42, NO. IO, OCTOBER 1994. 2511. Wavefield Modeling and Array. Processing, Part 111-Resolution Capacity. Miriam A. Doron, Member, IEEE, and Eyal Doron.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 42, NO. IO, OCTOBER 1994
2511
Wavefield Modeling and Array Processing, Part 111-Resolution Capacity Miriam A. Doron, Member, IEEE, and Eyal Doron
Abs@uct-This paper is the last in a series of three papers dealing with the application of wavefield modeling to array processing. We use here the wavefield modeling formalism in order to derive some fundamental limitationson the performance of passive arrays. We first derive upper bounds on both the ideal and effective resolution capacities, and show that the effective resolution capacity is bounded by the effective rank of the relevant sampling matrix, i.e., is determined by the array structure. We then study how the resolution capacity is affected when the sources are constrained to lie within an angular sector. We also look at the case where the source spectra are known a przori to be flat, and show that, contrary to the common opinion, if the array satisfies the spectral sampling condition, the effective wideband resolution capacity is bounded by the same bound as the narrowband one. We conclude by examining the interesting phenomenon of resonance in arrays. We show that these resonances, far from being mathematical artifacts, can manifest quite dramatically in the performance of arrays.
I. INTRODUCTION
T
HIS paper is the third in a series of three papers in which we present a new formalism for array processing, termed “wavefield modeling.” In the first paper [ 11, we presented the concept and mathematical foundations of wavefield modeling. We saw how, given a few assumptions which are almost always satisfied, one can write the signal dependent part of the array output as a product of a sampling matrix and a coefficient vector
where the sampling matrix G = [. . . , g-1, go, g l , . . .] is independent of the wavefield and the coefficient vector J, is independent of the array. G is given by an orthogonal decomposition of the array manifold ay
where y cp in 2-D and y (0,cp) in 3-D, r is the manifold of possible directions of arrival (DOA’s), and the {fn(y)} comprise a complete and orthogonal basis set on r. Manuscript received June 12, 1992; revised February 8, 1994. This work was supported by grants from the Royal Society, the Israel Academy of Science, the British Council, and the Rothchild Foundation. The associate editor coordinating the review of this paper and approving it for publication was Prof. John A. Stuller. M. A. Doron is with RAFAEL, Haifa, Israel. E. Doron is with the H. H. Wills Physics Laboratory, University of Bristol, Bristol, England. IEEE Log Number 9403726.
In this paper, we investigate the number of sources which can be uniquely resolved by a passive array. This number is sometimes termed the resolution capacity of the array. Bounds on the resolution capacity of an array were investigated previously [2]-[7] using a deterministic approach, i.e., the noise was ignored and infinite measurement and numerical accuracy were assumed. Thus, the problem was reduced to finding conditions for the existence of a unique solution to a set of nonlinear equations. The bounds that were derived using this approach were expressed in terms of the number of sensors, while the array structure and the angular spacing between the sources were not taken into account. We will refer to these bounds as the ideal bounds, since they are meaningful only under ideal conditions (no noise). Of more interest is the effective bound on the resolution capacity, which is defined as the number of sources above which the bearing accuracy begins to degrade rapidly, and therefore applies to realistic conditions (noisy measurements). Numerical studies have indicated that the effective bound on the resolution capacity is in fact affected by the array geometry [3] and by the spacing between the sources [8]. However, no analytic results were presented. In this paper we will use the wavefield modeling approach in order to quantify the effect of the array geometry on the effective resolution capacity. Our results apply to arbitrary arrays and extend to the case where the sources are known a priori to be confined to a limited angular sector. The latter extension is of special interest, since it enables us to investigate fundamental limitations on the ability of a passive array to resolve closely-spaced sources. In Section 11, we derive the ideal and effective bounds on the resolution capacity of an array. We show that the effective bound is determined by the rank of the sampling matrix, rather than by the number of sensors. These results are illustrated by numerical calculations in which we examine the effective resolution capacity via the CramCr-Rao bound (CRB) on the DOA estimation errors. In [3]-[5] and [9], it was shown that some wideband models can in principle relax the ideal bound on the resolution capacity. In Section 111, we show that these results do not necessarily extend to the effective bound. We investigate the case in which the source spectra are known a priori to be flat and show that, in sharp contrast to the ideal case, if the array satisfies the spectral interpolation condition at the highest frequency within the processing band, then the effective resolution capacity stays bounded by the number of sensors, regardless of the number of frequency bins used. Again, this is verified using numerical calculations.
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In the last part of this paper, we focus on the phenomenon of resonance, introduced in [l]. At resonant frequencies the information content of the array output drops, and we will show that this has the effect of lowering the resolution capacity of the array relative to that at neighboring frequencies. The data model which we will concentrate on in this paper is the Gaussian model, which is defined by the following. Consider an array of N sensors, sampling a wavefield generated by P wideband sources at locations 7 = [TI, 7 2 , . . . , yp], in the presence of additive noise. For the sake of convenience, we will assume that the sources are coplanar with the array, and therefore the locations of the sources reduce to their azimuth cp only. The extension to the 3-D case is straightforward, although a bit cumbersome. The observation time interval is sectioned into K subintervals of duration T d each, and a discrete Fourier transform is applied to each subinterval. For a sufficiently large subinterval length T d , we can assume that the discrete Fourier coefficients are uncorrelated, and write
An important tool in our derivations is N ( N ,q ) , defined as the number of independent real variables needed in order to uniquely specify a rank-q Hermitian N x N matrix H. H is uniquely determined by q real eigenvalues and q orthonormal complex eigenvectors. Each eigenvector is defined only up to a phase, and is therefore determined by 2N - 1 real variables. In addition, orthonormality implies q2 real constraints on the eigenvectors. We thus get
N ( N ,q ) = q + q(2N - 1) - q 2
= 2Nq - 4 2 .
(2.1)
A. The Ideal Resolution Capacity
In the following analysis, we ignore the noise and treat the problem as the deterministic problem of solving a set of nonlinear equations. Our derivation does not assume an infinite number of snapshots as in 121-[SI, but brings into account the number of frames averaged, via the rank of the sample covariance matrix. The derivation given in this subsection was originally presented in [7]. xk(wj) = Ay(wj)sk(wj) n k ( w j ) , j = 1,.. . , J , From (2.1), we see that the number of independent meaIC = 1,.. . , K (1.3) surements in R j is given by 2Nqj - q i , where q j is the rank where xk(wj),sk(wj), and n k ( w j ) denote vectors whose ele- of R j . The unknown parameters in our estimation problem are ments are the discrete Fourier coefficientsof the measurements the DOA’s (we assume that only the azimuth is unknown, since of the source signals and of the noise, respectively, at the the sources are coplanar with the array), and the crossspectral lcth subinterval and frequency w j . A y ( w ) is the N x P matrices of the sources {R,(wj)}J=l. When no a priori direction matrix, whose pth column is the array manifold knowledge on the structure of the matrices {R,(wj)} is vector a-,,(w) in the direction y p . The noise is assumed to available, they can be taken to be general P x P Hermitian be an ergodic and stationary zero mean Gaussian process, matrices, each containing P 2 real spectral parameters. A whose spatial covariance matrix at frequency w j is given by necessary condition for uniqueness is, then, that the number of vjI. The noise power v j and the number of signals P are unknowns be less than or equal to the number of independent known, and the source signal are stochastic Gaussian, possibly measurements. It is not always desirable to process all frequencies, since it correlated processes, with an unknown crossspectral density matrix {R,(wj)}&l. The wideband log-likelihood function may happen that the addition of a specific bin j will add more unknowns than independent measurements ( P 2 > 2Nqj - q;). for the Gaussian model is given by We would then be better off processing only a subset of bins. The counting procedure is therefore given by the following. Let J denote a subset of the set of indices { j } j ” = and let 1 3 1 denote the cardinality of J . Then, the necessary condition for where Rj is the data covariance matrix at frequency w j uniqueness, using the frequency bins { w j , j E J } only, can be written as
+
and R j = K-l E:=, x k ( w j ) x f ( w j ) is the sample covariance matrix at the frequency w j . 11. THE IDEAL AND EFFECTIVE RESOLUTION CAPACITIES
In this section, we derive bounds on the ideal and effective resolution capacities of a wideband array under the assumption of the wideband Gaussian data model. The method we use is based on the fact that from (1.4) one can see that the set of matrices {Rj}is a suficient statistic for the ML estimation of the spatial and spectral parameters of the wideband sources. We can, therefore, derive upper bounds on the resolution capacity by comparing the number of unknpwn parameters to the number of independent variables in {Rj}. Computing the resolution capacity itself (in contrast to its upper bound) is a different problem [2]-[6], which will not be treated in this paper.
Let PJ denote the maximal P for which (2.2) is satisfied. Then, the resolution capacity will be bounded by
P 5 max P J .
(2.3)
J
We now assume that the matrices
R j
all have equal rank
q. This is usually the case since in practice, almost always, rank Rj = min{N, K } = q. Equation (2.3) then becomes
J P 2 + P 5 J ( 2 N q - 4’).
(2.4)
Solving this quadratic equation, one gets
P 5 J(2J)-’
1 + 2Nq - q2 - 25
(2.5)
DORON AND DORON: WAVEFIELD MODELING-RESOLUTION
which implies
P
kR.
(2.13)
Suppose we now truncate E, and wy at a point L = C,kR, where C, > 1.Using the triangle inequality, the error in [e;]7L that would be induced by this truncation satisfies I
I
M
This truncation error would become negligible if C, satisfied . the Bessel funcJ J e ( k R ) ) E ~ / N MSince tion Je(kR) decreases superexponentially beyond e = k R , this is readily achieved for C, 1,and moreover C, = o( - log E ) . We now perform this truncation, and neglect the resulting error. Using the Cauchy-Schwarz inequality, we finally get N
n=p+l
Then, the effective rank r = rank, G is defined as the smallest integer r for which ~ G ( T 5 ) E. In order to make use of the effective rank, we first prove the following Lemma. Lemma I : Let r = rank,G, on,U,, and v, denote the effective rank and the nth singular value and left and right singular vectors of G , respectively, and let U,, V,, and ET be defined as UT = [UI,...,U,] vr = [Vl,...,V,] ET = diag[ol, . . . , or].
C.kR-1
e=-C,kR
Using similar considerations, we get for the error in the derivative CjkR-1
(2.9)
We write G in the form
G = U,X:,VF +
1v
o,u,v;
= G,
+ E,
(2.10)
n=r+l
and the corresponding array manifold vectors and their derivatives as (see (1.2)) a, = Gw, = G,w, a7 = GW, = G,W,
+ E,w, + E,W,
+ +
= aT, e; =1 ‘ ; eT,
(2.11a) (2.1 lb)
where for the sake of simplicity we have assumed (CLkR)’ >> 1. 0 We will now deviate from the ideal approach by allowing ourselves to neglect terms of order E . We do this by means of the following Lemma.
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Lemma 2: Let the reduced sample covariance matrices { R,} be defined as
R, = U;
(U,
1% U,
(U,
)
(2.17)
where R, is the sample covariance matrix at w,, and the Ur3( U , ) are defined as in (2.9), with r, = rank,G, being the effective rank of the sampling matrix G, at w,. Then, the set of matrices {R,}forms a sufficient statistic for the Gaussian model, up to O ( E )in the log-likelihood function. Proof: It is enough to prove the Lemma for a bandwidth consisting of a single frequency bin. Therefore, in the following, we will drop the frequency designation. Using the matrix inversion formula [lo], one can explicitly invert R in (1.4), giving
+A~ I )~- ~ R ,71 AH . (2.18) From Lemma 1 we see that U,U$A7 = Ay + O ( E ) ,and ~
-
=l - I 71 l[
(R
-
,
~
therefore tr(RR-')
= -1t r R -
71
71
+ 7111 -'R,GU,U,"
x [R&A7
71
71
From (2.23), we see that the effective resolution capacity of an array is bounded by the effective rank of its sampling matrices, rather than by the number of sensors. Since T 5 N , the bounds given in (2.22) and (2.23) are always more restrictive than or equal to the corresponding ideal bounds given in (2.6) and (2.7). Condition (2.23) is identical to the narrowband bound one gets from processing a single frequency bin j, provided rj = T . This bin can usually be taken as the highest bin in the frequency band (unless that bin is at a resonant frequency of the array; see Section IV). The characteristic function p~ ( p ) typically falls off superexponentially beyond a certain point (see numerical examples). Thus, there could be a large difference between the ideal and effective bounds, even at very high SNR. The results given in this section can also be used to compute the resolution capacity of an array for the case where the farfield sources are known to be confined to a limited angular sector. Let the sources be confined to the sector r' = [PO- cpc, 90 pC].G j may then be replaced by the modified sampling matrix G j = GjBH [l]. The effective resolution capacity of the array, given that the sources _are confined to r', is then bounded by the effective rank of Gj. The effective rank of G j will typically be lower than that of Gj, and therefore, the effective resolution capacity within a limited sector will be smaller than in the unrestricted case (see numerical examples given later on).
+
C. The Resolution Capacity and the CRB
1
+ 7111 -lR,A?U, + O ( E ) .
An alternative way to derive bounds on the resolution capacity is to study the CRB associated with the bearing (2.19) estimates, and locate the point where it diverges. This method has the potential for being more rigorous and quantitative Additive expressions of the form V-ltrR have no effect on than the counting approach used above. However, the bearing the ML estimates, since they do not depend on the signal accuracy of the ML estimates approaches the CRB only parameters (recall that 7 is assumed to be known). From (1.4) asymptotically ( K -+ CO), and, therefore, using this approach and (2.19), we therefore see that, up to terms of order O ( E ) , one can only derive the bounds on the resolution capacity in the ML estimates can be obtained by maximjzing a function this limit. Under the assumptions of the wideband Gaussian model, into which the measurements enter only via R. 0 The number of independent measurements contained in Rj the CRB is given by [ l l ] and [12] x [R,A?A7
is 2rj& -, @; where q j is the rank of Rj. Following the same arguments as in the previous section, a necessary condition for uniqueness is
P 5 maxP9
(2.20)
9
where PJ- is the maximal P satisfying
l.91P2
+p 5
(2Tjqj
(2.24) - q;).
(2.21)
i E 9
Let us define implies
4
= maxjqj and
P < J24."
T
= m a x j r j . Then, (2.20)
- 9"2.
(2.22)
If the matrices {Rj} are all full rank ( q j = r j ) , then, as in the ideal case, one can verify that condition (2.20) reduces to
P 0). The component of a plane wave at DOA cp which lies in the subspace of X*(r) is &J*,(lcr) exp(fincp). Since these functions have an appreciable magnitude inside M , their exclusion will cause large errors in interpolating from the boundary array into M . The effective rank of the sampling matrix G will therefore drop by 1 or 2 as we pass through the resonance. The array we examine here is a circular equispaced array, with 14 omnidirectional sensors, at an aperture range of 3.63 5 kR 5 4.03. In this range, there is a resonance at kR = 3.83171 ..., for which Jl(kR) = J-l(kR) = 0. The degeneracy of this resonance is therefore D, = 2. In Fig. 5(a), we plot in a solid line the characteristic function of the array at resonance. We see that the effective rank of G is 12, rather than 14, the number of sensors. This is due to the fact that the degeneracy of the resonance is Dp = 2, and thus, the effective rank of the sampling matrix at resonance drops by 2. As we showed earlier, 12 is too low a rank to enable the resolution of more than 11 sources. This fact is reflected in Fig. 5(b), where we show in a solid line the maximal CRB on the bearing errors for 13 sources of per element SNR 30dB, as a function of the array aperture. The source locations were chosen in an asymmetric fashion so as not to be affected by symmetry effects (we return to this point later). We see quite clearly that at resonance there is a huge jump in the bearing
3.1
3.75
3.8
3.85
3.9
3.95
Fig. 5 . Performance of a 14 omnidirectional element circular array as the frequency is swept through a resonance at koR = 3.83171 (solid line) and the performance of the same array with the addition of two dipole elements (dashed line). Thirteen independent sources are positioned slightly nonequispaced in [Oo,360'1: (a) Characteristic function p ( G ( k o (~p)) ; (b) maximal CRB on the bearing errors.
errors. In fact, the increase of the errors is so large that the effective resolution capacity decreases at the resonant aperture. A resonance may be removed if we add to the array sensors which are capable of measuring these components of the wavefield to which the rest of the array is blind, thus bringing the effective rank back to its maximal value of T M . In [ 13, we saw that one may remove a resonance by adding D, dipole sensors on d M , positioned so that they measure the normal derivative of !P(r). In Fig. 5, we show in a dashed line the characteristic function and the maximal CRB on the bearing errors for such a modified array, and one can see that the addition of the dipole sensors indeed completely removed the resonance effect. In the derivation of (2.27), we assumed that the steering vectors are independent. At resonance, for special DOA configurations, this condition may be violated. To illustrate this point, let us examine the case of a circular array of omnidirectional elements, measuring the emissions of P sources,
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equispaced on the interval ( - T ,
T ) . The
steering matrix is then
At a J * l ( k R ) resonance, the n = f l columns of G are identically zero. We can therefore also set the fl rows of W to zero. If P > L / 2 1 and the sources are equispaced, one can verify that the 41 rows of W are orthogonal to all the other rows. Thus, setting these two rows to zero causes the rank of W, and therefore that of Ay,to drop to P - 2. We see that, in this case, contrary to what is usually assumed, the steering vectors are not independent. This could give rise to false p e e s in the spatial spectrum. As an additional consequence, P l Ay does not vanish at resonance for P = *7 N - 2, if the sources are exactly equispaced, and so the CRB will not diverge at resonance. We thus see that the performance of an array may in special cases be highly sensitive to the exact locations of the sources. In conclusion, we would like to note the following. Resonances become increasingly common as the array size or operating frequency increases, since from Weyl’s law [ 131 the density of resonances (number of resonances in the range k 6 k ) increases with increasing aperture, typically as k R in 2-D and as (kR)’ in 3-D. Furthermore, resonances will be more pronounced in 3-D spherical arrays than in 2-D circular arrays, since spherical arrays possess a higher degree of symmetry. The resonance condition for a spherical array is
+
*
je(kR)= 0
for some lLl
< kR.
(4.4)
If this condition is satisfied, the wavefunctions x ( w , r)= je(kr)Yt,(O, cp) are unobservable by the array. Since -e 5 m 5 L, the degeneracy of a j e ( k R ) resonance of a spherical array is Dp = 21 1.
+
V. SUMMARY
of the processing band, the effective resolution capacity stays bounded by the number of sensors. We concluded this paper by examining the intriguing phenomenon of resonance. We studied in detail the example of a circular array, and showed that the effective rank of the sampling matrix drops abruptly at resonance, and with it the effective resolution capacity. This was also demonstrated by numerically computing the CRB on the bearing estimation errors, and comparing the resulting resolution capacity to the one predicted by the effective rank. The main message of this paper is that the effective resolution capacity is usually bounded by the array structure and/or the angular sector size, rather than by the number of sensors. Using the wavefield modeling approach, we were able to illuminate the mechanisms by which this comes about. ACKNOWLEDGMENT It is our pleasure to thank Dr. H. Messer and Dr. A. J. Weiss for their helpful advice and encouragement during the preparation of this work.
REFERENCES [ I ] M. A. Doron and E. Doron, “Wavefield modeling and array processing, part I-Spatial sampling,” ZEEE. Trans. Signal Processing, this issue, pp. 2549-2559. [2] Y. Bresler and A. Macovsky, “On the number of signals resolvable by a uniform line array,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 1361-1374, Dec. 1986. [3] Y. Bresler and A. J. Ficker, “On the resolution capacity of wideband sensor arrays,” in Communication, Control and Signal Processing (E. Arikan, Ed). New York: Elsevier Science, 1990, pp. 1553-1559. [4] Y. Bresler, “On the resolution capacity of wideband sensor arrays: Further results,” in Proc. ZCASSP ’9Z, May 1992, pp. 1353-1356. [5] A. J. Ficker and Y. Bresler, “Sensor-efficient wideband source location,” in Proc. 32nd Midwest Circuits Syst. Symp., Aug. 1989, pp. 586589. [6] 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. [7] M. A. Doron and H. Messer, “On the number of wideband Gaussian signals resolvable by a passive array,” in Proc. of the 26th SZCC, Mar. 1992. [8] B. Friedlander and A. J. Weiss, “On the number of signals whose directions can be estimated by an array,” ZEEE Trans. Acoust., Speech, Signal Processing, vol. 39, pp. 16861689, July 1991. [9] G. Su and M. Morf, “Modal decomposition signal-subspace algorithms,” ZEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 5 8 5 4 0 2 , June 1986. [IO] A. S. Householder, The Theory of Matrices in Numerical Analysis. New York Dover, 1975. [I 11 A. J. Weiss and B. Friedlander, “On the Cram&-Rao bound for direction finding of correlated signals,” ZEEE Trans. Signal Processing, vol. 41, pp. 495499, Jan. 1993. [12] P. E. Stoica and A. Nehorai, “Performance study of conditional and unconditional direction-of-arrival estimation,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 38, pp. 1783-1795, Oct. 1990. [13] H. P. Bakes and E. R. Hilf, Spectra of Finite Systems. Mannheim, Germany: B.I. Wissenshaftsverlag. 1976.
In this paper, we used the wavefield modeling formalism in order to derive some fundamental limitations on the performance of passive arrays. The main concept we used was that of the effective resolution capacity, which is the number of sources that can be uniquely resolved by a passive array in the presence of a finite (although possibly very small) amount of noise. This is in contrast to the ideal resolution capacity, which assumes zero noise. We first derived upper bounds on both the ideal and effective resolution capacities, and showed that the effective resolution capacity is bounded by the effective rank of the relevant sampling matrix. We then studied how the resolution capacity is affected when the sources are constrained to be within an angular sector. We also presented an alternate derivation of the effective bounds using the CRB on the DOA estimation errors. We finally looked at Miriam A. Doron (S’90-M’93), for photo and biography, please see the flat spectra model, which was previously shown to increase issue, p. 2559. the ideal resolution capacity according to the time-bandwidth product, and showed that this result does not necessarily extend to the effective resolution capacity. In particular, if the array satisfies the spatial sampling condition at the upper frequency Eyal Doron, for photo and biography, please see this issue, p. 2559.
this
