Applications Of Cumulants To Array Processing- Part III...

2252 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 9, SEPTEMBER 1997

Applications of Cumulants to Array Processing—Part III: Blind Beamforming for

Coherent SignalsEgemen Gonen and Jerry M. Mendel,Fellow, IEEE

Abstract—We provide a cumulant-based blind beamformingmethod for recovery of statistically independent narrowbandsource signals in the presence ofcoherent (or perfectly cor-related) multipath propagation. Our method is based on thefact that for a blind beamformer, the presence of coherentmultipaths is equivalent to the case of independent sources witha different steering matrix. Our approach is applicable to anyarray configuration having unknown response. Signal sourcesmust have nonzero fourth-order cumulants. There is no need toestimate the directions of arrival. Our method maximizes signal-to-interference plus noise ratio (SINR). A comparable result doesnot exist using just second-order statistics.

Index Terms—Array processing, blind beamforming, coherentsignals, cumulants, higher order statistics, signal separation.

I. INTRODUCTION

I N THIS PAPER, we address the problem of blind beam-forming for recovery of statistically independent sources in

coherent signal environments assuming no knowledge aboutthe array. Coherent signal environments are very likely inpractice when multipath propagation or smart jammers arepresent. Before presenting our approach, we first discussthe limitations of the existing covariance-based beamformingmethods for this problem and then state our assumptions.

There are a number of second-order-statistics-based cri-teria that have been proposed for obtaining the optimumbeamforming weight vector that combines the array sensormeasurements to recover desired signals while supressinginterferences. These criteria lead to the same general formfor the optimum weight vector [15], i.e., ,where

spatial covariance matrix of the received signal ;array response in the desired direction (look-direction);constant whose value depends on the criterion used.

Manuscript received May 6, 1996; revised August 23, 1996. This workwas supported by the Center for Research on Applied Signal Processing atthe University of Southern California. The associate editor coordinating thereview of this paper and approving it for publication was Prof. Michael D.Zoltowski.

E. Gonen is with Globalstar L.P., San Jose, CA 95134 USA (e-mail:[email protected]).

J. M. Mendel is with the Signal and Image Processing Institute, Departmentof Electrical Engineering—Systems, University of Southern California, LosAngeles, CA 90089-2564 USA (e-mail: [email protected]).

Publisher Item Identifier S 1053-587X(97)06440-4.

In the special case of MVDR [3], the array output power isminimized subject to a unity look-direction gain constraint

, which results in . It isclear that thearray responsein the desired signal directionmust either be known or estimated to implement the optimumbeamformer. If the array response or geometry is unknown,as in the blind beamforming problem, it is necessary tocalibrate the array to obtain the response information; however,array calibration is a very costly procedure. Calibration canbe avoided, and the array response can be estimated usingESPRIT [19]; however, ESPRIT requires translationally equiv-alent subarrays, which is often an impractical constraint, andas for other subspace-based methods, ESPRIT fails in thecoherent signals case. Even if the response function of thearray is known or array is calibrated, due to perturbationsin the geometry and the response of the array, the responsein the desired direction may be different than its calculatedvalue. Therefore, it becomes important to use the estimatedvalues of the array response to fine tune the received signals.There are a number of so-called property-restoring methodssuch as the adaptive CMA [24] that use second-order statisticsand rely on certain known properties of the source signals;however, the signals extracted by property-restoring methodsdo not necessarily have the same waveform as the actualsource signals.

The optimum beamformer using second-order statisticstends to cancel the desired signal, and it fails to performoptimally when there are signals coherent with the desiredsignal [18]. Moreover, it tends to cancel the desired signal inthe output [22]. A detailed explanation of signal cancellationphenomenon can be found in [22]. Several methods haveappeared in [1], [17], [22], [23], and [25] to overcomethe signal cancellation problem when coherent interferersare present. The methods of [1], [17], [22], and [23] arelimited to uniform linear arrays; in [25], some specific arrayconfiguration is required. None of these methods are directlyapplicable to the blind beamforming problem due to theirimplicit constraints on the array structure.

For a blind beamformer, on the other hand, the presenceof coherent multipaths does not make any difference. In otherwords, the case of coherent multipath signals is identical tothat of independent signals with no multipath because, asshown in Section II, each coherent multipath from a givensource causes only a reparameterization of the steering vectorof that source.

1053–587X/97$10.00 1997 IEEE

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GONEN AND MENDEL: APPLICATIONS OF CUMULANTS TO ARRAY PROCESSING—PART III: BLIND BEAMFORMING 2253

In the cumulant-based processing framework, the blindrecovery problem has received increased research interest.Adaptive solutions based on optimization of various cumulant-based criteria, or solutions depending on eigendecompositionof suitably defined cumulant matrices, were proposed (see[4], [5], [13], and [21] and references therein). In thesemethods, second-order statistics are used to whiten the signalpart of the received signals prior to applying cumulant-basedpost processing, which is the main drawback, because thecovariance matrix of the noise needs to be estimated or knowna priori. Besides, these methods have limitations. For example,the eigendecomposition-based method in [4] fails when thereare sources having identical kurtosis. An alternative is touse only higher than second-order cumulants. An iterativeapproach and a cumulant-enhancement method using onlyfourth-order cumulants were suggested by Cardoso (see [4]and references therein). In the fourth-order cumulant methodof Dogan and Mendel [6], it was assumed that the independentinterfering signals are Gaussian, whereas the sole desiredsignal is non-Gaussian. Cumulants were used to suppress theGaussian interferences and noise so that one is left only withthe desired signal statistics.

Here, we assume a more general scenario where theremay be multiple desired signal sources and interferences. Ourassumptions are as follows:

A1) The desired sources are statistically independentamong themselves and independent of the othersources, and all of the source signals may be subjectto multipath propagation.

A2) There is frequency-flat multipath propagation.A3) The desired source signals must have nonzero fourth-

order cumulants, but no such assumption is made aboutinterferences—if their cumulants are zero, they arealready suppressed by the virtue of cumulants; if not,they will be rejected by a beamformer.

A4) The array is a nonambiguous one, i.e., its response toa signal from a given direction is different from thatdue to another signal from a different direction.

Our earlier works on direction finding in the coherentsources scenario [11] and [12] provide a basis for our ap-proach.

The organization of this paper is as follows: We formulatethe problem in Section II. In Section III, a solution is pro-posed. Experimental results supporting our conclusions anddemonstrating our method are provided in Section IV. Finally,conclusions are presented in Section V.

Throughout the paper, lowercase boldface letters representvectors, uppercase boldface letters represent matrices, andlower and uppercase letters represent scalars. The symbol “”is used for conjugation operation, and the superscript “” isused to denote complex conjugate transpose.

II. FORMULATION OF THE PROBLEM

Consider a signal scenario in which there are several nar-rowband sources and interferences. Suppose that these signalsundergo frequency-flat multipath propagation producing sev-eral sets of delayed and scaled replicas, which are received by

an -element array having arbitrary and unknown responseand geometry. Let a total of signals from statisticallyindependent and narrowband sources, with multipathsignals for each source , impinge on the array.It is assumed that the number of sources is less than the numberof array elements (i.e., ). The collection of multipathsignals, which are scaled replicas of theth source, are referredto herein as theth group, and there are groups. The thgroup contains multipath signals of theth source and “smart”jammers, which are coherent with theth source signal. Thearray measurements are corrupted by additive noise whosespatial correlation structure is unknown. We assume thatsnapshots taken at time points are available.With these assumptions, the signal received by the array attime is

(1)

where ; is an unknownsteering matrix; is a wavefront vector; and isthe independent measurement noise vector that can be Gauss-ian, non-Gaussian symmetrically distributed, or a mixture ofGaussian and this type of non-Gaussian noise.

The coherence among the signals impinging on the arraycan be expressed by

......

......

......

(2)where

signal vector representing the coherent signalsfrom the th independent source ;

complex scaling vector for theth source;

.

The received signal vector, written in terms of independentsources, is

(3)

where . Columns of matrix are calledgeneralized steering vectors. The th generalizedsteering vector is then the combined response of the array tothe th group of signals. This result shows that the case ofcoherent multipaths is equivalent to the case of statisticallyindependent sources with a modified steering matrix.

Our objective is to blindly recover the source signalsby designing suitable beamformers. The best

bemforming vector for each source from an MVDR viewpointis the inverse of the spatial covariance matrix times thegeneralized steering vector for that source. However, dueto coherent multipaths, the generalized steering vector fora given source requires a multidimensional parameterizationand, hence, is difficult to estimate using only second-order



statistics, even if the array is calibrated. We provide a fourth-order cumulant-based method for estimating the generalizedsteering vector of each statistically independent source inthe first step of our solution. In the second step, we designbeamformers to recover each independent source signal, andwe correct the constellation rotation in the third step.

III. PROPOSEDSOLUTION

Our solution proceeds in three main steps.

1) Estimate the generalized steering vectors.2) Using the estimated generalized steering vectors in

the previous step, design beamformers to recover eachsource one at a time.

3) Correct the constellation rotation (which is inevitable)for communication signals. Since all of the signals areestimated one at a time regardless of which signals aredesired, temporal structures of the signals can be usedto differentiate one from the other.

Given two scalar processes and and an -vectorprocess , we define cum as the

matrix whose th entry is cumwhere and are the th and th components of

, respectively. The th element of will be denotedby .

A. Step 1: Estimation of Generalized Steering Vectors

First, the fourth-order cumulant matrix

cum

cum

cum

cum

(4)

is estimated from the data, where (the th generalizedsteering vector) is theth column of arethe fourth-order cumulants of the sources, anddiag . In the abovederivation, cumulant properties[CP1], [CP3], [CP5], and[CP6] in [14] were used. Note that the cumulant of theadditive Gaussian measurement noise is zero. The next to thelast line of (4) follows from the independence of the sourcesignals and[CP6], i.e.,

cumifotherwise.

(5)

Proceeding similarly, cum can beshown to be

cum

(6)

where diag .Note that due to assumptionA4), rank (i.e., full-

rank) when the signals arrive at the array from different angles.Here, we assume that the first two elements of the array havenonzero responses to each group, i.e., the first two rows ofare all nonzero. Under this assumption and assumptionA3),

and are nonsingular. The case when the first two rowsof have zero entries is treated in Section III-D.

Using (4) and (6), it is possible to estimate the matrixand columns of each to within a complex constant.

The solution is based on the idea of rotational invarianceof the underlying signal subspace, which is the basis ofthe ESPRIT algorithm [19]. Note that any non-Gaussianinterference source is treated similar to a desired source in(4) and (6), and thus, limits the number of resolvable desiredsource signals since .

In ESPRIT, the rotational invariance of the signal subspaceis induced by the translational invariance of the array, i.e., anidentical copy of the array that is displaced in the space isneeded. On the other hand, in our cumulant-based algorithm,the same invariance is obtained without any need for anidentical copy. In ESPRIT, the signal subspace is extractedfrom the eigendecomposition of the covariance matrix of theconcatenated measurements from the main array and its copy.Here, the signal subspace is extracted from the singular valuedecomposition of the concatenated matrix of (4) and (6),which, in turn, gives and the columns of , eachto within a complex constant. In the Appendix, we show howthe generalized steering vectors are estimated from (4) and (6)using the TLS ESPRIT algorithm.

In order to see the forest from the trees, we summarize thecomputational steps.

Step 1: From the array data, estimate thecumulant matrices

cum

cum(7)

and stack these matrices into a matrix

as .

Step 2: Perform SVD of . The number of nonzero singu-lar values gives the number of groups. (A moresophisticated detection algorithm for the number ofsources may be used here.) Keep the firstsubmatrix of the left singular vectors of, where

is the number of groups. Let this submatrix be.

Step 3: Partition into two matrices andas in (21).



Step 4: Perform SVD of . Stack the last rightsingular vectors of into thematrix denoted .

Step 5: Partition as , where and are

.Step 6: Perform eigendecomposition of ; keep the

eigenvalues. Let the eigenvector and eigenvaluematrices of be and , respectively.

Step 7: An estimate of is obtained to within a diagonalmatrix, as in (25).

Note that a comparable result is not possible using justsecond-order statistics for arrays having arbitrary shape andunknown response because a spatial array covariance-matrixdepends only on two arguments, whereas we need a statisticwith at least three arguments to obtain matrices similar to (4)and (6). If, on the other hand, an array consists of two identicalsubarrays displaced in space, two covariance matrices can beobtained that have a structure similar to (4) and (6). In thiscase, in addition to the two arguments of the covariance matrix,one extra argument is present due to the fact that responsesof identical but displaced arrays are identical up to a phaseterm, and the phase term serves as the extra needed argument.Nevertheless, even in this case, ESPRIT cannot be appliedto these two covariance matrices if some of the incomingsignals are coherent because the ranks of these matrices arethen less than the number of incoming signals, which violatesthe rank condition of the ESPRIT problem. It is possible torestore the ranks of these matrices using the spatial smoothingmethod [23]; however, spatial smoothing is applicable onlyto uniform linear arrays. Consequently,existing second-order-statistics-based methods cannot handle the coherent signalscase with arrays having arbitrary and unknown geometries.Our method can.

B. Step 2: Beamforming

Using the generalized steering vector estimatesobtained in the previous step and second-order statistics,

we can design beamformers to recover the source signalsto within a complex constant (one at a time) as

follows. The received signal at time pointcan be expressedas

(8)

where ; all the source signals except aretreated as interferences; is the generalized steering vectorof , and is thegeneralized steering matrix of the other sources

.Using (8), the spatial array covariance matrix can be written

as , where , andis the array covariance matrix of all other sources exceptand includes the noise.

A number of different criteria for optimum recovery of thesignals lead to the same beamformer structure thatis given by , where is the array covariancematrix, and the constant depends on the criterion beingused. The minimum-variance distortionless-response (MVDR)

beamformer weight vector is obtained by minimizing the arrayoutput power subject to the unitygain (distortionless response) constraint for thedesired signal. We denote this weight vector as . Thesolution for is [15]

(9)

where . This beamformer weight vec-tor also maximizes the signal-to-interference-plus-noise ratio(SINR), which is defined as

SINR (10)

The source signals are each recovered to withina complex constant by replacing in the above optimumbeamformers by its estimate obtained in Step 1 and replac-ing by its sample estimate so that , where

, and is either or .Step 2 can be done in parallel for all sources.

C. Step 3: Constellation Rotation Correctionfor Communication Signals

In the first step of our source recovery algorithm, thegeneralized steering vectors for each source are estimatedto within a complex constant [see (25)]. For communicationsignals, using these estimates in the above beamformers resultsin source estimates that are rotated arbitrarily from theiroriginal constellations. Since the choice of optimum decisionregions depends on the signal constellation, a method is neededto recover the actual constellation. In this section, we showhow this can be done to within a sign ambiguity for one-dimensional (1-D) signal constellations. The two-dimensional(2-D) case can be corrected using cumulants, which will bethe subject of another paper.

Let be the 1-D signal of interest and beits perfect estimate to within a complex constant ,i.e., . The constant accounts for boththe arbitrary scaling and the signal power so thatis normalized to have unit power. Since is real,

, and .Therefore,

, and . Usingthese results, and can be obtained as

(11)

and

(12)

Finally, the actual constellation of can be recoveredto within a sign ambiguity as , where

. The sign ambiguity comes from the fact thatand have equal powers. However, it is possible

to correct for the sign ambiguity by either using differentiallymodulated signals or by adding header bits to each signal.



D. Using Multiple Guiding Sensor Pairs

In the first step of the algorithm, we used only the first twosensors as the guiding sensor pair. Inspection of (4) and (6)reveals that the guiding sensor pair elements do not have tohave identical responses and that the choice of the guidingsensor pair is not unique in our method. To demonstrate thesetwo facts, consider the cumulants

cum (13)

cum

(14)

which are generated by using the sensorsand as the firstguiding sensor pair and and as the second guiding sensorpair, where is the generalized steering matrixdiagand diag . The derivations of(13) and (14) are identical to the derivations of (4) and (6).

Just as when we chose the first two sensors as the guidingpair, the two pairs of sensors andalso lead to two matrices, as shown in (13) and (14), whichare in ESPRIT form to estimate the generalized steeringvectors . This observation is useful for multiplepurposes. First, it suggests that the available data can beused efficiently by employing multiple guiding sensor pairs.Second, it provides a basis for a solution to a potential problemthat is associated with the practical implementation of ourmethod, as we will explain next.

In Section III-A, the first two sensors were chosen as theguiding sensor pair, and it was assumed that the first tworows of are all nonzero. The reason for this assumption isexplained as follows. Suppose that theth element inthe first row of is equal or close to zero. Then, ,which causes rank [see (4), (6) and (7)], and,consequently, the number of independent sources appears tobe one less than its actual value. As a result, all the sourcesbut the th will be separated. Similarly, each zero entry inthe second row of reduces rank by one, which in turnpartially destroys the rotational invariance between the signalsubspaces of and . In these cases, the availability ofmultiple candidates for the guiding sensor pairs proves to bea useful solution. A simple selection procedure for the “right”sensor pairs in such cases is proposed next.

A Simple Selection Procedure:1) Estimate the number ofgroups from the eigendecomposition of the array covariancematrix (sophisticated approaches such as MDL or AIC canbe used here). 2) Check the rank of estimated for

, and prepare a list of values offor which the rankof the estimated . This is a list of all the “safe” indicesfrom which the indices and of and canbe selected. The reason why we call this list “safe” is that foreach value of in this list, the th row of must have allnonzero entries because each zero entry in theth row ofreduces the rank of by one.

In general, we can choose the pair such thatin ways since interchanging and does notmatter. In addition, can be chosen in ways. Therefore, thesensor pairs and can be chosen in

Fig. 1. Array geometry used in the first experiment;L is the wavelength.

ways. Provided that these pairs are the “right” ones, for eachchoice, there corresponds an ESPRIT problem defined by thetwo matrices and . The solutions of theseproblems yield estimates of each generalizedsteering vector . Next, in order to improvethe generalized steering vector estimates, these estimates canbe averaged, or the principal component of the matrix withth column can be chosen as

an improved estimate of . Since the required computationsfor different choices of guiding sensor pairs are independent,they can be implemented inparallel; hence, this method doesnot require additional computing time. Of course, even formoderate size arrays (e.g., ), is quitelarge (e.g., 450).

We have found from extensive simulations that unfortu-nately, using multiple guiding sensor pairs and averaging doesnot lead to substantial improvements that warrant the extracomputations.

IV. SIMULATION EXPERIMENTS

A. Experiment 1

The scenario consists of three independent binary phaseshift keyed (BPSK) sources that are subject to multipathpropagation and arrive at the array in Fig. 1 from four,two, and three different directions, respectively. The arrivaldirections and propagation constants were chosen arbitrarilyas [50 , 70 , 90 , 100 ] and

and ; and, [45 , 65 ,85 ] and . Unity propagation constantsbelong to direct paths, and direct path SNR’s equal 10 dB.The array elements were assumed to be arbitrarily rotateddipole antennas. The array response to a signal from angle

is given by

. Three thousand snapshotswere taken. The problem of interest is to recover each sourcemessage one at a time.



Fig. 2. Cumulant-based and MVDR beamformer outputs for Experiment 1. SNR= 10 dB. “CBOB” refers to cumulant-based beamformer.

(a)

(b)

(c)

Fig. 3. Various beamformer outputs for two coherent signals near broadside from closely spaced directions {90�, 95�} at (a) SNR= 0 dB, (b) SNR= 10 dB, and (c) SNR= 20 dB.

We tested our cumulant-based beamforming method, whichassumes no information about the array geometry or response,and the classical MVDR beamformer for which we had toassume that arrival angles of the desired signals (the direct

paths from each source), and the array response in thosedirections are perfectly known. The beamformer outputs fromboth methods are presented in Fig. 2. Observe that whereascumulant-based beamformer outputs are localized around 1



(a)

(b)

(c)

Fig. 4. Various beamformer outputs for two coherent signals near endfire from closely spaced directions {0�, 5�} at (a) SNR= 0 dB, (b) SNR=10 dB, and (c) SNR= 20 dB.

and 1, the MVDR beamformer fails to recover the sourcemessages. The MVDR beamformer fails because of signalcancellation. Spatial smoothing, as explained in [23], is aremedy to signal cancellation in the MVDR beamformer forcoherent signals; however, spatial smoothing is applicable onlyto uniform linear arrays, whereas the array in this experimentis a nonuniform one.

This experiment supports our earlier claim that multiplecoherent signals received by an array of arbitrary geometry andunknown response can be recovered by our cumulant-basedblind beamformer.

B. Experiment 2

In this experiment, we compare our method to an MVDRbeamformer using the spatial smoothing method [23]. Sincespatial smoothing is limited to uniform linear arrays, werestrict ourselves here to this case, although our method isapplicable to any array. We assume that two coherent 3000bit BPSK signals of equal power and of zero relative phaseimpinge on a ten-element uniform linear array.

First, we assumed that the two signals arrive from broadsidefrom closely spaced directions {90, 95 } (measured withrespect to the endfire). For this scenario, we tested both ourmethod and the classical-MVDR (C-MVDR) and smoothed-MVDR (S-MVDR) beamformers. In our cumulant-based blind

beamformer, we used the first two sensors in the 0directionas the guiding pair and our rotation-correction algorithm.For the C-MVDR beamformer, we must assume that thedesired signal direction is either known or estimated; therefore,assuming the desired signal is the one arriving from 90,we designed the C-MVDR beamformer. For the S-MVDRbeamformer, we used a subarray of length 6 (subarray length

) for backward and forward smoothing.The outputs of these three beamformers for the desired signalat 0, 10, and 20 dB SNR’s with fixed noise power areshown in Fig. 3(a)–(c). As seen, the C-MVDR beamformerfails. On the other hand, the S-MVDR beamformer recoversthe signals as SNR is increased; however, for equal SNR’s,our cumulant-based beamformer is always better than theS-MVDR beamformer.

Second, we assumed that the two signals arrive from closelyspaced directions {0, 5 } near endfire. Reddyet al. [18]have shown that for this case, spatial smoothing loses itsdecorrelating power for moderate smoothing lengths and there-fore results in increased signal cancellation as SNR is in-creased. In our cumulant-based method, we used the pairs

and , where and are thefirst two sensor measurements in the 0direction. Assumingthe desired signal direction is 0, we designed the C-MVDRbeamformer. For the S-MVDR beamformer, we used the



Fig. 5. Output SINR of S-MVDR, E-MVDR, and our cumulant-based beamformer as a function of number of snapshots obtained from 10 Monte Carloruns. SNR= 10 dB. Signals are received from broadside. “o” denotes the cumulant-based beamformer; “�” denotes the S-MVDR beamformer; “�”denotes the E-MVDR beamformer.

same smoothing as before. Fig. 4(a)–(c) show outputs ofthe three beamformers for 0, 10, and 20 dB SNR’s withfixed noise power. The presence of coherence helps the C-MVDR and S-MVDR beamformers at low SNR’s, but thesebeamformers deteriorate as SNR is increased [18]. On theother hand, comparison of the first column of Fig. 4 with theother two columns indicates that our method is always betterthan the S-MVDR beamformer at equal SNR’s and that ourmethod improves as SNR is increased because our methodcombines coherent signal powers effectively instead of tryingto decorrelate them.

C. Experiment 3

We evaluate the output signal-to-interference-plus-noise-ratio (SINR) performance of our cumulant-based beamformerand compare it with those of the classical-MVDR (C-MVDR),smoothed-MVDR (S-MVDR), and an MVDR beamformer (E-MVDR), which uses the exactly known generalized steeringvector of the desired signal. Note that due to the multipaths,in reality, it is impossible to know the generalized steeringvectors prior to processing, even if the array is perfectly knownor calibrated; hence, the E-MVDR beamformer is rather ahypothetical one, which is designed as a benchmark for ourcumulant-based beamformer.

Assuming the same signal scenario and C-MVDR, S-MVDR, and cumulant-based beamformers as in Experiment2 and using the new E-MVDR beamformer, we performedtwo procedures.

1) Two 10-point Monte Carlo experiments for the case ofbroadside arrivals for SNR 10 dB and 0 dB:The output

SINR for S-MVDR, E-MVDR, and our cumulant basedbeamformer are plotted in Figs. 5 and 6 for each ofthese SNR’s. The cumulant-based beamformer performsbest even for a very small number of snapshots andconverges to the maximum possible output SINR valuequickly. The difference in the large-snapshot outputSINR between cumulant-based and S-MVDR is around19 dB at SNR 10 dB and 12 dB at SNR 0dB. More interestingly, the cumulant-based beamformeroutperforms the E-MVDR that uses the exact value ofthe generalized steering vector of the desired signal.Note that the only difference between our cumulant-based beamformer and the E-MVDR is that whereasthe E-MVDR uses the exact values of the generalizedsteering vectors, our beamformer uses the estimatedvalues of them. The simulation results show that ourblind beamformer “tunes” to the data better than theE-MVDR.

2) Two 10-point Monte-Carlo experiments for the case ofendfire arrivals for SNR 10 dB and 0 dB:The outputSINR for S-MVDR, E-MVDR, and our cumulant-basedbeamformer are plotted in Figs. 7 and 8 for each ofthe SNR’s. The cumulant-based beamformer performsbest even for very small number of snapshots, suchas 50, and converges to the maximum possible outputSINR value quickly. The difference in the large-snapshotoutput SINR between cumulant-based and S-MVDR isaround 11 dB at SNR 10 dB and 2.5 dB at SNR 0dB. Again, the cumulant-based beamformer outperformsthe E-MVDR.



Fig. 6. Output SINR of S-MVDR, E-MVDR, and our cumulant-based beamformer as a function of number of snapshots obtained from 10 MonteCarlo runs. SNR= 0 dB. Signals are received from broadside. “o” denotes the cumulant-based beamformer; “�” denotes the S-MVDR beamformer;“�” denotes the E-MVDR beamformer.

Fig. 7. Output SINR of S-MVDR, E-MVDR, and our cumulant-based beamformer as a function of the number of snapshots obtained from 10 MonteCarlo runs. SNR= 10 dB. Signals are received from endfire. “o” denotes the cumulant-based beamformer; “�” denotes the S-MVDR beamformer;“�” denotes the E-MVDR beamformer.

In this experiment, the output SINR’s of the classical-MVDR beamformer were very low, therefore, we do notdisplay them with the other three beamformers here. Theseresults suggest that not very many snapshots are needed

before excellent performance is obtained with our new beam-former.

Finally, note that whereas the smoothed-MVDR beam-former can utilize only the smoothing subarray, our method



Fig. 8. Output SINR of S-MVDR, E-MVDR, and our cumulant-based beamformer as a function of number of snapshots obtained from 10 MonteCarlo runs. SNR= 0 dB. Signals are received from endfire. “o” denotes the cumulant-based beamformer; “�” denotes the S-MVDR beamformer; “�”denotes the E-MVDR beamformer.

uses the entire array, i.e., our method uses a larger aper-ture.

D. Experiment 4

In this experiment, we compare our blind beamformingmethod to the spatial smoothing-based MVDR beamformerin terms of resolvable number of signals. We demonstratethat our beamformer can separate sources even if the totalnumber of incoming signals is more than the number ofsensors and show that spatial smoothing fails to separate thesources in this case. Since spatial smoothing is limited touniform linear arrays, we restrict ourselves here to this case,although our method is applicable to any array having arbitraryand unknown response. We assume four independent BPSKsignals of equal power and 3000 bits long, which are subjectto multipath propagation resulting in coherent signals. Thearray is assumed to be a ten-element uniform linear arraywith omnidirectional components. The four source signalsarrive at the array from two, three, four, and five differentdirections, respectively. Note that the total number of signalsimpinging on the array is 14, which is more than the numberof sensors. The signal arrival angles and propagation constantswere chosen as [55, 30 ] and ; [40 , 90 , 60 ]and ; [70 , 80 , 120 , 100 ] and

; [110 , 65 , 130 , 140 ,150 ] and .

For this scenario, we tested our cumulant-based beamformerand smoothed-MVDR beamformer. In our method, we usedthe first two sensors in the endfire direction as the guidingpair and our rotation correction algorithm. In the smoothed

MVDR beamformer, we had to assume that the desired signaldirections are either known or are estimated because MVDRdepends on the array response in the desired signal direction;therefore, we assumed that for each source, the desired signalis the direct path, and its arrival angle is known. Note thatfor our method, angle-of-arrival information is not needed.For the smoothed-MVDR beamformer, we used a subarrayof length 6 (subarray length ) forbackward and forward smoothing. Figs. 9 and 10 show outputsof both beamformers for 10 and5 dB SNR’s. Observe thatthe smoothed-MVDR method fails, whereas our method canseparate all of the four sources successfully.

V. CONCLUSION

We have developed a cumulant-based blind beamformer forrecovery of independent sources in the presence of coherentmultipath propagation, which is applicable to anyarbitraryarray configuration; it does not requireany knowledgeaboutarray response and reliessolelyon the measurements. There isno need to estimate the directions of arrival. Our approach isbased on the observation that by using cumulants of receivedsignals, two matrices can be formed that conform to theESPRIT architecture. In this approach, multipath powers areeffectively utilized instead of decorrelated.

The two matrices permit us to estimate the generalizedsteering vectors for each source blindly. Then, a number ofcumulant-based beamformers can be designed whose optimal-ity have already been shown in the second-order statisticsframework. Note that since the steering vectors are estimatedfrom the data, in some sense, the beamformer is tuned to the



Fig. 9. Cumulant-based (first column) and smoothed-MVDR (second col-umn) beamformer outputs for the fourth experiment. SNR= 10 dB.

data, thereby avoiding sensitivity problems associated withmismatch in the assumed steering vectors, which occurs inthe case of covariance-based processing. A comparable resultusing just second-order statistics does not exist for the theblind beamforming problem. Simulation results have verifiedour theoretical work.

APPENDIX

OBTAINING THE GENERALIZED STEERING VECTORS

Define a new matrix by concatenating , as

(15)

The singular value decomposition of yields

(16)

Fig. 10. Cumulant-based (first column) and smoothed-MVDR (second col-umn) beamformer outputs for the fourth experiment. SNR= �5 dB.

where diag ; ; and

. It follows, therefore, that

(17)

or, equivalently

(18)

Since is full-rank, (18) implies . Using the

fact that is orthogonal to , it follows that

span span (19)



Therefore, there exists a nonsingular matrix such that

(20)

or such that

(21)

where we partitioned exactly the same way as, i.e.,into two matrices and . Equation (21)establishes the signal subspace and its rotationally invariantcounterpart. Note that this rotational invariance is obtainedwithout requiring translational invariance of the array, asopposed to ESPRIT.

Having obtained this invariance, we follow the same stepsof ESPRIT in which we replace the signal eigenvectors of thecovariance matrix of the concatenated measurements by thefirst left singular vectors of the concatenated matrix ofand . For completeness, we present the standard ESPRITsteps in the following.

Equation (21) shows that and share a commoncolumn space of dimension; therefore, rank

. This last result implies [19] there exists a matrix

that is rank such that

(22)Since is full rank, (22) results in , whichis equivalent to

(23)

Equation (23) implies that the eigenvalues of must beequal to the diagonal elements of. To estimate the diagonalelements of , we therefore need a matrix that satisfies(22). Such a matrix can be obtained by performing a singularvalue decomposition of the matrix . Since

is rank , the last right singular vectors ofcan be selected as.

Using (20) and (23), columns of the steering matrixcanbe obtained to within a constant as follows. Let be theeigenvector matrix of . From (23), it follows that

, where is an arbitrary diagonal matrix withnonzero entries. Therefore, multiplying (20) by, we find that

(24)

and using the partitioning of (21) in (24), we find that, and , where was estimated as

explained previously. Finally, an improved estimate ofisobtained to within a diagonal matrix by averaging these resultsas

(25)

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewersfor their useful comments and suggestions.

REFERENCES

[1] Y. Bresler, V. U. Reddy, and T. Kailath, “Optimum beamformingfor coherent signal and interferences,”IEEE Trans. Acoust., Speech,Signal Processing, vol. 36, pp. 833–843, June 1988.

[2] D. R. Brillinger and M. Rosenblatt, “Asymptotic theory of estimatesof kth-order spectra,” inSpectral Analysis of Time Series, B. Harris,Ed. New York: Wiley, 1967, pp. 189–232.

[3] J. Capon, “High-resolution frequency-wavenumber spectral analysis,”in Proc. IEEE, vol. 57, pp. 1048–1418, Aug. 1969.

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[5] P. Comon, “Independent component analysis,” inProc. Int. WorkshopHigher Order Statist., Chamrousse, France, 1991, pp. 111–120.

[6] M. C. Dogan and J. M. Mendel, “Cumulant-based blind optimumbeamforming,” IEEE Trans. Aerosp. Electron. Syst., vol. 30, pp.722–741, July 1994.

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[10] E. Gonen, “Cumulants and subspace techniques for array signal pro-cessing,” Ph.D. Dissertation, Univ. Southern California, Los Angeles,Dec. 1996.

[11] E. Gonen, M. C. Dogan, and J. M. Mendel, “Applications of cumulantsto array processing: Direction finding in coherent signal environ-ment,” in Proc. Twenty-Eighth Asilomar Conf. Signals, Syst., Comput.,Monterey, CA, Oct. 1994.

[12] E. Gonen and J. M. Mendel, “Optimum cumulant-based blind beam-forming for coherent signals and interferences,” inProc. ICASSP,Detroit, MI, May 1995, pp. 1908–1911.

[13] L. Tong, Y. Inouye, and R. Liu, “Waveform preserving blind estima-tion of multiple independent sources,”IEEE Trans. Signal Processing,vol. 41, pp. 2461–2470, July 1993.

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[15] R. A. Monzingo and T. W. Miller,Introduction to Adaptive Arrays.New York: Wiley, 1980.

[16] C. L. Nikias and A. P. Petropulu,Higher-Order Spectra Analysis:A Nonlinear Signal Processing Framework. Englewood Cliffs, NJ:Prentice-Hall, 1993.

[17] S. U. Pillai, Array Signal Processing. New York: Springer-Verlag,1989.

[18] V. U. Reddy, A. Paulraj, and T. Kailath, “Performance analysis ofthe optimum beamformer in the presence of correlated sources andits behavior under spatial smoothing,”IEEE Trans. Acoust., Speech,Signal Processing, vol. ASSP-35, pp. 927–936, July 1987.

[19] R. Roy and T. Kailath, “ESPRIT—Estimation of signal parametersvia rotational invariance techniques,”Opt. Eng., vol. 29, no. 4, pp.296–313, Apr. 1990.

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[25] Y. L. Su, T. J. Shan, and B. Widrow, “Parallel spatial processing: Acure for signal cancellation in adaptive arrays,”IEEE Trans. AntennasPropagat., vol. AP-34, pp. 945–947, Mar. 1986.

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Egemen Gonenwas born in Izmir, Turkey, in 1970.He received the B.Sc. degree in electrical engineer-ing from Middle East Technical University, Ankara,Turkey, in 1991 and the M.Sc. and the Ph.D. degreesin electrical engineering from the University ofSouthern California (USC), Los Angeles, in 1993and 1996, respectively.

From August 1993 to September 1996, he wasa research assistant at the Signal and Image Pro-cessing Institute, USC. He joined Globalstar, L.P.,San Jose, CA, in October 1996, where he has been

working as a communication systems engineer. His present research interestsinclude antenna array processing, higher order statistics, and spread spectrumand satellite communications. He is co-author of the bookDigital SignalProcessing Handbook(Boca Raton, FL: CRC, 1997).

Jerry M. Mendel (F’78) received the Ph.D. degreein electrical engineering from the Polytechnic Insti-tute of Brooklyn, Brooklyn, NY, in 1963.

Currently, he is Professor of Electrical Engineer-ing, Director of Special Educational Projects for theSchool of Engineering, and Associate Director forEducation of the Integrated Media Systems Center,University of Southern California, Los Angeles,where he has been since 1974. He has publishedover 330 technical papers and is author or editor ofseven books, includingLessons in Estimation The-

ory for Signal Processing, Communications and Control(Englewood Cliffs,NJ: Prentice-Hall, 1987),Maximum-Likelihood Deconvolution(New York:Springer-Verlag, 1990), andA Prelude to Neural Networks: Adaptive andLearning Systems(Englewood Cliffs, NJ: Prentice-Hall, 1994). He is alsoauthor of the IEEE Individual Learning ProgramKalman Filtering and OtherDigital Estimation Techniques. His present research interests include higherorder statistics and neural networks applied to array processing and predictionof nonlinear time-series and fuzzy logic applied to prediction of nonlineartime-series, classification problems, and social science problems.

Dr. Mendel is a Distinguished Member of the IEEE Control SystemsSociety, member of the IEEE Signal Processing Society, the Society ofExploration Geophysicists, the European Association for Signal Process-ing, Tau Beta Pi, Pi Tau Sigma, and Sigma Xi. He is also a registeredProfessional Control Systems Engineer in California. He was President ofthe IEEE Control Systems Society in 1986. He received the SEG 1976Outstanding Presentation Award for a paper on the application of KalmanFiltering to deconvolution; the 1983 Best Transactions Paper Award for apaper on maximum-likelihood deconvolution in the IEEE TRANSACTIONS ON

GEOSCIENCE ANDREMOTE SENSING; the 1992 Signal Processing Society PaperAward for a paper on identification of nonminimum phase systems usinghigher order statistics in the IEEE TRANSACTIONS ON ACOUSTICS, SPEECH,AND SIGNAL PROCESSING; a Phi Kappa Phi book award for his 1983 researchmonograph on seismic deconvolution; a 1985 Burlington Northern FacultyAchievement Award; a 1984 IEEE Centennial Medal; and the 1993 ServiceAward from the School of Engineering at USC. He served as Editor ofthe IEEE Control Systems Society’s IEEE TRANSACTIONS ON AUTOMATIC

CONTROL.


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