IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 3, MARCH 2000
663
Broadband DOA Estimation Using “Spatial-Only” Modeling of Array Data Monika Agrawal and Surendra Prasad, Senior Member, IEEE
Abstract—Most of the existing techniques for DOA estimation of broadband sources use both spatial and temporal modeling. This may lead to increased complexity besides a large algorithmic delay. In this paper, we propose a technique that employs only spatial information in the form of a single spatial array covariance matrix. Assuming the source to have an ideal bandpass power spectral density, we formulate two subspace-based search functions for the estimation of DOA's of broadband sources. One of these employs a multidimensional search in the parameter space, whereas the other requires a MUSIC like one-dimensional (1-D) search. The multidimensional cost function is shown to be consistent, yields performance close to the CR bound, and is insensitive to correlation between sources. Both the proposed methods are shown to be robust to deviations from the assumption of ideal bandpass power spectral density used in their formulation.
I. INTRODUCTION
T
HE PROBLEM of estimating signal location parameters like the DOA's from observed sensor array broadband data has been of considerable interest to the signal processing community in recent years. The problem of wideband source location has applications in several fields such as radar and sonar, radio astronomy, and, more recently, speech processing via microphone arrays. The major recent contributions to DOA estimation of broadband sources include the works of Wax et al. [1], Wang and Kaveh [2], Bienvenu [3], Buckley and Griffiths [4], Grenier [5], Morf and Su [6], Doron et al. [7] and Agrawal and Prasad [8]. These methods variously attempt to carry out incoherent or coherent aggregation of the parameters via intermediate narrowband processing on the one hand [1], [2] and complex spatiotemporal processing on the other [3]–[8]. The recent work of Agrawal and Prasad [8] proposes a practical algorithm for the maximum likelihood DOA estimation of broadband sources via a uniform linear array. The various techniques outlined above can be classified into those that utilize a complete knowledge of the source power spectrum and those that do so only partially. In fact, one of the common factors that complicates the processing of wideband array signals may well be attributed to an implicit attempt to simultaneously effect a good model of each source, both in terms of its spatial location and its power spectrum. This complication manifests itself in different forms. Most of the formulations, including the rigorous (stochastic) maximum likelihood formulaManuscript received November 11, 1998; revised August 24, 1999. The associate editor coordinating the review of this paper and approving it for publication was Prof. Chi Chung Ko. The authors are with the Department of Electrical Engineering, Indian Institute of Technology, Delhi, New Delhi, India (e-mail:
[email protected]). Publisher Item Identifier S 1053-587X(00)01535-X.
tions, require use of several, frequency domain source covariance matrices (FDSCM's) or, equivalently, a covariance matrix of a large array of space-time samples (via, for example, the use of a tapped delay line following each sensor). Other methods, notably the signal subspace algorithms based on MUSIC etc., do not use the spectral knowledge explicitly but, nevertheless, require construction of devices like focussing matrices to map or aggregate the signal subspace at different frequencies onto a single central frequency. In terms of performance, it has been shown by Messer [16] via extensive study of the Cramér–Rao bound for different wideband scenarios that the performance improvement through the use of spectral information is potentially significant mainly when separation between sources is smaller than the beamwidth of the array pattern. Even in such conditions, the potential gain is mostly in the unrealistic case where the source spectrum is completely known and can be properly utilized. In all other cases of using partial knowledge of source spectra, the improvement is rather limited. These arguments justify adequately the use of signal space techniques [3]–[5], [8] for the wideband case as well, where source spectrum information, even where available, cannot be used completely. Motivated by these results, in this paper, we attempt to find simple yet robust techniques that may enable high-resolution DOA estimation of wideband sources without the need for constructing a complex spatio-temporal model in the process. More specifically, we attempt to obtain a spatial-only model for the array data, assuming all the sources to have a flat power spectrum over the region of their spectral support.1 This leads us to the formulation of two signal-subspace based techniques using a single measured spatial covariance matrix of the array data, unlike the multiple covariance methods used in [1]–[7]. One of these involves a MUSIC-like, one-dimensional (1-D) search over the null space of the observed covariance matrix, whereas the other involves a multidimensional search over the space spanned by a suitably constructed cross covariance vector. The formulations are general in the sense that these apply to arbitrary sensor geometries. Extensive simulation results show that the two methods work well over a wide range of difficult situations. The multidimensional search function proposed here is seen to achieve the Cramér–Rao bound asymptotically and is also shown to work equally well for both coherent and noncoherent wideband 1The idea of using a flat power spectrum model for each source has been suggested earlier by Buckley [9], where an attempt has been made to obtain a low-dimensional representation of the source via the Karhunen Leove (KL) expansion of the source output.
1053–587X/00$10.00 © 2000 IEEE
664
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 3, MARCH 2000
sources. Finally, it is demonstrated that the proposed methods are relatively insensitive to the assumption of a flat power spectral shape used in their formulation. This leads us to believe that these methods may be of value in several applications. The paper is arranged in the following manner. The problem of wideband DOA estimation from a single sample correlation matrix of array data is formulated in Section II. Two signal subspace-based solutions are proposed in Section III. One of these involves a multidimensional search, whereas the other involves a simpler single-dimensional formulation. Some properties of the proposed solutions, as well as their performance analysis, are summarized in Section IV. Results of extensive simulation studies that bring out detailed behavior of the two algorithms are discussed in Section V. II. PROBLEM FORMULATION In the broadband DOA estimation formulation proposed here, we model all broadband sources to have an ideal bandpass power spectrum over a given bandwidth. The signal received by the th sensor of an sensor array, from the th relative to the broadside of the array, can source located at then be written as (1) is the frequency support of the signal, and where denotes a measure of the signal spectrum at frequency “ .” denotes the propagation delay corresponding to Here, the th sensor with respect to a given reference sensor. For the , where case of a uniform linear array, denotes the sensor spacing. Hence, the received signal at the th sensor due to all the sources present is given by
different frequencies are mutually orthogonal. Substituting (5) into (4), the array covariance matrix due to the th source is given as (6) . Assuming the additive noise at where each sensor to be white Gaussian, the overall covariance matrix is given by (7) is given by (6). It is seen that the no longer has the convenient structure of the type , as is available in the narrowband case, which could lead directly to efficient subspace search techniques like MUSIC. However, the matrix composition as depicted in (6) clearly has information about the source DOA's imbedded in it. The problem of interest here is to devise a convenient method of finding the source DOA's from the elements of . In the next section, we propose two such approaches.
where matrix
III. SUBSPACE-BASED FORMULATIONS FOR DOA ESTIMATION Here, we present two formulations for DOA estimation from the matrix . The first one requires a -dimensional search, and the second needs a single-dimensional search procedure. 1) Multidimensional Formulation: We define a vector as (8) of cross covariances between the output of the first sensor and sensors. Thus, corresponds each of the remaining th element of the covariance matrix . Assuming to the the noise at the various sensors to be uncorrelated, it is easy to see that the vector , which consists of the first column of the matrix without the topmost element, can be written as
(2) (9) Assuming sources to be uncorrelated, the contribution to the th element of the spatial covariance matrix of the received signal from the th source is given by
where
is given by sinc sinc sinc
(3)
(10) Alternatively, we can write (9) in matrix form (11)
(4) Making use of the assumption of an ideal bandpass power spectrum for each source, we can write
where (12) and
is defined as the vector of
(5) denotes the power spectral density of the source Here, distributed uniformly over . We have with its power also made the usual assumption that the spectral components at
source powers i.e., (13)
Equation (11) implies that lies in the space spanned by the columns of the matrix i.e., span
(14)
AGRAWAL AND PRASAD: BROADBAND DOA ESTIMATION USING “SPATIAL-ONLY” MODELLING OF ARRAY DATA
A convenient mathematical way of expressing (14) is to write plus3ptplus3pt (15) where (16) is an orthogonal projection matrix that projects a given vector . Equivalently, we can write (15) as onto span
665
nature. In fact, its behavior is likely to depend a great deal on the separation between sources and their cross correlation coefficient. Therefore, no definite conclusions can be drawn about in the presence of correlated sources, as far as its convexity and minima are concerned. However, a constrained search, which is explicitly based on construction of , seems to enable us to handle correlated sources quite well. This is amply supported by extensive simulations carried out by us even for closely spaced sources. 2) A more general cost function involving more elements of the covariance matrix may be defined as
(17) is the orthogonal complement of and where . In our projects onto the orthogonal complement of span subspace formulation here, we use (17) to define a cost function to find the unknown directions. Given , we need to find a matrix (having a structure given by (12)) such that lies in the span of . Hence, we use the following -dimensional search function to estimate the DOA's (18) so that (18a) This cost function uses the inner product (and hence angle) of for given the observed with its projection onto the span as a measure of its closeness to the test span . A few other relevant remarks about this choice of the cost function are in order. Remarks: 1) First, we make the following very important observation regarding the usefulness of this cost function in a scenario with broadband correlated sources. Proceeding as in (1)–(9) for this scenario, it is easy to see that the vector can now be written as a more complex function of the kind (19a) is a measure of the cross-correlation between the where th and the th sources and where the vectors can be suitably identified. It is clear that now, lies in the union of the column spans of and a suitably defined matrix in terms of , i.e., span
span
(19b)
In any case, the vector continues to have a component in the span of . The use of (18a) now implies a constrained to yield a span that is highly correlated with search of (or, is close to) . Thus, even though, as implied by (19a), no longer lies completely in span , the search yields whose span is closest to the observed vector a matrix and thus leads to good estimates of DOA's in a correlated sources scenario. We hasten to add here, however, that it is that for the not at all obvious from the expression for correlated sources scenario, this function will have a convex
(20) where is a vector of cross covariances corresponding to the use of the th sensor as a reference sensor, and is the orthogonal projection matrix corresponding to , which is related to in a similar manner as is to in (11). Simulation studies show that the use of in can help improve the performance, especially lieu of at low signal-to-noise ratios. 3) It is clear that the number of sources should be less than in order for the criteria (18) and (20) to be nontrival. 4) Clearly, obtaining the value of requires a dimensional search for the minima of the nonlinear functions defined in (18a) and (20). This can be a computationally complex exercise. We note, however, that whereas the spatial-tem, poral modeling of [3]–[5] uses matrices of size where each sensor is followed by an FIR filter of order , the frequency domain approaches of [1] and [2] use frequency domain source covariance (FDSC) matrices of size . In comparison, the method proposed here uses a covariance matrix, thus partially offsetting single the higher cost of multidimensional search. Futhermore, it is possible to carry out this search rather efficiently via algorithms like the alternate projection method [11]. Hence, it can be said that the computational complexity of the search is no more than that of the search associated with associated with the narrowband maximum likelihood cost functions, which are similar to (18) in form. Alternatively, it is possible to reformulate the problem to estimate via a single-dimensional (MUSIC-like) suboptimal search procedure. This is discussed next. 2) Single-Dimensional Formulation [12]: We start by is a -dimensional rank deficient matrix noting that denote the null space of s.t. [5]. Let implies that . In the absence of noise, the null spaces are related via of and (21) Let
be an
-dimensional vector in the null space of
, i.e., (22)
Then, (21) implies that (22a)
666
Now, denoting the th column of the matrix
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 3, MARCH 2000
by
, we have
to prove this, we note that the covariance matrix from the observed snapshots
is estimated via
(23) and, In view of the explicit relationship between elements of to the direction as given in (6), the LHS of (23) can hence, be seen to be a function of “ .” This can be used to construct a search function for given in
(26) It is well known that bold
converges to w.p. 1
asymptotically, i.e., (27)
(24) (w.p. 1). Then, with the oband, in particular, served covariance vector, the cost function (18) becomes
where (24a)
(28)
. Hence, when plotted Note that would yield the required DOA's. If w.r.t. , the minima of the null space of the array covariance matrix spans a -dimensional subspace, then we may consider the use of the modified cost function given by
The RHS of (28) can be seen to be the norm of the projected . Since the norm of a vector is always greater than vector or equal to zero, it follows that
(25)
(29) Hence, the minimum value on the RHS (viz. “0”) can occur if and only if (30)
Thus, the single-dimensional search approach is comprised of first finding the null space of the covariance matrix, followed by the single-dimensional search for the minima of the scalar variable function given in (25). Clearly, the dimensionality of the null space has an important bearing on this cost function. Unlike the narrowband case, this dimensionality is not easy to predict in terms of numbers of sources and sensors. Grenier [5] has reported an interesting experimental study of the rank of a related (though larger) array covariance matrix due to broadband sources. It is shown that even for a single source, the rank is a function of direction of arrival, varying between 3 and 6. This makes it somewhat difficult to predict the rank and, hence, the dimensionality of the null space, as the number of sources increases. For the single, spatial only covariance matrix of interest here, this implies the loss of some degrees of freedom (i.e., reduction in the dimensionality of the noise subspace) and, hence, a reduction in the number of sources, which can be estimated using this criterion. As in the narrowband methods like MUSIC, the cost function also suffers from the loss of rank structure in the presence of correlated sources, making it unreliable in such scenarios. IV. ANALYSIS OF THE MULTIDIMENSIONAL SEARCH FUNCTION In this section, we present a brief analysis of the multidimen. An appropriate expression for the sional search function Cramér–Rao bound is also given here to serve as a benchmark on the performance. A. Consistency The multidimensional estimator proposed here can be shown to be consistent, i.e., it attains minima at actual source directions as the number of observations is increased to infinity. In order
or when (31) (w.p. 1) as , it follows from (15) that (31) Since will be satisfied asymptotically, and hence, it follows that (w.p. 1) as , provided the sources are uncorrelated. This is consistent. proves that obtained by minimizing A simple extension of this proof can be used to show that the also leads to a consistent estimate more general criterion of . The above argument does not hold when the sources are correlated since, as observed earlier in (19), no longer lies . However, as mentioned earlier, since completely in span continues to have component in the span , the criterion does is selected to correspond to exhibit a sort of minimum when the actual directions. However, it is no longer possible to argue that there is a unique minimum. B. Covariance Matrix of the Estimator is a highly nonlinear function of the The cost function unknown directions, and a general analysis of its performance is rather intractable. However, for the case of a single source, reduces to the typical MUSIC-like cost function used for the narrowband case. Thus, the analysis carried out by Porat and when considered Friedlander [13] becomes applicable for for the single source case. Using the results of [13], we can express the covariance of the estimate as cov cov
(32)
AGRAWAL AND PRASAD: BROADBAND DOA ESTIMATION USING “SPATIAL-ONLY” MODELLING OF ARRAY DATA
Furthermore, the value of cov [14]
is given for this case, via
cov where
and
667
(33)
are defined via the relation (33a)
The derivative terms in the expression (32) for the cost function can be evaluated in a straightforward way and are omitted here for the sake of brevity. C. Cramér–Rao Bound
Fig. 1. Performance of the proposed algorithms for DOA estimation of single broad band source at 10 with M = 8 and N (number of snapshots) = 100. The performance predicted from approximate analysis of J ( ) and Cramér–Rao bound (CRB) is also depicted.
The expression for the Cramér–Rao bound for the DOA estimation problem of broadband sources comprised of frequency components can be calculated by [16] CRB (34) where tr (35a)
Fig. 2. Mean square error (MSE) for a single source at 10 as a function of N , along with the asymptotic CRB for M = 8 and SNR = 10 dB.
tr (35b) tr (35c) is the source covariance matrix corresponding to where the th frequency. V. SIMULATIONS In this section, we give results of extensive simulation experiments carried out to study the behavior of the two proposed DOA estimators for broadband sources under various conditions of practical interest. Besides demonstrating the effectiveness of the two approaches (and their comparison), we also examine the performance of the two estimators under conditions of correlated sources as well as when there are significant deviations from the assumed flat power spectrum of the sources under consideration. The simulations reported here are around a uniform linear corresponding to the center array with a sensor spacing of frequency of the band. The spectral support of the signal is taken , representing to be the normalized frequency interval a b.w. of 40% of the center frequency. All performance curves given below are obtained by averaging the results over 100 independent trials. Fig. 1 shows the experimental mean square error (MSE) for the case of a single source placed at 10 with respect to the as a function of broadside of an array of eight sensors SNR. The corresponding curves obtained from the Cramér–Rao
Fig. 3. Mean square error (MSE) for a single source at 10 as a function of M , along with the CRB for N = 100 and SNR = 10 dB.
bound (34) and the approximate analysis equation (32) are also and shown. It is seen that use of all the criteria, viz., yield reasonable performance in this simple case, at least for high SNR's. However, the multidimensional search functions appear to approach the CR bound asymptotically, whereas the does not perform as single-dimensional search function and exhibit threshold behavior at lower well. Both SNR's as they appear to break down at SNR's of 5 dB or less. However, the threshold gets considerably extended for the mul. This may possibly be due to the tidimensional function makes a better use of all the statistical informafact that tion available in . However, there is little to choose between and at higher SNR's, both of which seem to be close . Figs. 2 to the approximate performance analysis based on and 3 show the behavior of the mean square error as a function of the number of snapshots and number of sensors, respec-
668
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 3, MARCH 2000
Fig. 4. Case of two sources: Mean-square-error in the estimation of DOA of two closely spaced, equal power, uncorrelated sources at 10 and 15 as a function of SNR for = 100 and M = 8. The mean-square-errors for the two sources are shown separately.
N
Fig. 6. Case of two correlated sources: MSE as a function of SNR with source directions of 10 and 15 for N = 100 and M = 8. The MSE's for the two sources are shown separately.
Fig. 5. Case of two sources: Mean-square-error in the estimation of DOA of a source as a function of its separation from another source present at 10 for N = 100; M = 8 and SNR = 10 dB.
tively. Once again, the functions and appear to yield does not. asymptotically optimal performance, whereas Fig. 4 depicts the performance in the presence of two closely spaced sources (with a separation of 5 ) for an eight element array. Once again, both and perform fairly well, remaining quite close to the Cramér–Rao bound. In exhibits a threshold addition, as in the single source case, does not. At high SNR's, behavior at low SNR's, whereas the performance of the two criteria is almost identical. Fig. 5 depicts the performance by plotting the MSE for the location of the second source as a function of its separation from a reference source fixed at 10 . It is satisfying to note that in or yield almost all cases, the methods based on results close to the CR bound. Next, we study the effect of source correlation on the performance of the two estimators. Fig. 6 shows the behavior of the MSE as functions of SNR when two fully correlated sources are present in the scenario. Like its narrowband counterpart (such as MUSIC), the 1-D cost function is seen to completely breakand , however, continue to perdown. The functions form well, even in the presence of source correlation. This corroborates our contention made earlier that the multidimensional or can handle both uncorreformulation based on lated and correlated sources with equal ease. Next, we compare the performance of the proposed method with that of Buckley and Griffiths' [4] spatio-temporal, eigenstructure-based method. This method uses a set of tapped delay elements and is based on higher dimensional matrices
(a)
(b) Fig. 7. Comparison with Buckley and Griffiths' (B&G) algorithm: Spatial spectrum for two equal power sources present at 10 and 15 for N = 100 and M = 8. (a) SNR = 20 dB. (b) SNR = 10 dB.
, where is the number of taps following each sensor. Its performance, therefore, depends on the number of tapped elfor comparison. Fig. 7(a) ements used. We have used and (b) show the comparative performance of our 1-D criterion with that of Buckley and Griffiths [4] for different combinations of source separations and source SNR's. The performance in all cases is comparable. It is seen that the proposed
AGRAWAL AND PRASAD: BROADBAND DOA ESTIMATION USING “SPATIAL-ONLY” MODELLING OF ARRAY DATA
(a)
669
Fig. 9. Effect of deviations from constant power spectrum assumption: MSE versus SNR for a single source at 10 with a triangular power spectrum for = 8 and = 100.
M
N
(b) Fig. 8. (a) Mean-square-error in the estimation of DOA's of two sources with unequal powers at 10 and 15. The weaker source is 10 dB below the stronger source. Abscissa represents of SNR of the stronger source. (b) Spatial spectrum of two sources with unequal powers at 10 and 15. The weaker source is 10 dB below the stronger source as the SNR's of the two sources are 20 dB and 10 dB respectively.
yields better or almost the same resolution performance at lower SNR's. It is, however, associated with somewhat larger variability as seen from the spread of the function plots for different runs. A complete graphical depiction of the comparative perfor, however, is more diffimance with the estimator based on cult, because of its multidimensional nature. In the same figures, therefore, we have shown the cloud (though the use of overlapping stars) associated with the estimates of the two directions . In all cases studied here, is seen to outperform via both the estimates obtained via Buckley and Griffiths' [4] ap. proach as well as by When two closely spaced sources have unequal power, use or is able to resolve the two sources well, but the of source with the smaller power exhibits a corresponding larger value of the mean square error [see Fig. 8(a)]. Fig. 8(b) shows the spatial spectrum obtained via the Buckley and Griffiths' for sources of unequal [4] approach and via the use of strength present at {10 15 } and SNR’s of 20 dB and 10 dB,
(a)
(b) Fig. 10. (a) MSE in the estimation of DOA's of two sources at 10 and 15 with a triangular power spectrum for = 8 and = 100 as a function of SNR. (b) Spatial spectrum, at SNR = 20 dB, of two sources at 10 and 15 with a triangular power spectrum for = 8 and = 100.
M
M
N
N
respectively. While the Buckley and Griffiths' [4] spectrum is still able shows significant error for the weaker source,
670
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 3, MARCH 2000
to resolve the two sources well, whereas gives very good estimates for both sources. Finally, we consider the effect of deviation of the power spectrum of the source from the assumed flat bandpass shape. A source at 10 is taken with the same spectral support as before but with a symmetrical triangular power spectrum density. The resulting MSE for both the 1-D and multidimensional cases are depicted in Fig. 9. The experiment was repeated for two closely spaced sources, with the same spectral shape (triangular) and support. Fig. 10 shows the MSE as a function of SNR for and . Note that the results for this case compare well with those of Fig. 4 showing a minor degradation. Fig. 10(b) shows while also comthe directional contour obtained from paring it with the Buckley and Griffiths method at an SNR of 20 dB. It can be seen that all the three formulations proposed here are relatively insensitive to this assumption, whereas the Buckley and Griffiths method starts to fail at this SNR. VI. CONCLUSION Using a spatial-only model for the broadband array data and assuming all the sources to have a flat power spectrum over a given passband region, we have proposed two simple formulations of signal subspace based techniques for DOA estimation of measured covariance broadband sources from a single matrix of the array data. It is demonstrated that the methods are not sensitive to the assumption of a flat power spectrum and that the multidimensional search-based method proposed here work quite well even for the case of coherent sources. REFERENCES [1] 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. [2] H. Wang and M. Kaveh, “Coherent signal subspace processing for detection and estimation of angle of arrival of multiple wideband sources,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, pp. 823–831, Aug. 1985. [3] G. Bienvenu, “Eigensystem properties of the sampled space correlation matrix,” in Proc. IEEE ICASSP, Boston, MA, 1983, pp. 332–35. [4] K. M. Buckley and L. J. Griffiths, “Eigenstructure based broadband source location estimation,” in Proc. IEEE ICASSP, Tokyo, Japan, 1986, pp. 1869–1872. [5] Y. Grenier, “Wideband source location through frequency-dependent modeling,” IEEE Trans. Signal Processing, vol. 42, pp. 1087–1096, May 1994. [6] G. Su and M. Morf, “Modal decomposition signal subspace algorithms,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 585–602, June 1986. [7] M. A. Doron, A. J. Weiss, and H. Messer, “Maximum likelihood direction finding of wideband sources,” IEEE Trans. Signal Processing, vol. 41, pp. 411–414, Jan. 1993. [8] M. Agarwal and S. Prasad, “DOA estimation of wideband sources using a harmonic source model and uniform linear array,” IEEE Trans. Signal Processing, vol. 47, pp. 619–629, Mar. 1999. [9] K. M. Buckley, “Spatial/spectral filtering with linearly constrained minimum variance beamformers,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 249–266, Mar. 1987.
[10] F. A. Giannella and P. M. Schultheiss, “Efficient location of closely spaced wide-band sources,” in Proc. IEEE ICASSP, 1990, pp. 2915–2918. [11] I. Ziskind and M. Wax, “Maximum likelihood localization of multiple sources by alternate projection,” IEEE Trans. Signal Processing, vol. 36, pp. 1553–1560, Oct. 1988. [12] C. S. Maelnnes, “DOA estimation of nonstationary coherent signals using signal vector randomization,” IEEE Trans. Signal Processing, vol. 46, pp. 1744–1749, June 1998. [13] B. Porat and B. Friedlander, “Analysis of the asymptotic relative efficiency of the music algorithm,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, pp. 532–543, Apr. 1988. [14] D. R. Brillinger, Time Series Data Analysis and Theory. New York: Holt Rinehart and Winston, 1975. [15] P. Stoica and A. Nehorai, “MUSIC, maximum likelihood and Cramér–Rao bound,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp. 720–741, May 1989. [16] H. Messer, “The potential performance gain in using spectral information in passive detection/localization of wideband sources,” IEEE Trans. Signal Processing, vol. 43, pp. 2964–2974, Dec. 1995.
Monika Agrawal received the B.Tech. degree in electrical engineering and the M.Tech. degree in electronics and communications engineering from the Regional Engineering College, Kurukshetra, India, in 1993 and 1995, respectively. She is currently pursuing the Ph.D. degree at the Indian Institute of Technology (IIT), Delhi. She is currently a Research Scholar at IIT. Her research interests include digital signal processing, array processing, and communications.
Surendra Prasad (SM’94) received the B.Tech. degree in electronics and electrical communication engineering from the Indian Institute of Technology (I.I.T.), Kharagpur, in 1969 and the M.Tech. and Ph.D. degrees in electrical communication engineering from I.I.T., New Delhi, in 1971 and 1974, respectively. He has been with I.I.T., New Delhi, since 1971, where he is presently a Professor of Electrical Engineering and coordinator of the Research and Training Programme in Telematics. He was a Visiting Research Fellow at the Loughborough University of Technology, Loughborough, U.K., from 1976 to 1977, where he was involved in developing algorithms for adaptive array processing for HF arrays. He was also a Visiting Faculty Member at the Pennsylvania State University, University Park, from 1985 to 1986. His teaching and research interests include communications and statistical and digital signal processing. He has been a consultant to a number of Government agencies as well as industry in these and related areas. Currently, he is engaged in research in various aspects of statistical signal processing and communications, including wireless communications. He has published more than 75 papers in these areas in reputed journals. He edited a special issue of the Journal of I.E.T.E. in March–April 1989 in the area of statistical signal processing and a book on signal processing for the I.E.T.E. book series. Dr. Prasad was the recipient of the Vikram Sarabhai Research Award in Electronics and Telecommunications for the year 1987, the Shanti Swarup Bhatnagar Award for Engineering Sciences for 1988, and the Om Prakash Bhasin Prize for Research in Electronics and Communications for 1994. He was a Co-Chairperson for the “Indo-U.S. Workshop in One and Two Dimensions” held in New Delhi in November 1989. He is a Fellow of the Indian National Academy of Engineering, the Indian National Science Academy, and the Indian Academy of Sciences.
