Coherent wide-band processing for arbitrary array ... - IEEE Xplore

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 41, NO. 1 , JANUARY 1993

Differentiating (11) with respect to 0 and using (17), we get, to first order J

lim

zfD(e0)=

K-m

Iim

P

C C

/’=I

K-mJ=l

1 -ff;,(eo)

( 1 8a)

v(4)

Let Q, be the N X P matrix composed of the eigenvectors of PA, corresponding to its nonzero eigenvalues. Then, according to Poincare’s separation theorem

h[QyR(J;)Q,l 5 h [ R ( f ; ) l ,

i

= 1,

*

. . , P.

642)

It can be verified that the P nonzero eigenvalues of PA,R($)PAJare equal to the eigenvalues of Q,”R(&)Q,, and therefore Substituting ( 1 1 ) into (15), and assuming independent Fourier coefficients yields lim cov

{f}=

K- m

lim

5 [

p

cov

~ + m / = l

1 -

(19a)

~(4)

P=I

A,[P,”,R(f;)PA,l 5 X,[R($)l,

K-m

(A3)

M . Wax, “Detection and estimation of superimposed signals,” Ph.D.

where aP, = lim

. . . , P.

REFERENCES

cov (19b)

-

= 1,

R( f i ) P A , .

J

C K-mJ=l

lim cov {zc) = lim

i

But for the true DOA’s, inequality (A3) becomes an equality, and therefore the true DOA’s maximize each individual eigenvalue of

q,,;

~ ( 4=)

K-m

lim 11,. K-m

From (18) and (19) we see that the asymptotic covariance of the DOA estimates is not affected by prior knowledge of the noise spectrum. Weiss and Friedlander showed in [7] that the narrow-band Gaussian CRB associated with the DOA estimates is not affected by a priori knowledge of the noise variance. Their result provides an alternative proof of Claim 3 for the narrow-band Gaussian case, since the Gaussian M L is an efficient estimator. The following claim provides conditions under which the asymptotic performance of the deterministic M L is equal to that of the Gaussian ML. Claim 4: The asymptotic covariance matrix of the deterministic wide-band M L DOA estimator is equal to that of the Gaussian ML, in the following cases: f o r p = I , . . . , P ; j = 1, . . . , J aP, >> v(&), (20)

dissertation, Stanford Univ., Stanford, CA, 1985. A. G . Jaffer, “Maximum likelihood direction finding of stochastic sources: a separable solution,” in Prnc. ICASSP, Apr. 1988, pp. 28932896. P. E. Stoica and A. Nehorai, “MUSIC, maximum likelihood and Cramer-Rao bound,” IEEE Trans. Acousr., Speech, Signal Processing, vol. 37, pp. 720-741, May 1989. P. E. Stoica and A. Nehorai, “Performance study of conditional and unconditional direction of arrival estimation,” IEEE Trans. Acousf., Speech, Signal Processing, vol. 3 8 , pp. 1783-1795, Oct. 1990. J . F. Biihme, “Array processing,” in Signal Processing, J . Lacoume, T. Durani, and R . Stora, Eds. Amsterdam, The Netherlands: Elsevier, 1987. P. M. Schultheiss and H . Messer. ”Optimal and suboptimal broadband source location estimation,” IEEE Truns. Acoust., Speech, Signul Processing, to be published. A . J. Weiss and B . Friedlander, “On the Cramer-Rao bound for direction finding of correlated signals,” IEEE Trans. Signal Processing, this issue, pp. 495-499. C . Radhakrishna Rao, Lineur Stutistical Inference and Its Applications, second ed. New York: Wiley, 1974, p. 64.

Coherent Wide-Band Processing for Arbitrary Array Geometry where C is any constant independent of the indices p a n d j . Proof: By substituting either condition (20) o r (21) into (18) and (19), and then substituting (18) and (19) into (16), we get equal asymptotic covariance matrices for the deterministic and the Gaussian estimators. Conditions (20) and (21) are, as expected, the asymptotic limit of conditions (13) and (14) given in Claims 1 and 2, respectively. However, Claims 1 and 2 are stronger than Claim 4 , since they deal with the individual noisy estimates and not with their asymptotic statistics. APPENDIX PROOFOF (1 7) We show that the true DOA’s maximize each individual eigenvalue of P , R ( A ) P , , separately. W e will use the Poincark separation theorem [SI, which states the following. Let B be an N x P matrix such that BHB = I . Then

X,[BHAB] 5 h , [ A ] ,

i

=

1, . .

. ,P

(AI)

where A, [ A ] denotes the ith eigenvalue of A , in decreasing order.

Miriam A . Doron, Eyal Doron, and Anthony J . Weiss

Abstract-We present a new method for coherent wide-band direction finding of far-field sources impinging on a two-dimensional array with a known arbitrary geometry. This method, termed array manifold interpolation (AMI), is based on obtaining the array manifold at a desired frequency!, by linear interpolation of the array manifold at a given frequency We use a separable representation of the array manifold vector, which separates the array geometry and the frequencyffrom the direction 0, in order to derive the required array manifold interpolation matrix. The AMI method is practical, computationally efficient, and robust. For the special case of a uniform circular array, we present a fast implementation of the AMI method, which utilizes the FFT algorithm.

.!

Manuscript received July 2 3 , 1991; revised April 16, 1992. M. A . Doron and A . J . Weiss are with the Department of Electronic Systems, Faculty of Engineering, Tel-Aviv University, Ramat-Aviv, 69978, Israel. E. Doron is with the Department of Nuclear Physics, The Weizmann Institute of Science, Rehovot 76 100, Israel. IEEE Log Number 9203358.

1053-587X/93$03.00 C? 1993 IEEE

IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL. 41, NO. I , JANUARY 1993

I.

INTRODUCTION

From (3), we can see that the array manifold vector has the form

In this correspondence we consider the localization of multiple wide-band sources via coherent signal-subspace processing. Several wide-band techniques, which are based on the coherent processing approach, have been suggested recently [ 11-[5]. However, most of them either require preliminary direction-of-arrival (DOA) estimates [1]-[3], utilize sector interpolation [4], o r require a large number of computations [5]. Computationally efficient coherent processing schemes, which do not require initial DOA estimates and are not limited to an angular sector, exist only for linear uniform arrays [6]-[8]. In this correspondence, we present a new method for coherently processing wide-band data received by a two-dimensional array with a known arbitrary geometry. This method, termed array manifold interpolation ( A M I ) , is based on obtaining the array manifold at a desired frequency fo by linear interpolation of the array manifold at a given frequency f. This interpolation is utilized to align the signal subspaces of the narrow-band covariance matrices, so that they may then be coherently combined into a single covariance matrix. The combined data matrix represents the same data reduction as the focusing method [I]-[3], and any signal-subspace o r ML direction finding procedure can be applied to it. 11.

A

SEPARABLE

REPRESENTATION OF

THE

415

ARRAYMANIFOLD

Consider an array of N sensors, sampling a wavefield generated in the presby P wide-band sources at locations 8 , , 82, . . ' , ence of additive noise. The observation time interval is sectioned into K subintervals of duration Td each, and a discrete Fourier transform is applied to each subinterval. For a sufficiently large subinterval length Td, we may write

eP,

forj= l;..,J;k=l;..,K

[as(f)l,

The vector a s ( f ) is referred to as the array manifold vector. 7,,(8,) is the propagation delay associated with the p th source and the m th sensor. Let the array be set in an arbitrary geometry in the plane. We define the origin of a polar coordinate system to be at the center of gravity of the array. The polar coordinates will be designated ( r , 4), and consequently the coordinates of the tn th sensor will be designated (r,,,,&). We assume that the sensors and sources are coplanar, and the sources are far enough from the array so that the signal wavefronts are effectively planar over the array. In this case (3)

[jkr,

COS ( + m

-

811

=

a 0 ( 4 , ~kr,)

(5)

m

eJzcos*

=

C n = -m

(6)

JAZ) ( j ) " e - I n *

where J J z ) is the n th-order Bessel function of the first kind. Applying (6) to ( 5 ) , we get m

ao(4m,kr,)

=

C " = -m

(j)nJ,(kr,)eJn'",e-Jno.

(7)

We now use (7) to write the array manifold vector in a slightly different way. We define the matrix G ( k ) and the vector #8 as [#'B I, = e-J"'B

[G(k)lm,,= ( j).Jn(krm)eIn',,,; where m = 1 . . . N a n d n = 0, f 1 , f 2 vector can then be written as

a s ( f ) = a,(k)

=

* * *

(8)

. The array manifold

G(k)#s.

(9)

We see that the array geometry and the wavenumber appear only in the matrix G, while the plane wave direction appears only in the vector #, The separable representation (9) of the array manifold vector is the main result of this section. In the following section, we will use this representation to derive the AMI algorithm. 111.

(1)

where X,(A), S,( A), and Nk(&) denote vectors whose elements are the discrete Fourier coefficients of the measurements, of the unknown source signals, and of the noise, respectively, at the kth subinterval and frequency&. A @ ( & )is the N X P direction matrix, whose p th column is

= exp

where the wavenumber k is defined as k 2 ( 2 x f ) / c . For conciseness of notation, we will henceforth refer only to dependence on k , instead off. The transformation (4) will therefore be treated as a transformation of the array manifold from k, to ko. We will use the series expansion of plane waves in polar coordinates [9]

THEAMI ALGORITHM

So far, the vector w e and the matrix G ( k ) have been infinite. However, from (7), we see that there is an effective cutoff for n >> kmaxrmax, where k,,, and rmaxdenote the maximal wavenumber in the processing bandwidth and the maximal sensor distance, respectively. This results from the fact that for n >> kr, the Bessel function J,(kr) decreases faster than exponentially [9]. Furthermore, the decay of the Bessel function begins when the order n of the Bessel function is equal to its argument. Let n, be defined by

I Jn(kmaxrmaJ I < E ,

for

>

ne

(10)

for some small E of our choice, and let G(k) and w s be defined as the finite matrix and vector, respectively, obtained by truncating G(k) and 0,at I n 1 = n,. Then, we can rewrite (9) as

~ s ( f )= as(k) = G(k)ws.

( 1 1)

Note that G ( k ) is an N x 2n, matrix. We require that

N

2 2n,.

(12)

Then, we see from (1 1) that a transformation satisfying (4) is given by

where c is the propagation velocity, and 8,, is the DOA of the p th source. We are interested in finding a transformation matrix q such that

qjas( A ) = a8(fo)

for all angles 8.

(4)

This transformation will be used to align the signal subspaces of the narrow-band covariance matrices, so that they may then be averaged into a single covariance matrix.

q

=

~(k~)~#(k,)

(13)

where G' = ( G H G ) - ' G Hdenotes the generalized inverse of G . Equation (13) holds only if G(k)HG(k)is full rank, and therefore we require condition (12). Condition (12) represents a generalized spatial sampling condition for the array, which enables us to perform the required manifold interpolation. Note that for N > Zn, (oversampling), the transformation q is singular.

IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL. 41, NO. I . J A N U A R Y 1993

416

The coherently combined covariance matrix 63 is now given by

where Rj = (1/K) E[= I Xk(J;)Xp(f;)is the narrow-band sample covariance matrix at the frequencyh. After implementing (14), any signal-subspace or M L direction-finding procedure can be applied to 63 (see e.g., [11-[4]). Furthermore, the computational load may be reduced by using the matrix 6 instead of 63,where 6 is the 2n, x 2n, matrix

Y

2

@

J

(? =

c G'(k,)R,(G'(k,))H.

(15)

/=I

e

From condition (12) we see that the dimensions of are smaller than or equal to those of 63. The application of e.g., MUSIC to the coherently combined matrix (? can proceed as follows: a) Perform EVD on e,"2eC,'/2, noise covariance matrix

where

/=I

@

e,,is the equivalent

J

e,,=

@

Fig. I . The geometry of the array

G'(k,)R,,(h)(G'(k,))H

(16)

and R,(f,) denotes the covariance matrix of the noise at&. b) Estimate the DOA's by locating the minima of the quadratic form d f ( k , ) UUHus(k,,), where u,(k,) = e ~ " 2 G ' ( k , ) a , ( k , ) ,and U is a matrix consisting of the noise eigenvectors.

nates were randomly generated. The resulting array configuration is plotted in Fig. 1. A processing band of 4 5 k 5 6 is spanned by J = 33 bins, and 50 snapshots are used. The central processing frequency is given by k,, = 5 . We examine a multiple cluster sccnario, with five far-field uncorrelatcd sources of equal power over I v . UNIFORM ClRCULAR ARRAYS the frequency band. The background noise is spatially and temporally uncorrelated. In this section, we will work out the procedure outlined above, In Fig. 2, we compare the AMI method with the incoherent widefor the special case of a uniform circular array of radius U . In this band MUSIC (see, e.g., [ I O ] ) , by plotting the results of six indecase, G takes the form pendent runs, for two cases. In the first case. plotted in Fig. 2(a), G = F3. (17a) the maximal senshr distance is rnrax= 0.94 m. Fork,,,,, = 6 , a 16element array satisfies the sampling condition (12), with t = 0.04 F is an N x 2n, matrix, whose elements are and n, = 8. In the second case, plotted in Fig. 2(b), we examined F,, = exp ( - j 27r n l ) . = 0.47 m. In this the case of an oversampled array, and took rnmsx case, we get n, = 4, and so a 16-element array oversamples the wavefield by a factor of - 2 . Note that in this case, (? is an 8 X n = 0, . . . , N - 1; L = -n, 1. . . . , n, (17b) 8 matrix instead of a 16 X 16 matrix. and 3 is a 2n, X 2n, diagonal matrix, whose diagonal elements are In Fig. 2(a), the first pair of sources are at -40" and -20" with an SNR of 4 dB, the second pair is at 5" and 20" with an S N R of I , . . . , n,. (17~) 3,! = (,j)'J,(ka), I = -n, I O dB, and the fifth source is at 80" with an SNR of -3 dB. In For the remainder of this section, we will take N = 2n,. F is then Fig. 2(b) the sources are at -80", -40", l o " , 40°, and 160", and the D F T matrix, and the transformation T,, given in (13), can be the SNR is unchanged. We see from Fig. 2 that in the first cluster, written as the AMI method detects and resolves the two sources, while the incoherent MUSIC fails to detect the second source. In the second T, = FD(k,)F-' (18) cluster, the incoherent MUSIC method barely resolves the two where D(k,) is a diagonal matrix whose main diagonal elements are sources, while the AMI method clearly resolves them. Finally, the the scaling factors [D(k,)],,,,= J , , ( k , l n ) / J , , ( k , ~Without ). loss of fifth weak source is detected more clearly by the AMI method than generality, we can take F to be unitary, in which case F - ' = F H . by the incoherent MUSIC method. We conclude that the AMI The coherently combined covariance matrix is now given by method is superior both in detection and resolution thresholds to the incoherent MUSIC method. J 63 = T,k,Tr = F D(k,)FHl?,FD(k,)F H . (19)

+

+

,?,

[/:I

I

From the RHS of (19). we see that the coherent processing can be implemented efficiently, using an FFT, followed by a simple scaling operation. v . SIMULATIONS In this section we describe Monte Carlo simulations carried out in order to demonstrate the performance of the AMI method. In our example, we consider a 16-element array, whose sensor coordi-

A N D CONCLUSIONS VI. SUMMARY

In this correspondence, we presented the AMI method for coherently processing wide-band data received by a two-dimensional array with a known arbitrary geometry. The AMI method is practical, computationally efficient, and robust. It does not require any iterations, and therefore has no convergence problems. It does not require prior knowledge of the DOA's, and therefore has no sensitivity problems. It does not perform sector interpolation, and

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 41, NO. I , JANUARY 1993

417

35 30

3 41 .-

25

20 15

c) 3

8

Q

10 5

0

Incoherent MUSIC

-5

-10

-60

-40

20

0

-20

40

80

60

100

Bearing [deg.] (a)

50

,

’

-10 -100

-50

0

50

100

150

200

Bearing [deg.] (b) Fig. 2. Comparison of the AMI and incoherent MUSIC methods for a five-source three-group scenario. Six independent trials are superimposed. The processing band is 4 5 k 5 6, spanned by 33 bins, for the 16 element array given in Fig. I : (a) r,,,.,x= 0.94; (b) rmAX = 0.47.

therefore is not limited to an angular sector. This method has an especially efficient realization via the FFT algorithm, for the case of a uniform circular array. We demonstrated the utility of the AMI method via simulations.

REFERENCES [l] H . Wang and M . Kaveh, “Coherent signal-subspace processing for

the detection and estimation of angles of arrival of multiple wide-band sources,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, pp. 823-831, Aug. 1985. [2] H . Hung and M. Kaveh, “Focusing matrices for coherent signal subspace processing,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASP-36, pp. 1272-1281, Aug. 1988. [3] M. A. Doron and A. J. Weiss, “On focusing matrices for wide-band array processing,” IEEE Trans. Signal Processing, vol. 40, no. 6, pp. 1295-1302, June 1992.

141 A. J. Weiss and B. Friedlander, “Performance analysis of coherent broad-band array processing,” IEEE Trans. Signal Processing, submitted for publication. [5] K. M. Buckley and L. J. Griffith, “Broad-band signal-subspace spatial-spectrum (BASS-ALE) estimation, ” IEEE Trans. Acousr., Speech, Signal Processing, vol. ASSP-36, pp. 953-964, July 1988. [6] G. Bienvenu, P. Fuerxer, G. Vezzosi, L. Kopp, and F. Florin, “Coherent wide-band high resolution processing for linear array,” in Proc. ICASSP ‘89, Apr. 1989, pp. 2799-2802. [7] H . Clergeot and 0. Michel, “New simple implementation of the coherent signal subspace method for wideband direction of arrival estimation,” in Proc. ICASSP ’89, Apr. 1989, pp. 2764-2767. [8] 1. Krolick and D. Swingler, “Focused wide-band array processing by spatial resampling,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 38, pp. 356-360, Feb. 1990. [9] M. Abramowitz and I . A. Stegun, Handbook of Mathematical Functions.” New York: Dover, ch. 9. [IO] M. Wax, T. J. Shan, and T. Kailath, “Spatio-temporal spectral analysis by eigenstructure methods,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-32, pp. 817-827, Aug. 1984.