A New Subspace Identification Algorithm for High-Resolution DOA Estimation

Transcription

1 1382 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 50, NO. 10, OCTOBER 2002 A New Subspace Identification Algorithm for High-Resolution DOA Estimation Michael L. McCloud, Member, IEEE, and Louis L. Scharf, Fellow, IEEE Abstract In this paper, we propose a new direction of arrival (DOA) estimator for sensor-array processing. The estimator is based on a linear algebraic connection between the standard subspace model of the array correlation matrix and a special signal-plus-interference model, which we develop in this paper. The estimator we propose is a signal subspace scaled MUSIC algorithm, which we call SSMUSIC. It is not a subspace weighted MUSIC, because the scaling depends on the eigenstructure of the estimated signal subspace. SSMUSIC has the advantage of simultaneously estimating the DOA and the power of each source. We employ a second-order perturbation analysis of the estimator and derive stochastic representations for its bias and squared-error. We compare the new DOA estimator with the MUSIC estimator, based on these representations. Numerical results demonstrate the superior performance of SSMUSIC relative to MUSIC and the validity of the perturbation results. Index Terms Bearing response function, direction of arrival (DOA), MUSIC, sensor array processing, subspace processing. support the small measurement length assumption. To this end we employ the perturbation methods of Li and Vaccaro [3] to analyze the bias and squared error of SSMUSIC and MUSIC. The bias of MUSIC was first derived in [4] and [5]. We also find a stochastic representation for the difference in squared error (SE) between SSMUSIC and MUSIC, averaged over the additive noise. This representation may be averaged by Monte Carlo techniques to compare the two algorithms, with greatly reduced complexity relative to simulation of the full estimators. This difference in SE favors SSMUSIC over MUSIC in all scenarios we have considered with reasonable length arrays. We further show that the SSMUSIC algorithm dramatically outperforms MUSIC for correlated signals in a two-signal example. This effect, attributable to the resolution of the signal subspace but not yet fully understood, is important for practical applications of DOA estimation in multipath environments. I. INTRODUCTION IN THIS PAPER, we propose a new algorithm for estimating the directions of arrival (DOAs) of multiple signals impinging on a sensor array. The algorithm is developed by considering the connections between the subspace model and a signal-plus-interference model for the array correlation matrix. The resulting DOA estimator, termed the signal subspace scaled MUSIC (SSMUSIC) algorithm, seeks the local maxima of a ratio of quadratic forms in the steering vector,. The denominator of SSMUSIC is the MUSIC functional [1] which measures the energy that resolves in the noise subspace, while the numerator is a bearing response function that resolves finer scale information about the signal subspace. Numerical experience indicates that the SSMUSIC algorithm is superior to MUSIC, particularly in the low signal-to-noise ratio (SNR) and small measurement support regimes. SSMUSIC also provides a power estimate for each source. In [2], it is shown that all detectors of the form shared by SSMUSIC and MUSIC have the same asymptotic performance (in the measurement length), but we notice significant improvements in SSMUSIC over MUSIC in precisely the opposite regime, namely the case of small measurement length. For this reason, we search for other analysis techniques which can Manuscript received February 11, 1999; revised August 19, This work was supported in part by the Office of Naval Research under Contract N and by the National Science Foundation under Contract ECS This work was presented in part at the 32nd Asilomar Conference on Signals, Systems, and Computers, Monterey, CA, November M. L. McCloud is with Magis Networks, Inc., San Diego, CA USA. L. L. Scharf is with Colorado State University, Ft. Collins, CO USA. Digital Object Identifier /TAP II. NOTATION AND PRELIMINARIES We will denote matrices and vectors with boldface type, using capital letters for matrices and lower-case letters for vectors. Given a matrix we denote the linear subspace of spanned by the columns of by. We will use the symbol to denote the orthogonal projection matrix with range space. We denote the orthogonal projection matrix with range space orthogonal to by. The oblique projection matrix with range space and null space is denoted by [6], the symbol denotes the Hermitian transpose operator. This matrix is idempotent, but not symmetric, and has the following resolution and nulling properties The two matrices and decompose the projection as follows III. FIRST- AND SECOND-ORDER MODELS FOR ARRAY PROCESSING Let us begin with the following model for signal-plus-noise in an (1) (2) -dimensional measurement sensor array (3) X/02$ IEEE

2 McCLOUD AND SCHARF: A NEW SUBSPACE IDENTIFICATION ALGORITHM FOR HIGH-RESOLUTION DOA ESTIMATION 1383 In this model is the steering vector to source at angle is the complex amplitude of source, and is complex white noise of covariance. The steering matrix is and the weight vector is. See [7] for a detailed derivation of this model. If we consider source to be the source of interest, then the signal-plus-noise model may be written as the following signalplus-interference-plus-noise model: In this model, the steering matrix contains the interfering sources and contains their complex amplitudes. The second order propagation model corresponding to (3) is is the diagonal matrix of powers for the uncorrelated sources:. Each term is a rank-1 covariance matrix for a radiating source. The secondorder model may also be written to be consistent with (4) In this equation is the diagonal matrix of the interfering sources powers. Equations (5) and (7) are model-based representations for the measurement covariance matrix. The corresponding model-free representation is is the signal subspace, is the orthogonal (or noise ) subspace, and. We denote the orthogonal projection matrices onto the signal and noise subspaces by and, respectively. From (5) (8) we may record three equivalent representations for the signal covariance Clearly the -dimensional subspaces, and are identical, but so far it is unclear how they should be extracted from the covariance matrix, especially when the latter must be estimated from random data. In order to illuminate this question, and to preview the method we shall propose, let us consider the least squares source separation of the component from the measurement (4) (5) (6) (7) (8) (9) The matrix is the oblique projection onto, along the direction of. The mean of is and the second moment is (11) When the angle between the subspaces and is small, then the noise gain can be large [6]. Nonetheless, (11) may be rewritten as (12) illustrating that the oblique projection extracts the rank-1 covariance for source from the signal covariance. Similarly, the rank covariance for the interfering sources may be extracted as Equations (12) and (13) may be combined to write (13) (14) Now, the essence of what we propose is to use an estimator of to extract the mode structure from an estimate of. To this end, we state the following lemma. Lemma III.1: The oblique projection may be written as is the pseudo-inverse. Moreover, this representation for provides the identity meaning. Proof: See Appendix A. Here is what this lemma suggests. Experimental data will bring an estimate of, and the estimate may be used to estimate and as (15) (16) is a test steering vector. Clearly is an oblique projection matrix, but moreover, it has the properties listed in the following theorem. Theorem III.2: The oblique projection matrix approximates in the following ways: (10) when for some

3 1384 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 50, NO. 10, OCTOBER 2002 when when Proof: See Appendix B. for some for some and, and consequently exploits information in both the signal and the orthogonal subspace. The weighted MUSIC estimators form the single quadratic form for some weighting matrix, and can only exploit information contained in the orthogonal subspace. One could imagine generalizations of the SSMUSIC algorithm of the form, which could exploit structure in the orthogonal subspace more finely. IV. SSMUSIC DOA ESTIMATION ALGORITHM We can now use the algebraic structure we have developed to suggest a new technique for DOA estimation. The idea is to sweep through candidate values of and measure the error in the estimation of the total signal power that results from. We know from Theorem III.2 that this error will equal zero whenever. This suggest that we use the measurements to build estimates of and and seek minima of this ratio of quadratic forms (or maxima of its inverse). We can additionally estimate the power of each detected signal with at the th detected DOA. We term the resulting DOA estimator the subspace scaled MUSIC algorithm. SSMUSIC Algorithm: Given a sequence of measurements,, compute the sample correlation matrix and its eigen-decomposition V. COMPARISONS WITH MUSIC In order to get some insight into the performance of the SS- MUSIC algorithm we develop a second-order (in the additive noise) representation for the bias and squared error of the SS- MUSIC and MUSIC algorithms in Appendix C. We use the methods of Li and Vaccaro in [3], with second-order expressions of the type derived in [9] replacing their first order perturbations. The bias of MUSIC was derived in [4] and [5]. We organize our received data into the matrix is the matrix of noise free signals from the sources and is a matrix of i.i.d. complex Gaussian noise samples with zero mean and variance. Let have the SVD 1 and the SVD (19) Estimate the noise power as the bearing response (17), and let. For each possible DOA,, form (20) We define, and. In Appendix C, the bias of each estimator is found to be (18) and choose the DOAs as the s that produce the largest peaks of. If estimates of the signal powers are desired, also return at these peaks. It is interesting to compare the SSMUSIC algorithm with the MUSIC algorithm. MUSIC employs the function and picks the locations of the largest peaks of. We see that if projects onto the true noise subspace the two techniques yield identical peaks. However, for small sample sizes, low SNRs, or correlated signals, may be poorly aligned with the true noise subspace, causing MUSIC to miss DOAs or pick spurious DOAs. The SSMUSIC algorithm deals with subspace mismatches by using a signal subspace functional in the numerator. This additional functional exploits more of the structure of the true correlation matrix and appears to be more robust to the effects of finite data length than MUSIC. Remarkably, it also appears to be more robust to the effects of nonzero signal cross-correlation than MUSIC, a point we clarify in our numerical examples. It should also be noted that the SSMUSIC estimator is not a weighted MUSIC estimator [8]. This is because the SSMUSIC estimator forms the ratio of the two quadratic forms (21) (22) and the expectations are taken with respect to the additive noise, and a and a denote the first and second derivatives of the vector a, respectively. In the preceding expressions we have made use of the following substitutions: (23) These are stochastic representations for the bias of the estimators in terms of the random, but noise-free, signal matrix. The utility of these representations is that they reveal a simple structure for the bias which contains only signal dependent random terms. We see that they differ only in the factor of in the second 1 The matrix Q Q is P and Q Q P with probability 1. 0 is an estimate of the eigenvalue matrix 3 0 I.

4 McCLOUD AND SCHARF: A NEW SUBSPACE IDENTIFICATION ALGORITHM FOR HIGH-RESOLUTION DOA ESTIMATION 1385 term. Numerical experience leads us to believe that this second term is dominant in the bias expression and consequently that the SSMUSIC estimator has a lower bias than MUSIC. Similarly, we find the average squared error,, for SSMUSIC to be related to the average squared error,,of MUSIC as (24) (25) With this stochastic representation for the performance difference between MUSIC and SSMUSIC, as a function of the signal matrix, we can easily perform Monte Carlo simulations to compare the two estimators. Notice that is not a random quantity since the noise and signal subspaces can be correctly identified from the noise free signal matrix with probability one ( with probability 1) as long as there are more measurements than signals and each realization of is independent and identically distributed. This means that and are the only stochastic terms which need to be generated in the simulations. This technique has the further advantage that it will not suffer from finite grid effects, unlike the Monte Carlo simulations of the full estimator, and it converges to its mean value after very few realizations. We have found that, averaged over several realizations, is negative for reasonable array lengths in all situations we have examined, suggesting that SS- MUSIC has smaller mean-squared error than MUSIC. VI. DOA ESTIMATION EXAMPLES To illustrate the performance of the SSMUSIC algorithm and demonstrate the validity of the analysis of the previous section we examine two beamforming examples. In each case, a uniform linear array (ULA) is employed with half-wavelength spacing between elements. The signals impinging on the array are assumed to be plane wave. A. Two Signals, Equal Power The first example we examine employs an sensor ULA with two equal power signals arriving at angles and. The Rayleigh resolution limit for this problem is (taking the limit in terms of the main-lobe width of the underlying sinc function for a conventional beamformer). In Fig. 1 we see the bearing response of the MUSIC and SSMUSIC algorithms for this problem, using measurements, at two different SNRs (defined as the ratio of the average signal power to the additive noise power in the signal subspace; that is SNR Tr. It can be seen that in the high SNR regime, each algorithm separates the two signals, while at low SNR only the SSMUSIC algorithm resolves both signals. Fig. 2 shows the average bias for the SSMUSIC and MUSIC estimators and the difference in SE for the two algorithms, as a function of SNR, when averaged over 2000 simulations. Also plotted in this figure are the averages of these quantities, based on a Monte Fig. 1. Normalized bearing response for the SSMUSIC and MUSIC estimators for the two signal example under two different SNRs, 5.5 db (column 1) and 15 db (column 2). (a) (b) (c) Fig. 2. Simulation results for DOA errors for the two signal example of Section VI-A. (a) shows the bias of the SSMUSIC algorithm for the first signal. (b) Shows the bias of the MUSIC estimator. Notice that the axes are different for (a) and (b). (c) Shows the difference in SEs (1 ) of the two techniques, averaged over 200 realizations. In each plot the dashed lines represent the sample average of the stochastic expressions from Section V (with 200 realizations) and the circles are the values obtained through the simulation of the array (averaged over 2000 simulations). The bias curves are plotted in radians and the MSE curve is plotted in radians squared. Carlo experiment that used just 200 realizations of the signal matrix in the stochastic representations of (21) and (25). Notice that the SSMUSIC algorithm outperforms MUSIC both in terms of bias and squared error. This can be seen in both the array response simulation results and the perturbation terms. Although the SSMUSIC algorithm was developed under the assumption of uncorrelated signals it is reasonable to ask how its performance compares to MUSIC when there is cross-correlation among the signals. In Fig. 3 we plot the mean squared error (MSE) of the two estimators as a function of the correlation

5 1386 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 50, NO. 10, OCTOBER 2002 Fig. 3. MSE of the SSMUSIC and MUSIC estimators for the two signal example versus correlation coefficient,. Each curve is the average of 2000 simulations with the SNR fixed at 15 db. Fig. 4. Normalized bearing response for the SSMUSIC and MUSIC estimators for the three signal example under two different SNRs, 07:5 db (column 1) and 10 db (column 2). term between the signals, meaning that we vary the parameter in the signal correlation matrix (26) (a) Notice that the SSMUSIC algorithm appears more robust to the correlation structure of the data than MUSIC. B. Three Signals, Unequal Power We now consider an example there are three signals impinging on a uniform linear array (ULA) with sensors and half wavelength spacing. The signals have angles of arrival and powers. The Rayleigh resolution limit for this problem is. Fig. 4 shows the bearing response functions of the the SSMUSIC estimator and the MUSIC estimator averaged over 100 Monte Carlo simulations with measurements and two different values of the SNR. For the low SNR case, only the SSMUSIC estimator resolves the two closely spaced signals. Fig. 5 shows the bias for the SSMUSIC and MUSIC estimators and the difference in SE for the two algorithms as a function of SNR, averaged over 2000 simulations, as well as the values predicted by the stochastic representations. As in the two signal example we see that the SSMUSIC algorithm outperforms MUSIC in terms of both bias and squared error. VII. CONCLUSION In this paper, we have presented a new DOA estimation algorithm which outperforms the MUSIC algorithm. Although not included in the numerical examples, we have seen similar performance gains over the so-called MUSIC2 algorithm examined in [2], in all of the examples we have examined. We have used a second-order perturbation analysis to find the bias and squared error of the SSMUSIC and MUSIC estimator. Through this analysis we have found a simple stochastic representation (b) (c) Fig. 5. Simulation results for the three signal examples of Section VI-B. (a) Shows the bias of the SSMUSIC algorithm for the first signal. (b) Shows the bias of the MUSIC estimator for this signal. Notice that the y axes are different for (a) and (b). (c) Shows the difference in SEs (1 ) of the two techniques, averaged over 200 realizations. In each plot the dashed lines represent the sample average of the stochastic expressions from Section V (with 200 realizations) and the circles are the values obtained through the simulation of the array (averaged over 2000 runs). The bias curves are plotted in radians and the MSE curve is plotted in radians squared. formed from the random signal components in the measurements and the array geometry, which models the difference in squared error between the two estimators. These representations are quite useful in analyzing the two algorithms as they allow for efficient Monte Carlo averaging and do not suffer from finite grid effects. Numerical examples demonstrate the superiority of the SSMUSIC estimator relative to MUSIC, for both uncorrelated and correlated signals.

6 McCLOUD AND SCHARF: A NEW SUBSPACE IDENTIFICATION ALGORITHM FOR HIGH-RESOLUTION DOA ESTIMATION 1387 APPENDIX PROOF OF LEMMA III.1 We begin by stating a result of [6] on the pseudo inverse of the matrix (27) APPENDIX PERTURBATION ANALYSIS Since we are interested in second-order expansions for the bias and squared error of MUSIC and SSMUSIC we need second-order expansions for the various terms in (20) with respect to. We use the results of [9] and some additional analysis to find the expansions 2 This allows us to expand the term as (28). Consequently, we see that and the claims of the lemma follow. APPENDIX PROOF OF THEOREM III.2 To establish the first claim we notice that as a consequence of Lemma III.1 (31) We can put SSMUSIC and MUSIC into the common framework of looking for local minima of the function 3 (32) for MUSIC, and, and for SSMUSIC, and. Proceeding as in [3], we assume that the SNR is sufficiently high so that the minima of are within one Newton iteration of the true s; that is, for a local minima of we assume (33) (29) denotes the th derivative of with respect to. Define and, and let and be and, respectively, with the, etc., replaced by their true values. Notice that. Then we find the difference between and to be since by the definition of the pseudo inverse and. The equality condition is clear since. The second claim of the theorem is a direct consequence of (29). To establish the third claim we simply plug in the value of (34) and are the th derivatives of and evaluated at. Notice that for we have The equality condition is a consequence of Lemma III.1. (30) (35) 2 We use the notation A = B to denote that A is equal to B up to a secondorder expansion in the noise matrix, N. We define the symbol = in an identical manner. 3 Notice that in Section IV we used d()=n() and sought local maxima of the resulting function.

7 1388 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 50, NO. 10, OCTOBER 2002 Letting and we find the secondorder expression for (36) To complete the perturbation expansion we find and to second order using the expansions in (31) and the derivatives from (35). The resulting expansions for the two estimators are (42). Notice that the first term in (42) is the SE expression that is found from first-order perturbation expressions [3]. Proceeding as before we find the average squared error for SSMUSIC to be (43) (37) (38) In the preceding expressions we have made use of the following substitutions (39) Using the second-order expectation results from Appendix D we find the bias of the estimators to be (41) and the expectations are taken with respect to the additive noise. Proceeding in the same manner we find the average (with respect to the additive noise) squared error (SE) for the MUSIC and SSMUSIC estimators using the results of Appendix D APPENDIX SECOND- AND FOURTH-ORDER EXPECTATIONS (44) In this appendix, we give closed-form expressions for the second- and fourth-order moments involved in computing the bias and squared error of the DOA estimation algorithms. The technique for finding the fourth-order moments relies on the extension of the result for zero-mean, real, and normal random variables that (45) This is also true for complex normal random variables as is easily shown by replacing each variable by its real and imaginary parts and applying the result for real random variables. Since we assume that the additive noise matrix is circularly symmetric, we only need worry about forms that involve or since any other second- or fourth-order combinations will have expectation zero, as will any third-order term. Table D lists all of the expectations necessary for the derivations in this paper. To show how they were derived we choose as an example the last entry in the table (46)

8 McCLOUD AND SCHARF: A NEW SUBSPACE IDENTIFICATION ALGORITHM FOR HIGH-RESOLUTION DOA ESTIMATION 1389 TABLE I EXPECTATIONS OF VARIOUS SECOND AND FOURTH ORDER FORMS IN THE i.i.d. COMPLEX WHITE GAUSSIAN MATRIX N Expanding the first term in the expectation (47) [3] F. Li and R. J. Vaccaro, Analysis of MUSIC and min-norm for arbitrary array geometry, IEEE Trans. Aerosp., Electron. Syst., vol. 27, pp , Jan [4] F. Li and Y. Lu, Unified bias analysis for DOA estimation algorithms, in Proc. ICASSP, vol. 4, Minneapolis, MN, 1994, pp [5] F. Li and Y. Lu, Unified bias analysis of subspace-based DOA estimation algorithms, in Multidimensional Signal Processing Algorithms and Application Techniques, C. T. Leondes, Ed. New York: Academic, 1996, vol. 77, pp [6] R. T. Behrens and L. L. Scharf, Signal processing applications of oblique projection operators, IEEE Trans. Signal Processing, vol. 42, pp , June [7] P. Stoica and R. Moses, Introduction to Spectral Analysis. New York: Simon and Schuster, [8] P. Stoica, A. Eriksson, and T. Söderström, Optimally weighted MUSIC for frequency estimation, Matanal, vol. 16, no. 3, pp , July [9] R. J. Vaccaro, A second-order perturbation expansion for the SVD, SIAM J. Matrix Anal. Appl., vol. 15, pp , Apr we have used (45) together with the statistics of the noise. Each of the remaining terms in (46) is found in exactly the same manner and after simplification we have Table I. REFERENCES [1] R. O. Schmidt, A signal subspace approach to multiple emitter location and spectral estimation, Ph.D. dissertation, Stanford University, Palo Alto, CA, [2] P. Stoica and A. Nehorai, MUSIC, maximum likelihood, and Cramer-Rao bound, IEEE Trans Acoust., Speech, Signal Processing, vol. 37, pp , May communication theory. Michael L. McCloud (S 92 M 00) received the B.S. degree in electrical engineering from George Mason University, Fairfax, VA, and the M.S. and Ph.D. degrees, in electrical engineering, from the University of Colorado, Boulder, CO, in 1995, 1997, and 2000, respectively. From 2000 to 2001, he was a Visiting Researcher and Lecturer at the University of Colorado. Since May 2001, he has been a Staff Engineer with Magis Networks Inc., San Diego, CA. His research interests include statistical signal processing and wireless

9 1390 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 50, NO. 10, OCTOBER 2002 Louis L. Scharf (F 86) received the Ph.D. degree in electrical engineering from the University of Washington, Seattle. From 1969 to 1971, he was a Member of the Technical Staff at Honeywell s Marine Systems Center, Seattle. From 1971 to 1981, he was with the Colorado State University as a Professor in the Departments of Electrical Engineering and Statistics. From 1982 to 1985, he was a Professor and Chair of the Department of Electrical and Computer Engineering,University of Rhode Island, Kingston. From 1985 to 2001, he was a Professor in the Department of Electrical and Computer Engineering, University of Colorado, Boulder. He is currently a Professor in the Departments of Electrical and Computer Engineering and Statistics, Colorado State University, Ft. Collins. He has held Visiting Professorships at Duke University, Durham, NC, at the University of Wisconsin, Madison. University de Paris Sud, Ecole Nationale Superiere d Electricite, and EURECOM (Sophia-Antipolis). His research interests include statistical signal processing, as it applies to communication and instrumentation. His current interests are subspace methods for adapting space-time transceivers in wireless communication, radar, and sonar. His recent contributions are to matched and adaptive subspace detectors; adaptive multiaccess communication; reduced rank Wiener filters for efficient coding and filtering; canonical decompositions for reduced dimensional filtering; and time-varying spectrum estimators for experimental times series modeling. Prof. Scharf is a past Member of the IEEE ASSP AdCom. He was Technical Program Chair for the IEEE International Conference on Acoustics, Speech, and Signal Processing in 1980, and Tutorials Chair for the same conference in In 1994, he served as a Distinguished Lecturer for the IEEE Signal Processing Society, in 1995, he received the Society s Technical Achievement Award, in 1998, he was recognized as a Pioneer of the Signal Processing Society, and in 2000, he received an IEEE Third Millennium Medal.