lems encountered in direction-of-arrival estimation of fully correlated signals. Simulation results that illustrate the performance of this scheme in conjunction with ...
806
TRANSACTIONS IEEE
VOL. ASSP-33, NO. 4, AUGUST 1985
ON ACOUSTICS. SPEECH, SIGNAL AND PROCESSING,
On Spatial Smoothing for Direction-of-Arrival Estimation of Coherent Signals TIE-JUNSHAN,
MAT1WAX,
AND
THOMASKAILATH,
FELLOW, IEEE
Abstract-We present an analysis of a “spatial smoothing” prepropresent simulation results that illustrate its performance cessing scheme, recently suggested hy Evans et al., to circumvent prohin conjunction with the eigenstructure technique. lems encountered in direction-of-arrival estimation of fully correlated signals. Simulation results that illustrate the performance of this scheme STATEMENT 11. PROBLEM in conjunction with the eigenstructure technique are described.
Consider a uniform lineararray composed of p identical sensors. Let y (q < p) narrow-bandplanewuves, centered I. INTRODUCTION at frequency wo, impinge on the array from directions {e,, recent years, there has been a growing interest in high * . . , e,}. Using complex (analytic) signal representation, esolution eigenstructure techniques for direction-of-arthe received signal at the ith sensor can be expressed as rival estimation. These methods, developed by Pisarenko 4 1121, Ligget [9], Owsley [11], Schmidt [14], Reddi [13], ri(t) = akSk(t)e-jw(i-1)sinkfic + ni(t) (1) Bienvenu and Kopp [ 11, Johnson and Degraff [8], and Wax k= 1 et al. [18], are known to yield high resolution and asymptotically unbiased estimates, even in the case that the where, in fairly common notation, sk( .) is the signal of sourcesarepartiallycorrelated. Theoretically, these the kth wavefront, akis the complex response of the sensor methods encounter difficulties only when the signals are to the kth wavefront, d is the spacing between the sensors, perfectly correlated. In practice, however, significant dif- c is the propagation speed of the wavefronts, and ni ( is ficulties arise even when the signals are highly correlated, the additive noise at the ith sensor. We assume that thesignals and noises are stationary and as happens, for example, in multipath propagation or in military scenarios involving smart jammers. The perfect ergodic complex-valued random processes with zero mean. In addition, thenoises are assumed to be uncorrelated with correlation case, referred to as the coherent case, serves the signals and uncorrelated between themselves, and to as a good model for the highly correlated case. have identical variance 2 . In spite of its practical importance, the coherent case Rewriting (1) in vector notation, assumingfor simplicity did not receive considerable attention until recently. Although a rather general solution was proposed by Schmidt that thesensorsareomnidirectional, i.e., ak 1, weob[ 141, the high computational complexityinvolved makes it tain 9 unattractive. Widrow et al. [ 191 and Gabriel [6], [7] der(t) = a(&) Si(t) n(t) (24 scribed two similar approaches, both aimed at “decorrei=l lating” the coherentsignals. The scheme in Widrow et al., called “spatial dither,” is based on mechanical “dith- where r(t) is the p X 1 vector ering” of the array, while Gabriel’s scheme is based on (2b) r(t) = [r,(t) * * , rp(t)lT “Doppler smoothing.” Recently, Evans et al. [4],[ 5 ] ,in an extensive study of direction-of-arrival estimation tech- and a(eJ is the “steering vector” of the array in the diniques, presented an attractive solution to the problem for rection Oi: the case of a uniform linear array. Their solution is based a(ei) = e - j w o n , . . . , e-jwoO, - 1)Ti I T? on a preprocessing scheme referred to as spatial smoothing that essentially “decorrelates” the signals and thus d T~ = - sin Bi. eliminates the special difficulties encountered with coherC ent signals. In this paper, we present a more complete analysis of To further simplify the notation, we rewrite (2) as thespatialsmoothingpreprocessingscheme. We also r(t) = As(t) n(t)
If”
e )
c
+
9
+
Manuscript received June 29, 1983; revised February 27, 1985. This work was supported in part by the Air Force Office of Scientific Research, Air Force Systems Command under Contract AF49-620-79-C-0058, and by the Joint Services Program at Stanford University under Contract DAAG29-81K-0057. The authors are with the Information Systems Laboratory, Stanford University, Stanford, CA 94305.
where s ( t ) is the q
X
1 vector
w = Csl(0, and A is the p
X
, s,(0lT
* *
q matrix
A = [a(U, *
0096-3518/85/0800-0806$01.00O 1985 IEEE
*
2
a(eJ1.
807
SHAN et al.: SPATIAL SMOOTHING FOR ESTIMATION OF COHERENT SIGNALS
It follows from our assumptions that
~ r ( trt(t): ) =R
=
A S A ~+
2z, ‘s: =
(4)
where denotes the conjugate transpose. Notice that the S is diagonal when the signals are uncorrelated, nondiagonal and nonsingular when the signals are partially correlated, and nondiagonal but singular when some signals are fully correlated (or coherent). Assuming that the spacing between the sensors is less than half a wavelength of the impinging wavefronts (d C 1/2, X. where X. = 2nc/bo), it follows that the columns of the matrix A are all different, and hence, because of their Vandermonde structure, linearly independent.Thus, if S is nonsingulur, then the rank of ASAt is q. If
{X, IX, *
-
*
and { u l , u2,
IX,}
-
* *
, up}
are the eigenvalues and the corresponding eigenvectors of R , then the above rank properties imply that 1) the minimal eigenvalue of R is equal to 2 with multiplicity p - q:
X,+,
=
X,,,
=
* *
-
= Xp =
2
2) the eigenvectors corresponding to the minimal eigenvalue are orthogonal to the columns of thematrix A , namely, to the “direction vectors” of the signals @,+I,
*
-
7
up> ‘1{ a ( h > ,
-
*
Y
a(e,)).
plicity of the smallest eigenvalue is p - (q - 1); 2) the eigenvectors corresponding to the minimal eigenvalue are orthogonal to the columns of the matrix A . Because of their Vandermonde structure, note that thefirst column of A in (5c) is no longer a legitimatesteering vector since no linear combination of two “direction vectors” can yield another steering vector. The results of a straightforward application of the eigenstructuretechniqueto R can now be easily understood. First, because the multiplicity of the smallest eigenvalue of R is now q - 1, the detection step will give q - 1 as the number of signals. Second, since only the , e,} are “directionvectors”corresponding to {e3, included in the “signal” subspace, only these directionsof-arrival will be resolved in the estimation step. In general, if m out of the q wavefronts are coherent, the application of the conventional eigenstructure technique will result in an inconsistency: while the number of 1, only q - m direcsignals detected will be q - m tions-of-arrival, corresponding to the the incoherent wavefronts, will be resolved. Thus, if only one group of coherent signals exists, the difference between the numberof signals detectedand the number of signals resolved will be indicative of the size of the coherentgroup. Realizing this, Schmidt [ 151 proposed the following procedure: if a coherent group of size m is detected, search for the linear combination of m “direction vectors” that is included in the “signal” subspace or, equivalently, that is orthogonal to the “noise” subspace. Unfortunately, because of the high dimensionality of this search involved, this solution is computationally unattractive; in the next section, we present a different solution.
- -
+
We shall refer to the subspace spanned by the eigenvec‘torscorrespondingtothe smallest eigenvalue asthe “noise”subspace,and to its orthogonalcomplement, spanned by the “direction vectors” of the signals, as the “signal” subspace. The high resolution eigenstructure techniques are based 111. THE SPATIALSMOOTHING PREPROCESSINGSCHEME on the exploitation of properties 1) and 2) above. UnforAs we have pointed out in theprevious section, the nontunately, these properties hold only when the covariance singularity of the covariance matrix of the signals is the matrix of the sources S is nonsingular. Different relations key to a successful application of the eigenstructure techhold when S is singular. Assume, for simplicity, that the nique. In this section, we present a preprocessing scheme, rank of S is q - 1. This implies that two signals, say the et al. [5], that guarantees this propintroduced by Evans first two, are coherent, i.e., s2(t) = asl(t),with CY denoterty even when the signals are coherent. ing a complex scalar describing the gain and phase relaLet a uniform linear array with L identical sensors { 1, tionship between the two coherent signals. In this case, , L } be divided into overlapping subarrays of size p , we can rewrite (2) as with sensors { 1, ,p } forming the first subarray, senr(t) = as@) n(t) (54 sors (2, * * , p ] forming the second subarray, etc. (see where s(t) is the ( q - 1) X 1 vector Fig. 1). Let rk ( ) denote the vector of received signals at the kth subarray. Following the notation of (3), we can s(t) = [(I + a ) S l ( 0 , s3(t), * * , s,(0lT (5b) write and A is the ( q - 1) X m matrix rk(t) = A D ( k - ’ ) #(t) nk(t) (74 A = [a(e,) + aa (e,), , u(~,)I. where D(k’denotes the kth power of the q X q diagonal From (5), it follows that the covariance matrix of r(t) also matrix can be written as D = ,diag { e-juo71, . . . e - j a o 7 q } . (7b)
- -
+
-
-
- -
-
+
4e3), . -
7
R
=
ASAt
+ $I.
(6)
The covariancematrix of the kth subarray is therefore Now S = E[s(t) ~ ( t ) ~the ] , covariance matrix of the mod- given by ified signals, is a (q - 1) X ( q - 1) nonsingular matrix R - &k1) SD?(k-1) A’ + 21 k (8) and A is of full column rank. Therefore,.in complete analogy to properties 1) and 2) above, we have 1) the multi- where S is the covariance matrix of the sources.
808
4, AUGUST 1985
ON ACOUSTICS, SPEECH, SIGNAL AND PROCESSING, ASSP-33, VOL. NO.
TRANSACTIONS IEEE
with C denoting the Hermitian square root of ( l / M ) S : 1 Cet=-S. M Clearly, the rank .of is equal to the rank of G . Thus, our task is to prove that G has rank q or, equivalently, using the rank operator p , to prove that p { G } = q. Recalling that the rank of a matrix is unchanged by a permutation of its columns, it can be easily verified that
s
- - - - - -
Ce--rL-m+l*
Fig. 1. Subarrayspatial smoothing
where cy is the ijth element of the matrix C and b; ( i = 1, . . . , q) is the 1 X M row vector bi = [I e-ja;Ti , . . . e - j ; ( M - 1 ) ~1i 3
The spatially smoothed covariance matrix is defined as the sample means of the subarray covariances:
i
=
1, * * * , q .
(14b)
To show that the matrix G is of rank q, namely, is full row rank, it suffices to show that each row of the matrix C has at least one nonzero element and that the vectors 1 is the number of subarrays. where M = L - p { b l , . - . , 6,) are linearly independent. The first fact folUsing (S), we can rewrite (9) as lows by contradiction. Assume that a row of C , say the kth, is composed of all zeros. This implies, by (13c), that the kth signal has zero energy, in contradiction to the definition of S as the covariance matrix of the nonvanishing signals. The linear independence of the vectors b l , . . * , or more compactly as b, follows by observing that for M 5 q , these vectors can R = AsA:' 22p (11a) be embedded in a Vandermonde matrix, which is known where S is the modified covariance matrix of the signals, to be nonsingular. The above result is stated in Evans et al. [5, pp. 2-24]. given by Their proof, however, is incomplete; they show, correctly, . M A that the matrix S ( t ) = 1/M [D(i-l)s(t)] . [ D ( i - ls(t)lt ) is nonsingular. Notice that ES = 1/M E . 1 = 1 [D(j-l)s(t)] [DCi-l)s(t)lt = 3, that is, the exWe shall now prove that when M 2 q , the number of pectedvalue of is equal to 3, the modified covariance signal sources S will be nonsingular regardless of the comatrix. Unfortunately, the nonsingularity of a random maherence of the signals. trix does not imply thenonsingularity of itsexpected Theorem: If the number of subarrays is greater than or value. Thus, the nonsingularity of the crucial element equal to the number of signals, i.e., if M 2 q , then the uponwhich the eigenstructure method hinges, does not modified covariance matrix of the signals is nonsingular. follow from the nonsingularity of S. Proo$ First, note that we can rewrite as It can be shown that in the special case that the covariance matrix of the sources is block diagonal, i.e., when there are several groups of coherent signals that are uncorrelated with each other, the number of subarrays can be reduced to the size of the largest group of coherent signals. Since the smoothed covariance matrix R has exactly the same form as the covariance matrix for the noncoherent case, one can successfully apply the eigenstructure methwhich can be further simplified to ods to this smoothed covariance matrix regardless of the coherence of the signals. However, this robustness comes at the expense of a reeffective aperture. To see this more quantitatively, duced where G is the q X M q block matrix consider the number of sensors needed to cope withq coherent wavefronts. Recallingthat the number of subar-
+
+
xE
s,
s
s
SHAN ef al.: SPATIAL SMOOTHING FOR ESTIMATION OF COHERENT SIGNALS
809
Fig. 2. Conventionalbeamformingmethod (with Hammingwindow) (six sensors; SNR = 3 dB; 500 “snapshots”; two coherentnarrow-band sources from 85”, 130°, one incoherent source from 70”). 30.00
25.00
1
A
Direction-of-Arrival
Fig. 3. Conventional MUSIC method (six sensors; SNR = 3 dB; 500 “snapshots”; two coherentnarrow-bandsourcesfrom 85”. 130”, one incoherent source from 70”)
+
The example we considered had three (q = 3) planar rays, given by M = p 7 m 1, must be greater than or 8 5 ” , 130”, and 70”. equal to q, and that the size of each subarray m must be wavefrontsatdirections-of-arrival 1, it follows thattheminimumnumber of The first two signals were coherent, while the third signal atleast q sensors needed is p = 2q. Comparing this t o p = q + 1 was not correlated with the others. Thearray was uniform for the conventional case, it is clear that we trade off half and linear, withSix elements a thirdwavelength apart. The signal-to-noise ratio was 3 dB, and the number of samples the effective aperture. (“snapshots”) taken from the array was 500. Applying the IV. SIMULATION RESULTS conventional beamforming method and the eigenstructure In this section, we present simulation results that illus- method of Schmidt [ 141, we obtained the results shown in trate the performance of the spatial smoothing scheme in Figs. 2 and 3, respectively. Only one dominant peak corresponding to the direction-of-arrival of the third signal is conjunction with the eigenstructure technique.
+
810
IEEETRANSACTIONS
-10.00
I .oo
ON ACOUSTICS.SPEECH,ANDSIGNALPROCESSING,
1
50.00
I
70
I
85
I
100.00
I
130
VOL. ASSP-33, NO. 4, AUGUST 1985
I
150.00
200.00
Fig.4. New method(sixsensors;subarraysize = 4; SNR = 3 dB; 500 “snapshots”; two coherent narrow-band source from 85”, 130”; one incoherent source from 70”).
ACKNOWLEDGMENT seen in both cases; the directions-of-arrivalof the two coherent signals were not resolved. However, first applying The authors wish to thank the referees for their helpful thespatialsmoothingpreprocessingschemewiththree comments. ( M = 3 ) subarrays of four (p = 4) sensors each, and then applying the eigenstructure method of Schmidt 1141 to the REFERENCES spatially smoothed covariance matrix yielded the results [l] G. Bienvenuand L. Kopp, “Adaptivity to background noise spatial shown in Fig. 4.In this case, the three peaks correspondcoherence for high resolution passive methods,” in Proc. IEEE ICASSP ’80, Denver, CO, pp. 307-310. ing to the directions-of-arrival of all the three signals are [2] J. P. Burg, “Maximum entropy spectral analysis,” Ph.D. dissertation, clearly seen. Stanford Univ., Stanford, CA, 1975. 131 J. Capon, “High resolution frequency-wavenumber spectral analysis,”
V. CONCLUDING REMARKS Proc. IEEE, vol. 57, pp.1408-1418,Aug.1969. [4] J. E. Evans, J. R. Johnson, and D. F. Sun, “High resolution angular A spatial smoothing scheme, introducedby Evans et al. spectrum estimation techniques for terrain scattering analysis and an[ 5 ]to circumvent problems encountered in the estimation gle of arrival estimation,” in Proc. 1st ASSP Workshop Spectral Estimation, Hamilton, Ont., Canada, 1981,pp.134-139. of the directions-of-arrival of coherent signals, was more “Application of advanced signal processing techniques to angle 151 -, completely analyzed. of arrivalestimationin ATC navigationand surveillancesystem,” Our emphasis was on the use of the spatial smoothing M.I.T. Lincoln Lab., Lexington, MA, Rep. 582, 1982. [6] W. F. Gabriel, “Spectral analysisandadaptive array superesolution scheme in conjunction with the eigenstructure technique. techniques,” Proc. IEEE, vol. 68, pp. 654-666, 1980. However, as pointed out by Evans et al., this scheme can [7] -, “Adaptivesuperesolution of coherent RF spatial sources,” in also be applied in conjunction with other processing techProc. 1st ASSP Workshop Spectral Estimation, Hamilton, Ont., Canada, 1981,pp.134-139. niques such as the minimum variance technique of Capon [8] D. H. Johnson, and S . R. Degraff, “Improving the resolution of bear131. It is also interesting to note, as again pointed out by ing in passivesonar arrays by eigenvalueanalysis,” IEEETrans. Evans et al., that the linear prediction technique of ClayAcoust., Speech, Signal Processing, vol. ASSP-30, pp. 638-647, 1982. 191 W. S. Ligget, “Passive sonar: Fitting models to multiple time series,’: ton and Nuttall [ lo], when used with a low-orderpredictor in Signal Processing, J. W. Griffith e f a l . , Eds. NewYork: Acain the spatialdomain, essentially performsthe spatial demic, 1973, pp.327-345. smoothing implicitly. In fact, it is the improved perfor- [IO] A. H. Nuttall, “Spectral analysis of a univariate process with bad data points, via maximum entropy and linear prediction techniques,” Namance observed for this method that apparently motivated val Underwater Syst. Cen., New London, CT, Tech. Rep. 5303, 1976. Evans et aE. to investigate the spatial smoothing scheme. 1111 N. L. Owsley, “Spectral signal set extraction,” in Aspects ofSignal Processing,Part II, G . Tacconi, Ed.Dordecht,The Netherlands: The extension of the spatial smoothing scheme to more Reidel, 1977, pp. 469-475. difficult scenarios arising in arrayprocessing, e.g., to nar[I21 V. F. Pisarenko, “The retrieval of harmonics from a covariance funcrow-band signals with unknown center frequency and to tion,” Geophys. J. Roy. Astron. Soc., vol. 33, pp.247-266,1973. wide-band signals, follows straightforwardly from Wax et [13] S . S . Reddi, “Multiple source location-A digital approach,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-1.5, pp.95-105,1979. al. [18]. A modification of the idea for adaptive beam- [I41 R. 0.Schmidt, “Multiple emitter location and signal parameter esformingincommunication applications is described by timation,” in Proc. RADC SpectralEst.Workshop, 1979, pp. 243258. Shan and Kalath [ 161.
SHANet
al.: SPATIAL SMOOTHING FOR ESTIMATION OF COHERENT SIGNALS
[15] -, “A signal subspace approach to multiple source location and During 1975-1980 was with he RAFAEL, Israel, spectral estimation,” Ph.D. dissertation, Stanford Univ., Stanford, where conducted he research development and CA, 1981. [16] T. J. Shan, and T. Kailath, “Adaptive beamforming for coherent sigIEEE Trans. Acoust., Speech, Signal Pronals and interference,” c a s i n g , vol. JuneASSP-33, 1985. [17] T. J. Ulrych,and R. W. Clayton, “Times series and maximum entropy,” Phys. Earth Plan. Ins., vol. 12, pp. 188-200, 1976. [18] M. Wax, T-J. Shan, and T. Kailath, “Spatio-temporal spectral analysis by eigenstructure methods,” IEEE Trans. Acoust., Speech, Signul Processing, vol. ASSP-32, pp. 817-827, 1984. [19] B. Widrow, K. M. Duvall, R. P. Gooch, and W. C . Newman, “Signal cancellationphenomena in adaptiveantennas:Causesandcures,” IEEE Trans. Antennas Propagat., vol. AP-30, pp. 469-478, 1982.
811
of communication systems, tracking system, and po1 sitionlocationtechniques.From 1981 to 1983 he was Research a Assistant in Department the of where University, Stanford engineering, Electrical he conducted research in the area of detection and estimation in sensor arrays. In 1984 he was a Visiting Scientist in IBM Research Laboratories, San Jose, where he was working on statistical modeling, pattern recognition, and image compression.
