anonparametric detection method for array signalprocessing in correlated noise fields,” IEEE Trans.Signal Process., vol. 44, no. 4, pp. 1030–1032,Apr. 1996.[11] D. Cochran, H. Gish, and D. Sinno, “A geometricapproach to multiple channel signal detection,”IEEE Trans. Signal Process., vol. 43, pp. 2049–2057, Sep. 1995.

Canonical correlations for target detection in a passive radar network

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Canonical correlations for target detection in a passive radar network

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