1396 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. …...ﬁltering, a connection tomutual information forWSS Gaussian random processes, as well as providing a generalization of the

1396 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 6, MARCH 15, 2014

Detection of Spatially Correlated Time Series From aNetwork of Sensor Arrays

Nick Klausner, Student Member, IEEE, Mahmood R. Azimi-Sadjadi, Senior Member, IEEE, andLouis L. Scharf, Life Fellow, IEEE

Abstract—This paper addresses the problem of testing for theindependence among multiple ( ) random vectors. The gener-alized likelihood ratio test tests the null hypothesis that the com-posite covariance matrix of the channels is block-diagonal, using ageneralized Hadamard ratio. Using the theory of Gram determi-nants, we show that this Hadamard ratio is stochastically equiva-lent to a product of scalars, which are independently drawn froma beta distribution under the null hypothesis. This result is thenused to derive an asymptotic null distribution, which can be usedto identify an appropriate threshold when the sample support islarge. These results are then extended to the problem of detectingthe presence of spatially correlated time series when each observeremploys an array of sensors. Assuming wide-sense stationary pro-cesses in both time and space, the likelihood ratio is shown to in-volve a Hadamard ratio of an estimated cross-spectral matrix atevery frequency/wavenumber pair. The proposed detector is com-pared to several alternative detectors, using simulated space-timefields.

Index Terms—Broadband coherence, coherence, cross-spectralmatrix, generalized coherence, generalized likelihood ratio test(GLRT), multichannel coherence, multichannel signal detection,spatio-temporal processing.

I. INTRODUCTION

D ETECTING the presence of a common but unknownsignal among two or more data channels is a problem

that finds its uses in many applications, including collaborativesensor networks [1], geological monitoring of seismic activity[2], radar [3], and sonar [4], [5]. Some detection systemsin such situations use decision fusion to combine individualdetection decisions into one global decision. However, thisdetection paradigm is sub-optimal as local decisions are basedon the perspective of a single sensory system. Thus, methodsthat capture the coherent or mutual information among multipledata sets are needed.Multichannel detection has been considered in [6], [7], and

[8]. In [6], a geometric approach to multi-channel detection is

Manuscript received March 30, 2013; revised August 01, 2013 and August01, 2013; accepted December 12, 2013. Date of publication January 09, 2014;date of current version February 24, 2014. The associate editor coordinating thereview of this manuscript and approving it for publication was Dr. Antonio DeMaio. This work was supported by the Office of Naval Research (ONR) undercontracts N00014-12-C-0017 and N00014-12-1-0154 and the Air Force Officeof Scientific Research (AFOSR) under contract FA 9550-10-1-0241.N. Klausner and M. R. Azimi-Sadjadi are with the Department of Electrical

and Computer Engineering, Colorado State University, Fort Collins, CO 80523USA (e-mail: [email protected]; [email protected]).L. L. Scharf is with the Departments of Mathematics and Statistics, Colorado

State University, Fort Collins, CO 80523 USA (e-mail: [email protected]).Digital Object Identifier 10.1109/TSP.2014.2298833

proposed by defining the generalized coherence (GC) amongmultiple channels, which is shown to be a natural extensionof the magnitude-squared coherence (MSC) for more than twochannels. Under the assumption that the observations from eachchannel contain white, complex normal noise, the authors de-rived closed-form expressions for the null distributions of boththe MSC and three-channel GC measures. This leads to a re-cursive formulation for finding the null distribution as one addsadditional channels. A similar technique was considered in [7]by forming a Generalized Likelihood Ratio Test (GLRT) andusing the assumption that observations are zero-mean, complexnormal random vectors. Given multiple independent realiza-tions of this random vector, the GLRT involves testing whetherthe sample covariance matrix has diagonal structure under thenull hypothesis versus any arbitrary, positive-definite (PD) co-variance structure under the alternative. In both [6] and [7], thedetection statistic applies to temporally white but spatially cor-related Gaussian sequences, and is given by the determinant ofthe sample covariance matrix over the product of its diagonalelements, i.e., a Hadamard ratio.The work in [6] and [7] was recently extended in [8] by con-

sidering the detection of both spatially and temporally corre-lated time series. Given multiple independent realizations of avector-valued time series, the GLRT of [8] tests whether or notthe space-time covariance matrix is block-diagonal. The GLRTis a generalized Hadamard ratio involving the sample covari-ance matrix. Assuming temporally wide-sense stationary pro-cesses, and allowing the length of each time series to grow large,the test statistic is shown to be a function of frequency-depen-dent Hadamard ratios for narrowband cross spectral matrices. Ateach frequency this Hadamard ratio is a narrowband coherencestatistic that measures linear dependence among the time seriesat that frequency. The log of each such narrowband coherenceis integrated over the Nyquist band to produce the broadbandcoherence statistic. This GLRT is shown to exhibit many ap-pealing properties including invariance to channel-by-channelfiltering, a connection to mutual information for WSS Gaussianrandom processes, as well as providing a generalization of theMSC spectrum for more than two channels [8].In this paper, we consider the detection problem addressed

in [8]. Using the theory of linear prediction and its connectionwith Gram determinants, we show that the likelihood ratio canbe written as a product of independent beta random variablesunder the null hypothesis that the covariance matrix is block-di-agonal. This extends the result in [6] to the case of temporallycorrelated time series and makes it possible to set thresholdsfor false alarm control. Assuming that the number of indepen-

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KLAUSNER et al.: DETECTION OF SPATIALLY CORRELATED TIME SERIES 1397

dent realizations used to construct estimates of the covariancematrix grows large, we derive an asymptotic null distributionwhich may also be used to determine a threshold to approxi-mately achieve a given false alarm probability. We then turnour attention to the detection of spatially correlated time seriesusing a network of sensor arrays. Assuming WSS processes inboth time and space, the likelihood ratio is shown to convergeto a broadband integral of a log Hadamard ratio of a cross fre-quency-wavenumber spectrum, when the length of each timeseries and the number of sensors in each array grows large. Thisextends the frequency domain technique developed in [8] to thefrequency/wavenumber domain when each observer employsan array of sensors. Although this result is asymptotic, it sug-gests frequency/wavenumber implementations of the GLRT de-tector even for low space and time support for measured signals.The paper is organized as follows. Section II gives a brief

overview of the GLRT and several of its invariance properties.Section III derives an equivalent expression for the likelihoodratio using Gram determinants and presents the asymptotic nulldistribution. In Section IV we derive the likelihood ratio fortemporally/spatially WSS random processes and demonstrateits performance through simulation in Section V. Concludingremarks are made in Section VI.

II. REVIEW OF THE LIKELIHOOD RATIO

The problem considered here is testing for indepen-dence among random vectors with each vector

representing a length timeseries captured at sensor as depicted in Fig. 1. Assuming thiscollection of random vectors to be zero mean, the compositevector has covariance matrix

......

. . ....

with . This matrix not onlycharacterizes the second-order information for each channel in-dividually but also captures the interdependence between everypair of channels. If the set of random vectors is jointlyproper complex normal, testing for independence among allchannels becomes a test of whether for all .Consequently, the null hypothesis is the hypothesis thatis block-diagonal versus the alternative that it is not.We now assume we are given an experiment producing

iid realizations of the random vector from eachchannel , where

The composite vectors are orga-nized into a data matrix :

.... . .

...

Fig. 1. Inmany cases, the random vector represents the collection of a lengthtime series at one sensor location.

The probability density function (PDF) of is

In this expression is the estimated composite covariancematrix

......

. . ....

and is an sample estimate of the cross-covariancematrix . The GLRT for this problem computes the likelihoodratio [8]

(1)

where denotes the set of all PD Hermitian matrices anddenotes the set of all matrices in which are block-diagonal.Also, the matrices and aremaximum likelihood (ML) estimates of under the alternativeand null hypotheses, respectively, andis referred to as the coherence matrix [8]. Note that thematrix is a square-root matrix of such that

. Then and.

Finally, assuming that all time series are jointly WSS andusing results on determinants of asymptotically large block-Toeplitz matrices, the authors in [8] extended this likelihoodratio to the frequency domain by noting that as ,

(2)


with , , an estimated composite powerspectral density matrix:

......

. . ....

Here, is a quadratic estimate of the cross power spec-trum between channels and at frequency . The likelihoodratio given in (2) is referred to as broadband coherence as thetest statistic is the log of the narrowband Hadamardratio, , integrated over the Nyquistband.

A. Invariance Properties

Under suitable choices for the matrix , the hypothesistesting problem and the likelihood ratio statistic given in (1)remain unchanged upon replacing the random vector with .Two examples of such linear transformations are given below.• The set of all matrices such that

with any invert-ible matrix. This invariance property was noted in [8]and shows us that there exists no channel-by-channelinvertible linear transformation, including scaling andfiltering, that moves a covariance from to or viceversa.

• The set of all matrices such that with anypermutation matrix. The invariance of the likelihood

ratio to this class of linear transformations is easily seenby recalling properties of the determinant of a Kroneckerproduct. This invariance property shows that the orderingin channel index has no influence on likelihood.

III. STOCHASTIC REPRESENTATION OF THE LIKELIHOODRATIO STATISTIC

In this section we derive a formula for the likelihood ratiogiven in (1) consisting of a product of normalized predictionerrors. The virtue of this formula is that it leads to an exact sto-chastic representation for under the null hypothesis – arepresentation which may be used to set a threshold for falsealarm control. This is then used to derive an asymptotic distribu-tion valid for large which may also be used to determine thethreshold needed to approximately achieve a given false alarmprobability.

A. The GLR Revisited

We’ll begin by noting that, for any and any, the data matrix can be partitioned as follows

...

where the matrix contains all realizationsof the time-series , up to sensor :

......

. . ....

The matrix contains all realizations of thetime-series at sensor up to temporal sample ,

......

. . ....

and the vectorcontains all realizations of random variable . With thispartition in the data matrix, the northwest corner of the Grammatrix has the structure

.... . .

with entries defined as follows

Gram determinants [9] are a technique commonly used to testwhether or not a collection of vectors in an inner product spaceare linearly independent. Namely, a set of vectors are linearlyindependent if and only if the determinant of their Gram matrixis non-zero. Using this result, it is straightforward to show thatthe determinant of the estimated composite covariance matrixcan be decomposed into a product of scalars as follows

Using the definition of these matrices given above, this term canbe written

where denotes the projection onto the di-mensional subspace spanned by the columns of matrix

. Moreover, using results for the inverse of a 2 2


Fig. 2. Orthogonal decomposition of the projection into and.

block matrix, one can derive yet another equivalent expressionfor this term

(3)

where and denote theprojection onto the dimensional subspace spanned by thecolumns of matrix and the projection onto the di-mensional subspace spanned by the columns of matrix ,respectively.To compute the determinant of the block-diagonal matrix

in the denominator of the likelihood ratio, one can take a verysimilar approach to show that

Recalling the relationship given in (3), note that this term canalternatively be expressed as follows:

The geometry of this decomposition is depicted in Fig. 2.

Using the decompositions of the determinants of these ma-trices, it is then straightforward to see that the likelihood ratiogiven in (1) can finally be written

(4)

Each term within the product of this expression is the ratio ofthe squared residual from two different linear least squares prob-lems: the numerator, , is found by regressing ontoall the random variables previous to it while the denominator,

, is found by regressing onto those random vari-ables associated with channel only. Recalling the second in-variance property in Section II-A, one must keep in mind thatthe order in channel index used when constructing this sequenceof estimation problems ultimately has no effect on the likelihoodratio. That is, switching orders of channels will lead to differentestimation problems generally leading to different individual ra-tios within the product in (4). However, the product as a wholeis invariant to order.Fig. 2 depicts the geometry of the projections and

used to construct each term of the likelihood ratioin (4). The squared length of the vector in essencerepresents the increase in mean-squared error incurred by ex-cluding channels from the estimation problem.The smaller the length of this vector relative to the length of

, the more evidence in support of the null hypothesisof independence as it indicates that one can just as accuratelyestimate by ignoring the previous channels. Although theexpression of the likelihood ratio given in (4) is no easier tocompute than that in (1), it shows that (1) can be written as aproduct of scalar random variables whose null distribution isthe subject of the upcoming section.

B. Stochastic Representation Under the Null Hypothesis

To characterize the distribution of the likelihood ratiounder the null hypothesis , we begin by imposingthe assumption that for any

. Recalling the first invari-ance property in Section II-A, we note that under thesecircumstances we can always apply the linear transformation

, a pre-whitener,to the random vector without any consequence to the like-lihood ratio. Thus, there is no loss in generality to assumethat (note that ) or equivalently that

.


Fig. 3. Monte Carlo Results.

Looking again at Fig. 2, it is clear that the two projectionsand lie in two orthogonal subspaces of ,

i.e., . A straightforward applicationof Cochran’s Theorem [10] then shows that and

are statistically independent chi-squared randomvariables with degrees of freedom and

, respectively, where

Noting that if and represent two independent chi-squaredrandom variables with degrees of freedom and , respec-tively, then the random variable is distributed accordingto a beta distribution with parameters and , it thenfollows that

(5)

where , all distributed independently ofone another. A discussion of the independence in these randomvariables is given in the appendix. Equation (5) says, “under thenull hypothesis, the likelihood ratio statistic is distributed as theproduct of independent beta random variables, .”Note that if the assumption of a complex normal distributionfor the data channels is replaced with a real-valued multivariatenormal, we can modify the above statements accordingly bysimply halving the parameters of these beta random variables,i.e., .For both the real and complex-valued versions of the GLR

and with , , and , Fig. 3 displayshistograms of Monte Carlo trials generated in two differentfashions: the darker plots are generated by drawing data from a

normal distribution, forming sample covariance matrices, andcomputing the generalized Hadamard ratio given in (1) whilethe lighter plots are generated by sampling from the appropriatebeta distributions and forming the product given in (5). Thesefigures show good agreement in the histograms, illustratingstochastically the mathematical fact that the likelihood ratiostatistic in (1) is equal in distribution to a product of indepen-dent beta random variables. Fractions of threshold exceedancesmay then be counted to determine which thresholds returnwhich false alarm rates.

C. Asymptotic Null Distribution

Given the definition of the beta function and its con-nection with the gamma function

every random variable within the product given in (5) hasraw moment

where denotes the PDF of a beta random variablewith parameters and . Using this fact along with the factthat these random variables are independent, one can see thatthe likelihood ratio has the following moments under the nullhypothesis

If we define with an arbitraryreal number, the characteristic function of the random variable

can be written

Its cumulant generating function is


To investigate the properties of this cumulant generatingfunction for large , we employ the following asymptoticexpansion [11], [12]

where denotes an order Bernoulli polynomial. Thisseries converges as provided that .Assuming that the variable does not go to zero as becomeslarge (in fact we’ll find that will be chosen to be

), then the expression given above can be used toexpand the log-gamma functions in the cumulant generatingfunction for large . Using the 2nd order Bernoulli polynomial

and after a bit of algebra, one obtains theasymptotic expression

Carefully counting the number of independent real parametersthat must be estimated under the alternative and null hypotheses,respectively, one finds that and

. Thus, it is clear that the value defined above is re-lated to the dimensions of these two spaces through the simpleexpression

The purpose behind the variable in this story is to manipu-late the higher order terms in the expansion so that one obtainsa more accurate approximation [13]. Namely, it is clear that ifwe choose the following value for

then the first order term in the asymptotic expansion ofcan be made to vanish as . This effectively producesan approximation whose error is compared to an errorthat is if one were to set . Letting tend toinfinity and exponentiating the resulting cumulant generatingfunction, we find that

Fig. 4. Asymptotic Empirical False Alarm Probabilities,for and .

which one recognizes as the characteristic function of a chi-squared random variable with degrees of freedom. Thus, forlarge it follows that

With and , Fig. 4 displays empiricalfalse alarm probabilities versus for the random variable

when (shown by a darker line) and when(shown by a lighter line). Here the threshold is chosen

from the asymptotic chi-squared distribution and set to achievea desired false alarm rate of (shown by a dashedline). From the figure we can see that, by incorporating ascaling and allowing it to deviate from unity, one can achievefalse alarm rates that are closer to the desired value for anyfinite by selecting the appropriate value for . Even formoderate values of and , however, one can also see thatthis approximation requires very large values for to achieveconvergence of the false alarm probability to the desired valueof 0.05. This suggests that numerical integration of a stochas-tically generated product of betas as discussed in Section III-Bor other approximation techniques such as saddlepoint approx-imations [14] are a practical alternative.

IV. EXTENSIONS TO VECTOR-VALUED TIME SERIES

In certain examples of multi-channel detection applications,one may have the opportunity to observe multiple time seriesfrom each channel when deciding if a phenomenon common toall channels exists. One such example is a situation where sev-eral platforms each employ an array of sensors to take advantageof the spatial diversity such a sensing paradigm might have. Asbefore, the problem considered here is to test for the indepen-dence among random vectors, but we now assume that therandom vector from each channel


contains samples from a -dimensional vector-valued timeseries

The likelihood ratio used to solve this problem is essentially thegeneralized Hadamard ratio given in (1). The only differenceis that each block of the composite covariance matrix, , isnow rather than . In fact, all of the resultsof Sections II and III readily generalize to this situation uponreplacing with .As described in [8], the extension of the GLRT to the fre-

quency domain can be accomplished by first independently ap-plying the linear transformation , with de-noting an DFT matrix, to the data from each channel.Define the length- DFT vector

and the matrix . Then thelinear transformation

simply corresponds to a unitary DFT analysis of the channelat frequency , . Recalling the firstinvariance property in Section II-A, it follows that both sets ofsignals, and , share the same likelihood ratio so that(1) can be written

Introducing a permutation to the rows and columns of the matrixinside this determinant, the GLRT can be written as

.... . .

...

is a matrix that not only captures the second-order informa-tion between channels but also between frequencies. The matrix

is an block matrix consistingof blocks of the form

where and we use the convention.

We now assume that all channels are temporally WSS in thesense that, for any pair of channels and , there exists amatrix-valued sequence, , such that

Using results on large block-Toeplitz matrices [15], the ma-trix becomes asymptotically equivalent to the block-diagonalmatrix

so that as and grow large but and remain fixed theGLRT becomes

(6)

Again, the matrix is an block-struc-tured matrix consisting of submatrices of the form

which is a quadratic estimate of the cross power spectral densitymatrix between channels and at frequency and we use the

convention . In other words, the matrix

is a cross spectral matrix of cross spectral

matrices, with the cross spectral matrix .

The likelihood ratio given in (6) is a direct extension of theresults in [8] to account for the situation being considered hereand is not a particularly interesting result in that it simply cor-responds to replacing every scalar-valued power spectral den-sity estimate in (2) with a matrix. Although this result isperfectly general in that nothing has been assumed about thesevector-valued time series other than that they are temporallyWSS, we proceed under the context of multiple-array detectionin which case a notion of space can be ascribed to the time seriesof each channel.To take advantage of the spatiotemporal properties of the

problem, we now consider independently applying the lineartransformation to each channel (instead of thematrix considered earlier) with denoting a

DFT matrix. Note that pre-multiplying the vector bythe matrix simply corresponds to the application of a 2-di-mensional DFT, one applied temporally and the other spatially.For any frequency , we can then introduce a permutation ofthe rows and columns of the previously defined matrixso that

(7)

where

.... . .

...

(8)

and . Define the length- DFTvector at frequency for as follows


Then the matrix has entries of theform

(9)

When the entries of correspond to time series at differentspatial locations, the frequency variable is often referred toas the wavenumber and, to avoid confusion with the variable, we will adopt this terminology. Equations (7) – (9) define afrequency/wavenumber formula for null hypothesis testing in ansensor network of -sensor arrays.We now impose additional structure on the problem at hand

by assuming that all channels are not only temporally WSS butspatially WSS as well so that the multivariate covariance func-tion, , considered earlier now corresponds to a sequenceof Toeplitz matrices. That is, for any pair of channels and ,we now assume that there exists a two-dimensional sequence,

, such that

with a temporal lag and a spatial lag. An example of whenthis model would hold would be a set of sensor suites, laidout in an arbitrary geometry, but with their respective -elementarrays laid out co-linearly.Again invoking results on large block-Toeplitz matrices, it

follows that the matrix becomes asymptotically equiva-lent with the block-diagonal matrix

so that as , , and grow large but remains fixed the GLRTbecomes

(10)

The matrix has elements

which is a quadratic estimate of the cross power spectral den-sity between channels and in the frequency/wavenumber do-main. Thus, we see that the GLRT involves the computationof a Hadamard ratio at each frequency/wavenumber pair ( ),followed by broadband integration of its logarithm – a broad-band-broadwavenumber coherence.

Fig. 5. Detection of a Source using Multiple Linear Arrays.

V. SIMULATION RESULTS

In this section we provide simulation results to demonstratea situation where the results of Section IV apply, and to demon-strate the improvement in detection performance that can beachieved under such a situation. Let’s consider a network of

sensor arrays, each of which is a uniform linear array(ULA) of sensor elements. Our aim is to generate aquite arbitrary field at sensor array 1, and then propagate thisfield to the other two sensor arrays as depicted in Fig. 5.The propagating signal for is

assumed to be a zero-mean WSS random process. As such,it follows that there exists an orthogonal increment process

such that [16]

The random measure , which may be treated as a nar-rowband component of the signal at the instantaneous fre-quency , is a complex normal random variable with covariance

and is the power spectral density

This signal is then propagated to each sensor array so that thesignal measured at the sensor array, , may be written

with a bulk propagation delay and it may be assumed withoutany loss in generality that . For this simulation, it is as-sumed that arises from a first-order autoregressive processwith coefficient and white noise variance so that the spec-tral density may be written [16]

The signal received at each array is then propagated as aplanewave among its elements. At each -element array a tem-porally colored nonpropagating noise component is added inde-pendently of all sensors so that the observation at the array,

, may be written


where the noise vector has the cross spectral matrix

Also, the vector denotes the array response or steeringvector

with a propagation delay dependent on the properties of themedium, the distance between sensor elements, and the Direc-tion-of-Arrival (DOA) of the source. Consequently, eachblock of the frequency-dependent spectral density matrix of thecomposite observation can then be written as follows

For this simulation, the sensor noise is generated by passingunit-variance white noise through a 5th-order FIR filter withcomplex weights so that the noise spectral densitymay be written

Upon collecting all measurements at each sensor ele-ment, the data record is temporally partitioned into non-over-lapping copies of a time series of length . The like-lihood ratio given in (10) (denoted “Frequency/WavenumberDomain GLRT”) is then used to discriminate situations wherea source is present from those in which each sensor array ob-serves its own correlated noise field only. The performance ofthis detector will be compared to the classical likelihood ratiogiven in (1) (denoted “Time Domain GLRT”) as well as its fre-quency domain version given in (6) (denoted “Frequency Do-main GLRT”). First, however, we demonstrate the stochastic na-ture of the squared residuals in the Time Domain GLRT.As mentioned briefly at the beginning of Section IV, all of the

results of Section III generalize to the situation considered hereby simply replacing with in which case the stochasticrepresentation given in (5) becomes

With realizations of the composite vectorunder the null hypothesis, Fig. 6 plots the ratio of squared resid-uals, to , given in (4)

for when (a) the channel index isand (b) when it is . The dashed lines in both of theseplots show a 95% confidence interval for each beta randomvariable , i.e., the interval such that

Fig. 6. Ratio of squared residuals under . (a) Channel Index . (b)Channel Index .

. Likewise, with the presence of a sourcewith signal-to-noise ratio , Fig. 7displays the same for one realization from the alternative hy-pothesis. Comparing these two figures, it is clear that this in-terval traps with high probability under the null hypothesis,but does not under the alternative hypothesis wheremany valuesfall below the interval, signaling a deviation from independence.With and a source, Fig. 8 displays the

Receiver Operating Characteristic (ROC) curves for all threedetection methods considered here. From Fig. 8 we can see thatthe Frequency/Wavenumber Domain GLRT exhibits a perfor-mance that exceeds that of the Frequency Domain GLRT whendiscriminating these two hypotheses while the performance ofthe Time Domain GLRT is particularly poor. This is most likelydue to the fact that the Time Domain GLRT does not exploitwide-sense stationarity and its manifestation in a Toeplitz struc-ture for the or the blocks of . A true GLRT


Fig. 7. Ratio of squared residuals under . (a) Channel Index . (b)Channel Index .

for this case would use an ML estimate for Toeplitz matrices, anintractable problem with no analytical solution. So the time-do-mainGLRT, while generally applicable, is actuallymis-matchedto the WSS problem. On the other hand, the frequency-domainand frequency-wavenumber domain GLRTs, while not as gen-erally applicable, are better matched to the WSS case. More-over, these forms estimate cross-spectral matrices, which areapproximately block-diagonal in the WSS case, and use onlytheir block-diagonals. In other words, they exploit the assumedwide-sense stationarity by using only diagonal blocks of thecross spectral matrix. Asymptotically, this approaches a GLRTthat is faithful to the assumptions of wide-sense stationarity.Finally, Figs. 9 and 10 compare the performances of the Fre-

quency/Wavenumber and Frequency Domain GLRTs for a suc-cessively smaller number of copies but with a higher powersource. Note that the Time Domain GLRT has been excludedfrom these two studies because of insufficient sample support,i.e., is too small to construct positive-definite covariance

Fig. 8. Detection Performance with and .




estimates. Similar to the results of Fig. 8, we can again seethat the Frequency/Wavenumber Domain GLRT outperforms ineach case. Again, this is likely due to the fact that the likeli-hood ratio given in (10) is better matched to the (spatially) WSScase versus its alternative given in (6) which, while more gen-erally applicable, does not fully exploit wide-sense stationarity.Thus, by taking advantage of the spatiotemporal properties ofthe problem at hand, we can see that the GLRT given in (10)presents an appealing likelihood ratio that exhibits improveddetection performance when compared to the two alternativesconsidered in this paper.

VI. CONCLUSION

Detecting the presence of common characteristics amongtwo or more data channels is a problem that finds its uses in awide range of applications. This paper considers the GLRT of[8] which tests whether or not a composite covariance matrixis block-diagonal, through the use of a generalized Hadamardratio. Using the theory of Gram determinants and their connec-tion with linear prediction, this generalized Hadamard ratio isshown to be equivalent to a product of ratios involving errorvariances of linear predictors. Assuming that the underlyingcovariance matrix is truly block-diagonal, we then concludethat the likelihood ratio is statistically equivalent to a product ofindependent beta random variables under the null hypothesis.Building on this, we derive an asymptotic null distributionwhich can be used to select an appropriate threshold when thenumber of samples used to construct estimates of the covari-ance matrix grows large.These results are then extended to the detection of spatially

correlated time series from a network of sensor arrays. ApplyingFourier analysis to each channel in both time and space and as-suming both temporally and spatially WSS random processes,the GLRT is shown to be a Hadamard ratio of an estimatedcross spectral matrix at every frequency/wavenumber pair whenthe length of each time series and the number of sensors ineach array grows large. To demonstrate the proposed detectionmethod, simulations consisting of 3 spatially separated ULAssensing the presence of a wideband source were conducted.Through these simulations, it was shown that the proposed fre-quency/wavenumber domain technique presents an appealingversion of the GLRT that can provide improved detection per-formance, especially in situations with low sample support.

APPENDIX

The purpose of this appendix is to establish the mutual in-dependence of the random variables given in (5). Consider therandom data matrix with , and withoutloss of generality ,

......

. . ....

where for and. One can think of the row of this matrix as an

-sample surrogate for the random variable . As describedin Section III, we may partition X

...

with containing all realizations of randomvariables and containing all real-izations of random variable . The determinant of the Grammatrix may then be written

Now, conditioned on the data matrix , the random scalaris a quadratic form involving standard

complex normal random variables with a known, deterministicidempotent matrix so that . This condi-tional probability distribution is dependent on the number ofrows ( ) and columns ( ) of but in no way dependent onwhat value this data matrix actually takes making statisti-cally independent of . As the sequence of preceding randomvariables are all a function of , this alsoimplies pair-wise independence between and .By induction on , it then follows that the entire sequence ofrandom variables are mutually independent sothat the random variable is distributed as theproduct of independent chi-squared random variables withdegrees of freedom . Aless heuristic proof of this fact can be found in [17].In a very similar manner we can now see that, although the

random variables and given in (4) are func-tionally dependent on the data matrix through theconstruction of the projection matrices and , respec-tively, they are in fact statistically independent of this data ma-trix making the sequence of random variables given in (5) mu-tually independent.

ACKNOWLEDGMENT

The authors would like to thank Dr. D. Cochran andDr. R. Butler for their invaluable comments and suggestions.

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Nick Klausner (S’08) received the B.S. degreesin electrical engineering and economics from theColorado School of Mines, Golden, CO, USA, in2008 and the M.S. degree in electrical and computerengineering from Colorado State University, FortCollins, CO, USA, in 2010. Since 2010, he has beenworking towards the Ph.D. degree in electrical andcomputer engineering at Colorado State Universityunder the direction of Dr. M. R. Azimi-Sadjadi andDr. L. Scharf.His current research interests include statistical

signal processing, digital signal processing, and digital image processing.

Mahmood R. Azimi-Sadjadi (M’81–SM’89) re-ceived the M.S. and Ph.D. degrees from the ImperialCollege of Science and Technology, Universityof London, London, U.K., in 1978 and 1982,respectively, both in electrical engineering withspecialization in digital signal/image processing.He is currently a Full Professor with the Elec-

trical and Computer Engineering Department,Colorado State University (CSU), Fort Collins, CO,USA. He also serves as the Director of the DigitalSignal/Image Laboratory, CSU. His main areas

of interest include statistical signal and image processing, target detection,classification and tracking, sensor array processing, learning systems, anddistributed sensor networks. He is the co-author of the book Digital Filteringin One and Two Dimensions (New York, NY, USA: Plenum, 1989).Dr. Azimi-Sadjadi has served as an Associate Editor of the IEEE

TRANSACTIONS ON SIGNAL PROCESSING and the IEEE TRANSACTIONSON NEURAL NETWORKS.

Louis L. Scharf (S’67–M’69–SM’77–F’86–LF’07)received the Ph.D. degree from the University ofWashington, Seattle, WA, USA.From 1971 to 1982, he served as Professor of

Electrical Engineering and Statistics with ColoradoState University (CSU), Ft. Collins, CO, USA. From1982 to 1985, he was Professor and Chairman ofElectrical and Computer Engineering, Universityof Rhode Island, Kingston, RI, USA. From 1985 to2000, he was Professor of Electrical and ComputerEngineering, University of Colorado, Boulder,

CO, USA. In January 2001, he rejoined CSU as Professor of Electrical andComputer Engineering and Statistics. He is currently Research Professor ofMathematics at CSU. He has held several visiting positions here and abroad,including the Ecole Superieure dElectricite, Gif-sur-Yvette, France; EcoleNationale Superieure des Telecommunications, Paris, France; EURECOM,Nice, France; the University of La Plata, La Plata, Argentina; Duke University,Durham, NC, USA; the University of Wisconsin, Madison, WI, USA; and theUniversity of Tromso, Tromso, Norway. His interests are in statistical signalprocessing, as it applies to radar, sonar, power, cyber-security, and wirelesscommunication.Prof. Scharf was Technical Program Chair for the 1980 IEEE International

Conference on Acoustics, Speech, and Signal Processing (ICASSP), Denver,CO; Tutorials Chair for ICASSP 2001, Salt Lake City, UT, USA; and Tech-nical Program Chair for the 2002 Asilomar Conference on Signals, Systems,and Computers. He is past-Chair of the Fellow Committee for the IEEE SignalProcessing Society. He has received numerous awards for his research contri-butions to statistical signal processing, including a College Research Award, anIEEE Distinguished Lectureship, an IEEE Third Millennium Medal, the Tech-nical Achievement and Society Awards from the IEEE Signal Processing So-ciety, and the Donald W. Tufts Award from the IEEE Underwater AcousticSignal Processing Workshop.

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