34 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 1. JANUARY 1988

Superresolution by Structured Matrix Approximation

RAMDAS KUMARESAN, MEMBER, IEEE, AND ARNAB K. SHAW

Abstract-High-resolution bearing estimation with arrays of limited aperture is often accomplished using the eigenvalue-eigenvector decom- position of the spatial correlation matrix of the received data. The bearing estimation problem is formulated as a matrix approximation problem. The columns of a matrix X are formed by the snapshot vectors from an N-element array. The matrix X is then approximated by a matrix X,,, in the least square sense. The rank as well as the partial structure of the space spanned by the columns of XM are prespecified. After X,,, is computed, the bearings of the sources and, consequently, the spatial correlation of the source signals can be estimated. The performance of the proposed technique is then compared with two existing methods by simulation. The comparison has been done in terms of bias, mean- squared error, failure rates, and confidence intervals for the mean and the variance estimates for the three methods at different signal-to-noise ratios. When the sources are moving slowly and the number of snapshot vectors available for processing is large, a simple on-line adaptive algorithm is suggested.

I. INTRODUCTION

ONSIDER an equally spaced linear array of N sensors. C Suppose that plane waves from M-point sources from distinct directions impinge on the array. The received signal samples will be correlated in both time and space resulting in a two-dimensional signal. Usually this signal is converted into a one-dimensional signal by using a narrow-band filter before attempting spatial domain processing. The received signal is sampled simultaneously at the N sensors. At the kth instant in time, N such samples form a vector x k , called the “snapshot” vector which is defined as follows:

x k = [ x k ( o ) x k ( l ) * * ‘ X k ( N - 1 ) I T ( 1 )

where “T” denotes the matrix transpose. K such snapshot vectors are observed at K equally spaced time instants. The observed samples are composed of signal from the M-point sources, sk = [ s k ( o ) s k ( l ) . . . S k ( N - I ) ] and noise wk =

X k = S k $ W k , k = l , 2 , .“, K . (2 )

[ w k ( o ) w k ( l ) W k ( N - That is

The signal samples at the nth sensor element for the kth snapshot is modeled as follows [ 2 ] :

M

where

M assumed number of point sources, d spacing between the sensor elements, h wavelength of radiation of the received signals, N number of elements in the array: N > 2M, Om bearing of the rnth source, Amk amplitude of the mth source at the kth snapshot, (Ymk phase angle of the mth source at the kth snapshot.

In a simplified form, the signal samples are written as follows:

where w, and amk are defined as

27rd wm=- sin Om

h

and

See [ 2 ] , for example, for details of the above model. wk(n) is the noise sample at the nth sensor element at the kth snapshot time instant and it represents the observation noise and the noise introduced in the medium. The noise is assumed to be uncorrelated with the signal. It is assumed that the number of snapshots used for processing, K , is small enough that the bearings do not change considerably in this duration. This assumption also applies to the adaptive algorithm described in Section V.

Using (4) in ( 2 ) , x k can be written as

X k = T a k + W k , k = l , 2 , “ ‘ , K ( 5 )

where T is an N x M Vandermode matrix defined below:

\ \ m = I

n=O, 1 , . a . , N - 1 (3)

Manuscript received January 27, 1986; revised May 5, 1986. This work was supported by the Office of Naval Research under Contract N00014-86-K- 0094.

R. Kumaresan is with the Department of Electrical Engineering, Kelley Hall, University of Rhode Island, Kingston, RI 02881.

A. K. Shaw is with the Department of Electrical and Computer Engineer- ing, Wright State University, Dayton, OH 45431.

IEEE Log Number 8717996.

and

ak= [ a k ( l > , ak(2)Y ’ . a k ( M ) ] ‘. (7)

The elements of the vector ak, which are the complex amplitudes, are usually modeled as samples of a stationary

0018-926X/88/0100-0034$01.00 O 1 9 8 8 IEEE

KUMARESAN AND SHAW: SUPERRESOLUTION BY STRUCTURED MATRIX APPROXIMATION 3 5

stochastic process because of the unpredictability of the behavior of the sources [2]. The spatial correlation matrix of the observed vector x is

R = E [ x x’] (8)

where “ + ” denotes complex conjugate transpose and E represents the expectation operator. Since the signal and the noise are uncorrelated

R = T R , T + + u : I (9)

where R, is the (M X M ) correlation matrix of the vector ak in (7) and U’, is the variance if the noise is white. R, is diagonal if the source signals are uncorrelated or noncoherent [2] and then TR,T+ is Toeplitz, otherwise, it is non-Toeplitz.

Often, only a finite number of snapshots, xl, x2, * * e , xK are available for processing, instead of R . An estimate of R , called R , can be obtained from the K snapshots as follows:

where X is an N x K matrix in which the snapshot vectors are embedded as follows:

x= [XI x2 * - XK].

Thus given xI, x2, * - a , and XK or R , the problem is to estimate the angles of arrival (or the elements of the T matrix in (6)) and akr k = 1, 2, . . . , K , from which R, can be estimated. The noise samples wk(n) are assumed to be white, so that only the noise power U’, needs to be estimated.

The problem stated above has received considerable atten- tion in recent literature ([ 11-[3] and references therein). The two areas, spectrum analysis and array signal processing, have significant common ground, and the recent (10-15 years) spurt of enthusiasm for high-resolution spectrum analysis has spilled over to the area of array signal processing as well. Some of the significant contributions in adaptive array processing include the Howell-Applebaum adaptive algorithm [4], Widrow’s LMS algorithm [5], Capon’s maximum-likelihood method [6], the sample covariance matrix inversion method of Reed et al. [7], and Burg’s technique (as applied to the array processing problem) [SI. Comparisons of some of these techniques can be found in Gabriel [l], Johnson [9], and McDonnough [lo].

A related, but somewhat different approach is through the so-called eigenvalue-eigenvector methods [ 1 11-[ 171. These methods make use of a property of the true correlation matrix R in (9) that the signal part of the correlation matrix TR, T+ is of rank M ( < N ) . This knowledge of the signal is used to “separate” the “signal subspace” from the “noise sub- space.” This is achieved by fitting a matrix of rank M , called RM, in the least squares sense to the observed correlation matrix R in (10). That is, an error El defined as follows:

El= \lZ?-l?Mlli (1 1)

is minimized by finding the best R M . 1 1 a 1 1 2 denotes the Euclidean norm. It can be shown that the matrix RM which

minimizes El is given by [25]

where Xk and v k are the eigenvalues and the corresponding eigenvectors, respectively, of R and X I , X2, . . . , AM are the M largest eigenvalues of R . The rationale of this approach is that RM is much less noise-corrupted than R and is closer to TR, T+ . In a series of papers, Tufts and Kumaresan [ 181, [ 191 have shown the benefits of using RM in place of R . They have also illustrated that instead of fitting RM to R , one could as well fit an 2~ to X , using singular value decomposition (SVD) [18], [19]. The SVD-based methods are briefly reviewed in the following paragraph.

First RM or XM is estimated using (12) or SVD of X , which provides a vector or vectors orthogonal to the columns of RM or xM, i.e., vectors that lie in the noise subspace spanned by V M + ~ , v M + 2 , . . -, v N , where vi = [u;l u;2 * ujN] T. Since the vector(s) will be approximately orthogonal to the columns of the T matrix, a polynomial or polynomials formed with the elements of these vectors as coefficients, i.e.,

N

V,(z )= U ; k z - k + l k = I

computed at z = e’”, will have nulls approximately at e j w k , k = 1, 2, e , M , from which the source bearings can be estimated. The different eigenvalue-eigenvector based meth- ods [13]-[17] differ in the way these vectors are calculated from the noise subspace eigenvectors. Kumaresan and Tufts [ 171-[20] proposed a “minimum-norm’’ vector which has some desirable properties. If the value of M is not known a priori, it is usually estimated from the size of the Xk.

The above said eigenvalue-eigenvector decomposition based methods implicitly use the information that the dimen- sion of the signal subspace, i.e., the rank of zM is less than N. This is done by minimizing the error El in (1 1). In this paper the above problem is treated as a structured matrix approxima- tion problem where we also make use of the knowledge that the array data are composed of exponentials with unknown spatial frequency components. Effectively, the matrix X is approximated by a matrix zM of not only specified rank M , but also the space in which the columns of 2~ can lie is partially specified.

The paper is organized as follows. In Section 11, the formulation is presented as a matrix approximation problem. The approximation error is reparametrized in terms of a convenient set of parameters and this leads to an iterative technique that minimizes the error. However, it should be noted that this reparametrization makes this technique suitable only for uniformly spaced linear arrays. In Section 111, the two phases of the iterative technique to estimate the bearings are discussed and also the signal amplitude and spatial covariance estimates are computed from the bearing estimates. Simulation results and comparison with other methods are presented in Section IV. In Section V, a simple on-line adaptive algorithm is suggested to estimate the bearings when the locations of the

36 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 1, JANUARY 1988

sources (i.e., the angles of arrival) are varying slowly. Section VI consists of some concluding observations.

11. MATRIX APPROXIMATION IN A SUBSPACE

In the following we will model the observed snapshots xk, k - , K , embedded in a matrix X = [xl x2 . . * xK], by = 1,2 ,

a matrix xM defined as

xM= T[al a2 . - aK], (13)

that is, the model in ( 5 ) will be invoked directly. The error criterion to be minimized is given by the following Euclidean norm:

E2 = 11 X - 2~ 11 = tr [( X - 2 ~ ) + ( X - 2M)] ( 14)

by finding the best xM. At this point, it is important to note that the criterion in (1 1) only requires that the approximating matrix be of rank M , whereas the one in (14) also specifies that the columns of 2 M should lie in the span of the columns of the Vandermonde matrix T. Also, if M > K (which is unlikely, because the number of snapshots almost always far exceeds the number of sources), the criterion in ( 1 1 ) leads to a trivial answer, whereas (14) is still meaningful.

If T i n (13) is known, the minimization of E2 in (14) will be the usual linear least squares problem. However, in (13) both T and [a, a2 aK] are unknown. It turns out that situations of this type arise in a special class of nonlinear least squares problems in which the parameters separate into linear parame- ters ak's, and nonlinear parameters in the elements of T. Several numerical analysts [2 11-[23] have studied such prob- lems in a more general setting. The key point made in these references is that the underlying parameter sets [al * * . aK] and T are separable so that the error can be reformulated to depend only on the nonlinear parameters in T and still possess the same minima. A brief summary of this approach is presented below.

If the bearings of the sources, i.e., T is known, then the corresponding amplitudes [al a2 . . * aK] are found by the usual least squares solution [25]

[al a2 aK]=[T+T]- l T+ [xl x2 - * e xK]

= [ T+ TI-' T+ X . (15)

Plugging (15) in (13)

ZM= T[T+T]- ' T+ X. (16)

For convenience, an error matrix E is defined now as

E = X - 2 M

= [ I - T [ Tt TI ' T+ ] X . (17)

Using (17) in (14), the new error criterion can be written as

E 3 = t r [ E t E ] = t r [ X + ( I - T[T+T]- 'T+)X] (18)

since I - T[T+ TI - T' is idempotent. Golub and Pereyra [21] have shown that E3 and E2 have the same minima if [al a2 . . aK] is indeed estimated as in (1 5). Note that the explicit

entry of the amplitudes al a2 . aK from the error expression

has been eliminated in E3. The main objective is to obtain the globally best T that minimizes E3 in (1 8).

The above expressions are not new. Similar expressions are found in [21]-[23], [34]. Tufts and Kumaresan [26] among others [27] have obtained similar expressions (but the number of snapshots was assumed to be one, i.e., K = 1). But minimizing E3 requires an exhaustive search procedure over all possible T matrices and is tedious [26].

The following point is interesting and was observed by the first author earlier in [28]. See also our related work [24]. We are interested in finding an alternate expression for the error E3 in (18) which is easier to minimize. In order to do that, let B ( z ) = bo + blz-' + + bMz-M be an Mth-degree polynomial with roots at ejwl, eJw2 . . . eJwM. In Appendix I it is shown that the error norm E3 in (1 8) can be reformulated to depend only on the coefficients bo, bl, . , bM, so that

E 3 = b + ( 5 Y ; ( B B + ) - ' Y k ) b k = 1

where b, Yk, and B are defined in (27) and (35). The idea behind this reformulation is that the minimization of E3 with respect to the coefficients of B ( z ) can be achieved by an iterative algorithm instead of the multidimensional search that is needed in minimizing (18) directly with respect to T.

If a vector b that indeed minimizes E3 in (19) is found, there is no guarantee that the roots of B ( z ) will lie on the unit circle as desired in the model in (5) . In order to ensure this, certain constraints need to be imposed on the coefficients of the polynomial B(z) . Thus the following constrained minimiza- tion problem needs to be solved:

minimize K

with respect to E3= b + Y l (BB+)- 'Ykb bo, bi, b,w k = 1

(20) subject to

B(z)=O, at J z l = l .

There are more than one equivalent necessary and sufficient conditions [31] that can be imposed on the coefficients of B(z) . One such condition, which helps simplify the computa- tional algorithm described in the next section, is that the coefficients should obey conjugate symmetry, i.e., bk =

bh-k for k = 0, 1, e . . , M ("*" denotes complex conjugation). The second condition is that B ' ( z ) , the deriva- tive of B ( z ) with respect to z , should have all its roots inside the unit circle. The first condition is easy to impose whereas the second condition is not. However, numerical experience confirms that not imposing the second condition may not be a serious drawback.

111. COMPUTATIONAL ALGORITHM

An error criterion similar to the one in (19) (with K = 1 and without the special constraints) occurs in a filter design problem discussed in [32]. The algorithm presented here is a modified version of the ones discussed in [32] and [24]. The modification is to impose the coefficient symmetry constraints on the polynomial B ( z ) discussed above and also a fast

KUMARESAN AND SHAW: SUPERRESOLUTION BY STRUCTURED MATRIX APPROXIMATION 37

technique proposed in an earlier work [24] to invert the BB + can be obtained as follows:

matrix has been used. This fast technique requires inversion of only a 2 M x 2 M matrix at each iteration step instead of the

.. 1 K R,=- I k 6;. (24)

full N - M x N .- M matrix BB+. The algorithm is k = l

iterative and consists of two phases. The first phase minimizes a simplified error criterion where the variability of the middle matrix (BB)-l in (20) is ignored. At convergence, the first phase provides a good starting point for the second phase where the complete error criterion (20) is minimized.

The magnitude of the off-diagonal terms indicate the spatial correlation between the received signals from which the information regarding multipath or the range of the sources can be inferred.

IV. SIMULATION RESULTS Phase I

In this phase of the algorithm the error E', defined below, is minimized at the ith iteration step

K E' 4 b ' + Y : ( B l - l B ' - l + ) - l Y k b '

k = I

At the ith iteration step, E' is minimized by setting the derivatives of E' with respect to the real and the imaginary parts of the elements of the b vector to zero. In this phase of iteration, variations in the middle matrix (BB+) is not taken into account. This phase of the algorithm is described in Appendix I1 in detail.

Phase I1 The first phase of the algorithm does not, in general, obtain

a b f which corresponds to the minimum of E3. This is because the matrix (BB+)-' in the error expression (21) is assumed to be constant at each iteration. In this phase of the algorithm, the variation of the middle matrix (B'B'+)-I is also considered when the gradient of E' is set to zero. The details of this phase of iteration is described in Appendix 111. In both the phases, the inverse of matrix (B,B:) is obtained by inverting a 2M x 2M matrix instead of an N - M x N - M matrix [24]. This is possible because of the banded Toeplitz structure of BIB,+.

From the b vector obtained at the termination of Phase 11, the polynomial

M

B ( Z ) = x bkZ-k k = O

is formed and its roots (or zeros) are computed. The roots should lie on the unit circle (except at very low signal-to-noise ratios (SNR's)). Using these roots, the ?=matrix (the estimate of T as in (6)) is formed. Then the amplitude vector estimates iil, iiz, e , iiK are computed as follows:

If the signal estimates are dlesired, then

xM= T[?=+T]-I?=+x (23)

is calculated. Note that the estimate of' the spatial correlation matrix R,

Performance of the proposed algorithm has been compared with two existing methods by simulations. The model used for simulation has two plane waves (M = 2) that arise from two incoherent point sources and are incident on a linear array consisting of 8( = N ) equally spaced sensor elements. For this part of the experiment, the sources are assumed stationary making l8"( = 0,) and 22"( = 0,) angles at the center of symmetry of the sensor array. An observed snapshot vector xk

= [&(I), xk(2), * 1 , x k ( 8 ) ] 'is composed of signal and noise components as defined in (2)

xk=sk$wh

where wk is generated in a computer as zero-mean unit variance, white complex Gaussian noise. The elements of s k ,

i.e., sk(n), n = 1, 2, e , 8, are generated using (3), where. d = Xl2, N = 8, 6 , = 18", and O2 = 22" have been used. c f l k

and (Y2k are independent random phase angles uniformly distributed in the interval [ - a , a] . A l k = AZk = 3 1.62 for 30-dB SNR and 3.98 for 12-dB S N R .

Using the above source and receiver model, the proposed method has been compared with two existing methods, MUSIC [ 161 and minimum-norm SVD [ 171, [ 151.

For the present method, the error expression in (19) is minimized in two phases (as explained in Section I11 and Appendices I and 11) to find the optimum b vector. Then the polynomial B ( z ) = bo + biz-' + b;z-* is formed, where bl is real. Eight independent trials were conducted at SNR values 30 and 12 dB and the plots of 1/\B(z)l2, z = eJw, w varying from 0 to 2 a are shown in Fig. l(a) and (a ') , respectively. The results with the same noise realizations for the minimum-norm SVD and MUSIC methods are plotted in Fig. l(b) and (b'), 1 (c) and (c '), respectively.

Somewhat more quantitative results comparing the three methods are given in Tables 1-111. 500 trials with different noise epochs were performed for each SNR values in the range of 10-30 dB. For the minimum-norm SVD and MUSIC methods, the two largest peak locations of the angular spectra P(w) [17] are determined and their locations are used to estimate the bearing angles O1 and Oz. These two methods are said to "fail to resolve" the two direction of arrivals if P(o) had only one distinct peak in the vicinity of O 1 and Oz. The present method "fails to resolve" if at least one of the estimates of O1 and 02, obtained from the roots of B(z ) , is k 5 ' from its true location. Tables 1-111 list the bias, the mean squared error, the confidence intervals [35] for the estimates, and the number of times the methods "failed to resolve" for the three different methods. The confidence intervals were computed using the successful trials only. The present method

38 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 1 , JANUARY 1988

-I

I I 1

0 0

0 ID

0 0

10 -N

^ .

a 0

0 N

'0.00 2.09 4 . 1 9 6.28 R A D I A N S

N

I

(a')

1 30 2.09 4 . i 8 6.28

R A D I A N S (b ')

0 0

0 'n

* 00 zo -0 N - -0

a0 N

'0.00 2.09 4 . ; 8 6.28 R A D I A N S R A D I A N S

(C ') (C)

Fig. 1 . Comparison with minimum norm and MUSIC methods in terms of angular spectra P(w), w varying from 0 to 2 ~ . Each figure shows eight independent trials. Ten snapshots are used in each trial. Eight sensor outputs comprise one snapshot. (a)-(a') Proposed method: (a) SNR = 30 dB. (a') 12 dB. (b)-(b') Min. norm: (b) SNR = 30 dB. (b') 12 dB. (c)-(c') MUSIC method: (c) SNR = 30 dB. (c') 12 dB.

TABLE I STATISTICS OF THE ESTIMATE OF B,( = 18") USING PRESENT METHOD

___________ ~ _ _ ~

95% Confidence 95% Confidence SNR i n B i a s in HSE in No. of F a i l u r e s dB Degrees Degree2 (500 T r i a l s ) Interval for Mean I n t e r v a l f o r Variance

- _ _ _ ~ _ _ _ ~ _ _ _ 30. -.00115 ,00952 0 17.9903 - 18.0074 .00845 - ,01083

0 25. -.00215 .0305 17.9825 - 18.0132 ,02709 - .03472

0 .08880 - .I1381 17.9682 - 18.0237 20. -.00406 ,100

0 17.9440 - 18.0492 .)I914 - .40903 15. -.00338 .360

4 17.9375 - 18.0581 .41627 - .53400 14. -.0022 ,470

35 17.9852 - 18.0653 .68624 - .E8734 12. - .0148 .777

17.7631 - 18.0091 1.0548 - 1.4446 I89 IO. -.I14 1.24

Uniformly spaced linear array is composed of eight sensor elements. Sources are at 18" and 22". Similar results may be obtained for the estimate of e,( = 22"). Confidence intervals are computed using successful trials. Method converged at all iterations.

KUMARESAN AND SHAW; SUPERRESOLUTION BY STRUCTURED MATRIX APPROXIMATION 39

TABLE I1 STATISTICS OF THE ESTIMATE OF el( = 18") USING MINIMUM NORM SVD METHOD

.~ ~- . .

-. -. ~~

SNR i n B i a s i n MSE in2 No. of F a i l u r e s 95X Confidence 95% Confidence dB Degrees Degree (500 T r i a l s ) I n t e r v a l f o r Mean I n t e r v a l f o r Var iance

30.0 0.00108 0.0123 0

25.0 0.00297 0.0386 0

20.0 0.012I 0.129 I

15.0 -0.0108 0.416 36

14.0 -0.059 0.507 65

12.0 -0.185 0.741 I04

10.0 - . 3 2 5 1.253 139

0.01092 - 0.014 17.9914 - 18.0108

17.9858 - 18.0202 0.0342 - 0.0438

0.1147 - 0.1471 17.9805 - 18.0436

0.3677 - 0.4755 17.9305 - 18.0479

17.8744 - 18.0077 0.4427 - 0.5774

0.6186 - 0.8169 17.7326 - 17.8983

0.9966 - 1.3354 17.5642 - 17.7862

Uniformly spaced linear array is composed of eight sensor elements. Sources are at 18" and 22". Similar results may be obtained for the estimate of e,( = 22"). Confidence intervals are computed using successful trials.

TABLE I11 STATISTICS OF THE ESTIMATE OF el( = 18 ") USING MUSIC METHOD

SNR i n s l a s LC, MSE in N O . o f F a i l u r e s 95% Confidence 959. Confidence d B o e g r e e s Degree' (500 T r i a l s ) I n t e r v a l f o r Mean I n t e r v a l f o r Variance

30.0 0.037 0.0459 0

25.0 0.074 0. I I 4 19

20.0 0.148 0.304 I I 3

15.0 -0. 104 0.729 299

14.0 -0.190 0.918 326

12.0 -0.427 1.246 374

10.0 -0.553 I .389 408

18.0148 - 18.0555 0.03953 - 0.05067

0.0963 - 0.12402 18.0445 - 18.1035

18.0947 - 18.2006 0.24618 - 0.32632

17.7996 - 17.9918 0.59649 - 0.88271

0.72300 - I . 10074 17.6703 - 17.9495

17.3550 - 17.7900 1.23047 - 2.02018

1.4733 - 2.6341 17.163 - 17.7306

Uniformly spaced linear array is composed of eight sensor elements. Sources are at 18" and 22". Similar results may be obtained for the estimate of e,( = 22"). Confidence intervals are computed using successful trials.

has a better performance on all counts over the other two methods. I1 combined.

quired to attain convergence was about 4 to 8 for Phases I and

For the MUSIC and the minimum-norm based SVD methods. 8N3 floating-point operations (flops) are required to V. A SIMPLIFIED ON-LINE ADAPTIVE ALGORITHM - - compute the eigendecomposition explicitly. The angular spectra P(w) need to be computed with a fine grid to avoid

In many practical situations, the number of antenna ele- ments N is extremely limited whereas the number of snapshots

introduction of bias in the bearing estimates. For the present method, the N - M x N - M matrix B B f needs to be inverted at each iteration step. Since this matrix has a Toeplitz (banded) structure, computation of the inverse can be reduced by using a technique suggested in [24] or by using Cholesky decomposition which requires (N - M ) 3 / 6 flops. The bearing estimates are easily obtained in the present method by rooting the polynomial B(z) .

For the present method, the iterations converged for all the noise realizations attempted. The number of iterations re-

available is very large. Also, the sources which generate the plane waves move slowly in their angular locations. Adaptive methods based on eigendecomposition have been reported in recent literature [36]-[38] for this problem. A variety of suboptimal techniques can be designed based on the error criterion of (20). One such technique which is computationally very simple is suggested below.

The snapshot vectors x k are assumed to be continuously available to the processor and the bearings of the sources are assumed to remain constant within a window of P snapshots

40 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36. NO. I , JANUARY 1988

(for sufficiently small P) . Then one can minimize the criteria Ekr Ek+ ( k is the time index of the present snapshot) defined below

k

Ek= x;B+(BB+)-'Bx,,. (25) n=k-P+1

This, of course, is equivalent to fitting the plane-wave model to the past P snapshots. This error can be minimized similarly as in Section 111. However, since the bearings are unlikely to change significantly between snapshots, the following "rea- sonable" criteria can also be minimized instead:

minimize k

with respect to Ek= X,' Bl (Bk- 1B:- I)-'BkXn n = k - P + I bk

(26) subject to

Bk(l)=O, at 121 = 1

where the matrix Bk-, is formed with the elements of the bk- vector obtained while minimizing Ek- 1. Rewriting the error as in (19) (and using (35), i.e., Bkxn = Ynbk)

minimize with respect to Ek = bk+ ( 5 y,' Wk+-iwk-l Yn) bk

bk n=k-P+I

where

wk- 1 = (Bk- i Bl - ) - I / ' Or Bz- (Bk- 1Bl- 1)- I .

The advantage of the above criterion is that, since the term inside the bracket is independent of bk, Ek can be minimized by solving a set of linear equations similar to those in Section 111, Phase 1, as each snapshot comes in. Usually no further iterations are involved at each step, but a second iteration certainly improves the estimate at the cost of a little more computation. In the second iteration at each step, the estimate of b obtained in the previous iteration is used to compute Bk.

Performance of this adaptive algorithm has been tested with a limited number of simulations. The same source and receiver model as in the stationary case has been used except that in this case the sources are assumed to be moving, i.e., the bearing angles O i and O2 are changing at every sampling instant.

For Figs. 2(a) and 3(a), the source locations were changed monotonously keeping the separation at 4 between them. Initial bearing angles were O i = 2" and 0 2 = 6" and at every sampling instant, angles were changed by A0 = 0.002", i.e., 500 snapshots were processed for a total change of 1" in bearing angles. A window length of 10( = P ) past snapshots was used to adapt the b vector at every sampling instant. For SNR values of 30 and 20 dB, the results are shown in Fig. 2(a) and Fig. 3(a), respectively. Plots of li(B(z)l2, z = e lw , w varying from ~ 2 7 r were taken after every 500 samples, i.e., after a total change of 1 in the bearing angles. A total change of 20" for each source is shown in the plots. The instantaneous numerical values indicate an average bias of about kO.3 percent from the true bearing positions.

For Figs. 2(b) and 3(b), the bearing angles were changed in

I 3 . 1 4 4.19 5 . 2 4 6l.28 Z!-14 4l.13 51.24 61.28

R A D I A N S R A D I A N S (a) (b)

Fig. 2. (a) Monotonously changing sources. Total change of 20" shown. Angular position changed by 0.002" at each snapshot. Plots were taken after processing every 500 snapshots. SNR = 30 dB. (b) Zig-zag change in source locations. Direction of change reversed after 2500 snapshots. SNR = 30dB.

3 ' . , . 4 4 ' . i 3 5 ' 2 . 3 8 ' . 2 8 R A D l A N S

Fig. 3. (a) Monotonously changing sources. Total change of 20" shown. Angular position changed by 0.002" at each snapshot. Plots were taken after processing every 500 snapshots. SNR = 20 dB. (b) Zigzag change in source locations. Direction of change reversed after 2500 snapshots. SNR = 20dB.

a zig-zag manner. The separation was kept at 4" . At every sampling instant, angles were changed by \A01 = 0.002' except that after every 2500 samples, the direction of change was reversed. The SNR was 30 dB for Fig. 2(b) and 20 dB for Fig. 3(b).

VI. CONCLUSION A new iterative technique for accurate bearing estimation is

described. The bearing estimation problem has been formu- lated as a matrix approximation problem which utilizes the inherent structure in the data received by a uniformly spaced array. A constrained least squares error criterion is minimized after rewriting the error criterion in terms of the coefficients of a polynomial. The minimization is achieved by a computa- tionally efficient iterative algorithm. A detailed performance

KUMARESAN AND SHAW: SUPERRESOLUTION BY STRUCTURED MATRIX APPROXIMATION

comparison of this method with other contemporaries based on simulation results shows the superior accuracy of the proposed method. A modification of the method for source location tracking is also proposed.

APPENDIX I Let B(z) = bo + b1z-I + s . 0 + b M z M be an Mth-

This implies degree polynomial with roots at eJw I , e J w 2 , . a , e J w M .

M b M - , , e + J w m n = O , for m = l , 2, M.

n = O

This also implies M

b M - , e + J w m ( " + q ) = O , f o r m = l , 2, * . - , M a n d a n y q . n=O

Writing these identities in matrix form, we obtain

. .

. . e;(N-I)wl . . e;(N-I)wZ

where B is the matrix on the left. Thus the columns of the matrices B + (rank = N - M ) and T (rank = M ) are orthogonal. From a well-known theorem on projection matri- ces [30, p. 109, theorem 5.31 it follows that

Z=B+(BB+)- IB+ T ( T + T ) - ' T + .

Thus

B+(BB+)- lB=(Z- T ( T + T ) - ' T + ) . (28)

If the above identity is plugged in (17), the error matrix becomes

E = B+ ( B B + ) - ' BX.

Thus E 3 = t r E + E = t r ( X + B + ( B B + ) - ' B X ) . (29)

The following forms of the error norm E3 are equivalent:

E 3 = t r [X+(Z- T ( T + T ) - ' T + ) X ] (30)

= t r [ X + ( B + ( B B + ) - ' B ) X ] (31)

= tr [ B + ( B B + ) - ' B X X + ] (32)

= K tr [B+(BB+)-IBR^] (33) K

= X: B+(BB+)- 'Bxk k = 1

(34)

41

Thus the error criterion E3, which is intended to be minimized, is reparameterized in terms of the coefficients of a polynomial B(z ) , instead of T and ak's. Note that Bxk in (34) can be rewritten as follows: B X k

4 Ykb. (35) Thus substituting Ykb in place of Bxk in (34) another form for E3 is obtained

K

E3 = b + Y: (BB+)- l Ykb k = I

= b + ( 5 Y;(BB+)-IYk) b. k = I

APPENDIX I1

In this Appendix the first phase of the computational algorithm which attempts to minimize the error criterion in (20), is described. A constraint needed to force the zeros of B ( z ) on the unit circle is used in the algorithm derivation. But for this constraint, this Appendix largely follows Evans' thesis [321.

The error criterion to be minimized is given below:

E 3 = b + (i k = I Y:(BB+)-IYk) b. (37)

The Yk matrix is defined in (35). Since the B matrix is a function of the terms in the b vector and the initial estimate for b is not available, as a first step, (37) is minimized with the assumption that

( B B + ) - ' = I .

So that in the first step, the following criterion is minimized:

(38)

In order to facilitate imposition of the conjugate symmetry constraints on the elements of the complex valued vector b, i.e., bk = b$-k, k = 0 , 1 , 2, * e , M , the elements of the b vector are now written in terms of the real and imagimry parts, bkr and bki, as follows:

r 1

b =

1 j 0 0 o . a . 0 0 0 1 j o . . * o

0 0 0 0 0 s . . 1

0 0 l - j O . . . O 1 - j o 0 o . . * o

. . . . . . . . .

. . . . . . . . .

L

Cb, -J

(39)

42 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 1 , JANUARY 1988

where C is the matrix consisting of 1's and -+j's ( j = a) and b, is the vector consisting of the real and the imaginary parts of the first half of the elements of b. Without loss of generality, M has been assumed to be even. Using (39) in (38), the error expression becomes

where

R I P C+(CY: Yk)c (42)

is a real-positive semidefinite (M + 1 X M + 1) matrix. The quadratic form in (41) can now be minimized with respect to the elements of the real vector b,. In order to avoid a trivial solution, the vector norm I( b,J12 can be set to 1, in which case it results in an eigenvalue-eigenvector problem; that is, b, is given by the eigenvector corresponding to the minimum eigenvalue of R I [33]. Alternatively, one of the elements of b,, say bo,, may be set to 1. In that case, only a linear set of simutaneous equatons need to be solved as described below. In the following equation, the right hand side of (41) has been rewritten with bor = 1 and b2 has been formed with the rest of the terms (unknown) in b,. Also, RI has been partitioned accordingly.

where

b T= [ bo, 1 b l] = [ 1 1 b T] and

b2= [ h i blr b11 . * * b [ ( ~ / 2 ) - 1 1 r b [ ( ~ / 2 ) - r ~ i b ~ ~ / 2 ) ~ 1 (44)

containing M unknowns, R2 is an M x M matrix, and rl is an M vector and ro is a scalar. Rewriting (43)

(45) E; = ro + b,T rl + r r b2 + b: R2 b2.

Taking a derivative of E with respect to b2 and setting it to zero

a E; -=2r l+2R2 b2=0 (46) a b2

from which b2 is obtained as follows:

b2= -R;I r l . (47)

Once b2 is found from (47), the initial estimate of b, namely b", can be found using (39) as shown below

- -

After the initial estimate of b, i.e., b(O) is formed, the BCo) matrix, composed of the elements of the b(O) vector (see (27),

for definition of the B matrix) can be formed and the error can be written as follows:

The minimization of this error proceeds exactly in the same manner as the initial iteration step, except that at (i + 1)th step of iteration, R I in (42) has the following form:

where

(52)

At each iteration step, R f is formed using (50) and the steps from (43) through (52) provide a new estimate of b, namely b('+') which is formed similarly as in equation (48). The iterations continue until b(1) G b('- I ) .

wi 4 ~ i + ( ~ i ~ i + ) - l

APPENDIX I11

Phase II: In this phase, the derivative of (BB +) - is taken into account. The complete error expression in the Phase I is from (36)

where W' is defined in (52). Note that though ( B B + ) - ' was included in the expression of error in the second stage of Phase I, it was lumped in the R I matrix (see (51)) and while proceeding with the minimization of E, R I was assumed to be constant (see (44)-(46)) throughout Phase I .

In Phase I1 of the algorithm, variation of (BB +) with respect to the terms in b, vector is taken into consideration to achieve a better estimate of b.

To simplify notation, the error expression is rewritten below from (53)

where

(54)

e: 4 W'YkCb, ( 5 5 )

and

(56) ,i r e f T er . . . e ; ] T.

For the rest of the derivation, the superscript i will be

~

43 KUMARESAN AND SHAW: SUPERRESOLUTION BY STRUCTURED MATRIX APPROXIMATION

suppressed. Taking partial derivatives of E3 with respect to each of the M variable terms in b, we get

is used in (57), then (61) can be written as

- -

e + l l

e + l2

e + l M

- -

%= [E] + e + e + [E] = 2 Re [[E] + e] a brj

(57)

where the subscript rj denotes the j t h of the M variable terms in b, and Re [ a ] represents the real parts of the terms inside the brackets, and

. . . 51’ ( 5 8 ) a b,

where

= O . (62) +

The vector e in (56) may also be written as

WYI c WY2C In (59), abr/abrj produces an (M + 1) vector with a “ 1 ” in

the position where the brj variable occurs. For example, b, 4 Zb,. (63) e =

- [ O O * . . 1 0 . . . O I T , a br ~-

a brj

with a “1” at the position of brJ and

aw a -=- [B+[BB+]- I ] a brj a brj

= [E] [ B B + ] - ’ - W

+ B [e] + ] [BB+]-I

Since, for any complex number x, x + x* = 2 Re (x) , (61) can be written as

l & Z 1 br

Re

B+

br=O. (64) where aB/abrJ are N - M x N matrices with a “1” or a ‘‘ &a’’ (depending on whether brJ is the real or the imaginary part) occurring in the B matrix wherever the element 6, is present and zeros elsewhere. For M unknown variables, bo,, b l r , . . . , bM/2r, (57) will produce M partials. Putting these M partials on top of each other an M-vector can be formed as shown below:

gM1 g M 2 * gM,M+I

Now defining an M x M matrix G and an M x 1 vector g as

and

g12 * g l , M + I

g 2 2 . . g 2 , M t I . . . . . .

g M 2 * . g M , M + I

= 0. G “

Since bf S [llbTlT’, as defined in (44), bz can be found as

b2 = [ G ‘GI Gg . (66)

At the “i”th iteration of Phase 11, b(‘) is formed using (66) Also, (58) will produce M partials for each of the unknowns. If these vectors are denoted as 1 1, 1 2 , * * , lM and this notation

44 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 1 , JANUARY 1988

and (64). The B(‘) matrix is formed using the elements of the b(‘) vector in (27). At the (i + 1)th iteration step this B(’) is used, as for example, in (60), This minimization phase continues until b(’+ I ) E b(‘) is reached and this optimum b vector corresponds to a minimum of the error surface of E3.

It vector contains 2(M + 1) unknown terms, the conjugate symmetry con- straints were imposed on the b vector and bo,* was set to unity in both the phases. These enabled the minimization of the error to be carried out with respect to only kif unknowns at every iteration steD.

for a class of non-linear models,’’ Techrometrics, vol. 15, no, 2 , pp. 209-218, May 1973.

1231 H. D. Scolnik, “On the solution of non-linear least squares problems,” Ph.D. dissertation, Univ. of Zurich, Zurich, Switzerland, 1970.

1241 R. Kumaresan, L. L. Scharf, and A. K. Shaw, “An algorithm for pole- zero modeling and spectral analysis,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, June 1986. C. L. Lawson and R. J . Hanson, Solving Least Squares Problems. Englewood Cliffs, NI: Prentice Hall, 1974. D. w. Tufts and R. Kumaresan, “Improved spectral resolution 11,” in Proc. IEEE Conf. on Acoustics, Speech and Signal Processing (Denver,

[271 R. Birgenheir, “Parameter estimation of multiple signals,” Ph.D. dissertation, UCLA, Los Angeles, CA, 1972.

be noted here that though the [251

[261

1980), pp, 592-597.

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A. Cantoni and L. C. Godara, “Resolving the directions of sources in a correlated field incident on an array,” J . Acoust. Soc. Amer., vol. 67, no. 4 , pp. 1247-1255, Apr. 1980. S. S. Reddi, “Multiple source location-A digital approach,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-15, no. I , pp. 95-105, 1979. R. 0. Schmidt, “Multiple emitter location and signal parameter estimation,” in Proc. RADC Spectral Estimation Workshop (Rome,

R. Kumaresan and D. W. Tufts, “On estimating the angles of arrival of multiple plane waves,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-19, no. 1, pp. 134-139, Jan. 1983. D. W. Tufts and R. Kumaresan, “Frequency estimation of multiple sinusoids: Making linear prediction perform like maximum likeli- hood,” froc. IEEE, vol. 70, no. 9, pp. 975-989, Sept. 1982. R. Kumaresan and D. W. Tufts, “Singular value decomposition and spectral analysis,” in froc. 1st IEEE Workshop on Spectral Estimation (Hamilton. Ont., Canada, Aug. 1981) pp, 6.4.1-6.4.12. R. Kumaresan, “On the zeros of the linear prediction error filter for deterministic signals,” IEEE Trans. Acoust., Speech, Signal Proc- essing, vol. ASSP-31, no. 1, pp. 217-220, Feb. 1983. G . H. Golub and V. Pereyra, “The differentiation of pseudoinverses and nonlinear problems whose variables separate,” SIAM J. Numer. Anal., vol. 10, no. 2, pp. 413-432, Apr. 1973. 1. Guttman, V. Pereyra, and H. D. Scolnik, “Least squares estimation

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R. Kumaresan, “Estimating the parameters of exponentially damped and undamped sinusoidal signals,” Ph.D. dissertation, Univ. of Rhode Island, Kingston, RI. Aug. 1982. C. R. Rao and S . K. Mitra, Generalized Inverse ofMatrices and Its Applications. M. Marden, Geometry of Polynomials (Math Surveys, no. 3, American Math. Soc., Providence, RI), 1966, ch. IO. p. 206. A. G. Evans and R. Fischl. “Optimal least squares time-domain systhesis of recursive digital fillers,” IEEE Trans. Audio Elec- troacoust., vol. AU-21, no. 1, pp. 61-65, Feb. 1973. A. G . Evans, “Least-squares system parameter identification and time domain approximation,” Ph.D. dissertation, Drexel University, Phila- delphia, PA, 1972. K. Hoffman and R . Kunze, Linear Algebra. Englewood Cliffs, NJ: Prentice-Hall, 1971. D. C. Rife and R. R. Boorstyn, “Multiple tone parameter estimation from discrete-time observations,” Bell Syst. Tech. J. , vol. 55 , pp. 1389-1410, 1976. P. I. Bickel and K. A. Doksum, Mathematical Statistics. San Francisco, CA: Holden-Day, 1977, ch. 5 . R. J . Vaccaro, “On adaptive implementations of Pisarenko’s harmonic retrieval method,” in Proc. ICASSP 84, Mar. 84. Y. H. Hu, “Adaptive methods for real time Pisarenko spectrum estimate,” in froc. ICASSf 85, Mar. 85. T. P. Bronez and J . A. Cadzow, ”An algebraic approach to superresolution array processing,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-19, no. 1, Jan. 1983.

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Ramdas Kumaresan (S’78-M’79-S’79-M’82) re- ceived the Ph.D. degree from the University of Rhode Island, Kingston, in August 1982

He was a member of the technical staff at GTE Business Communication Systems from September 1982 to June 1983 He rejoined the University of Rhode Island as an Assistant Professor in the Department of Electrical Engineering in July 1983 where he is currently an Associate Professor His areas of research interests include spectrum analysis and architectures for signal processing computers.

Arnab K. Shaw was born in Calcutta, India, on April 2, 1955. He received the B.E.E. degree from Jadavpur University, Calcutta, in 1979, the M.E.E. degree from Villanova University, Villanova, PA, in 1983, and the Ph.D. degree from the University of Rhode Island, Kingston, in 1987.

He is currently an Assistant Professor in the Department of Electrical and Computer Engineer- ing at Wright State University, Dayton, OH. His research interests are in spectral estimation, numeri- cal analysis, and applied mathematics