On Unitary and Forward–Backward MODE

Digital Signal Processing 9, 67–75 (1999) Article ID dspr.1999.0332, available online at http://www.idealibrary.com on

On Unitary and Forward–Backward MODE1 Alex B. Gershman*,2 and Petre Stoica† *Communications Research Laboratory, Department of Electrical and Computer Engineering, McMaster University, Hamilton, Ontario, Canada; †Systems and Control Group, Uppsala University, Sweden Alex B. Gershman and Petre Stoica On Unitary and Forward– Backward MODE, Digital Signal Processing 9 (1999), 67–75. A new unitary (real-valued) formulation of the popular MODE directionof-arrival (DOA) estimator is considered. Our unitary MODE algorithm has a reduced computational complexity because it is based on the eigendecomposition of a real-valued covariance matrix. We prove its exact equivalence to the forward-backward MODE (FB-MODE) estimator derived by Stoica and Jansson. This property sheds a new light on the usefulness of FB-MODE. r1999 Academic Press

1. INTRODUCTION The problem of reducing the computational complexity of eigenstructure methods via real-valued formulations has drawn recently a considerable attention in the literature [1–4]. In this paper, we develop a real-valued variant of the popular MODE direction of arrival (DOA) estimator [5, 6] using a unitary transformation [1]. Then, we show that the proposed unitary MODE cost function is exactly equivalent to the forward–backward MODE (FB-MODE) cost function that has been recently derived and investigated by Stoica and Jansson [7]. Although both estimators are shown to be equivalent in performance, the unitary variant of MODE developed here has a reduced computational complexity as compared with FB-MODE in its original formulation [7]. In this context, the performance loss of FB-MODE relative to conventional MODE in coherent source scenarios [7] can be viewed as a natural price for the improved computational efficiency of unitary MODE. Additionally, we show via simulations and a real data example that this performance loss is 1 Supported in part by DFG under Grant Bo 568/22-1, INTAS under Grant INTAS-93-642-Ext, and the Senior Individual Grant Program of the Swedish Foundation for Strategic Research. 2 Corresponding address: A. B. Gershman, Communications Research Laboratory, Department of Electrical and Computer Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4K1. E-mail: [email protected].

1051-2004/99 $30.00 Copyright r 1999 by Academic Press All rights of reproduction in any form reserved.

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compensated by the superior threshold performance of the FB-MODE estimator in uncorrelated source scenarios.

2. REAL COVARIANCE MATRIX Let a uniform linear array (ULA) be composed of n sensors and let it receive q (q ⬍ n) narrowband sources impinging from the directions ␪1, . . . , ␪q. Assume that there are N stationary snapshots x(1), x(2), . . . , x(N ) available. The array observation vector can be modeled as [2, 5, 7] x(t) ⫽ As(t) ⫹ n(t),

(1)

where A ⫽ [a(␪1 ), . . ., a(␪q )] is the n ⫻ q matrix of source direction vectors,

1

5

a(␪) ⫽ 1, exp j

␻ c

6

5

d sin ␪ , . . . , exp j

␻ c

62

T

d(n ⫺ 1) sin ␪

(2)

is the n ⫻ 1 steering vector, s(t) is the q ⫻ 1 vector of source waveforms, n(t) is the n ⫻ 1 vector of white sensor noise, ␻ is the center frequency, c is the propagation velocity, d is the interelement spacing, and (·)T stands for transpose. The n ⫻ n covariance matrix is given by R ⫽ E5x(t)x(t)H6 ⫽ ASAH ⫹ ␴2I,

(3)

where S ⫽ E5s(t)s(t)H6 is the q ⫻ q source waveform covariance matrix, I is the identity matrix, ␴2 is the noise variance, and (·)H stands for Hermitian transpose. In the uncorrelated source case, S is a diagonal matrix, and the matrix (3) is known to be centro-hermitian [4]. To ‘‘double’’ the number of snapshots and decorrelate possibly correlated source pairs in the case of arbitrary matrix S, the centro-hermitian property is often forced by means of employing the so-called forward–backward sample covariance matrix [1, 3, 4] ˆ ⫹ JR ˆ *J), ˆ FB ⫽ 1⁄2(R R

(4)

instead of the conventional (forward-only) sample covariance matrix

ˆ ⫽ R

1

N

兺 x(k)x(k)

H

N k⫽1

.

(5)

Here, J is the matrix with ones on its antidiagonal and zeros elsewhere, and (·)* stands for complex conjugate. The n ⫻ n real-valued sample covariance matrix is given by [1, 3, 4] ˆ Q, ˆ ⫽ QHR C FB

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(6)

where Q is any unitary, column conjugate symmetric n ⫻ n matrix. According to [3], any matrix Q is column conjugate symmetric if JQ* ⫽ Q holds. For example, the sparse matrices [1, 3, 4]

Q⫽

1

冑2

3

4

I

jI

J

⫺jJ

,

Q⫽

1

冑2

3

I

0

jI

0T

冑2

0T

J

4

(7)

0 ⫺jJ

can be chosen for arrays with an even and odd number of sensors, respectively. In (7), the vector 0 ⫽ (0, 0, . . . , 0)T. The signal- and noise-subspace eigenvectors and eigenvalues of the matrices (4) and (6) are related as [1, 4] ˆ S, E ˆ N ⫽ QHU ˆ N, ⌳ ˆ S ⫽ ⌫ˆ S, ˆ S ⫽ QHU E

ˆ N ⫽ ⌫ˆ N, ⌳

(8)

where the eigendecompositions of the matrices (4)–(6) are defined in a standard way [3, 4], ˆ V ˆH ˆ ˆ ˆH ˆ ⫽V ˆ S⌸ R S S ⫹ VN⌸NVN,

(9)

ˆ ⌳ ˆ ˆH ˆ ˆ ˆH ˆ ⫽U R FB S SUS ⫹ UN⌳NUN,

(10)

ˆH ˆ ˆ ˆH ˆ ⫽E ˆ ⌫ˆ E C S S S ⫹ EN⌫NEN.

(11)

The reader can readily check that the matrices (7) satisfy QHQ ⫽ I and JQ* ⫽ Q. Checking that (6) is real-valued and that (8) holds is also straightforward.

3. UNITARY MODE The complex-valued MODE algorithm exploits a specific Toeplitz matrix B with the property BHA ⫽ 0.

(12)

The conventional MODE cost function can be expressed as [5, 6] ˆ WV ˆH f (B) ⫽ tr5PBV S 1 S 6,

(13)

PB ⫽ B(BHB)⫺1BH,

(14)

ˆ S ⫺ ␴ˆ 2I)2⌸ ˆ ⫺1, W1 ⫽ (⌸ S

(15)

where

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and ␴ˆ 2 is an estimate of the noise variance. The source DOAs are directly obtained from the estimate of B [5, 6]. Using the fact that Q is a unitary matrix, let us reformulate (12) as ˜ ⫽0 BHQQHA ⫽ BHQA

(16)

˜ ⫽ QHA describing the array manifold after the unitary with the new matrix A transformation C ⫽ QHRFBQ ⫽ 1⁄2 QH (R ⫹ JR*J)Q

(17)

⫽ QHAS˜AHQ ⫹ ␴2QHQ

(18)

˜ S˜A ˜ H ⫹ ␴2I, ⫽A

(19)

where S˜ ⫽ 1⁄2 (S ⫹ DS*DH),

(20)

D ⫽ diag5e⫺j(␻/c)d(n⫺1) sin ␪1, . . . , e⫺j(␻/c)d(n⫺1) sin ␪q6.

(21)

Employing (16), we can define the cost function of a unitary MODE, as opposed to the conventional MODE function in (13) and explained below. Let us exploit the eigendecomposition of the matrix (6) instead of the matrix (5) in Eqs. (13)–(15) and take into account that, according to (16), the matrix QHB should be employed instead of B. Since QHB(BHQQHB)⫺1BHQ ⫽ QHPBQ,

(22)

the cost function of unitary MODE can be defined as ˆ SW2E ˆ H6, fU(B) ⫽ tr5QHPBQE S

(23)

ˆ S ⫺ ␴ˆ 2I)2⌳ ˆ ⫺1. W2 ⫽ (⌫ˆ S ⫺ ␴ˆ 2I)2⌫ˆ S⫺1 ⫽ (⌳ S

(24)

where, according to (8),

The cost function (23) is based on the eigendecomposition of the real-valued matrix (6), which has a reduced computational complexity (approximately by a factor of 4 [2]), as compared to the eigendecomposition of the complex-valued matrices (4) or (5). Note that, owing to the sparse structure of (7), the left and right multiplications by Q and QH in (6) and (23) have a negligible contribution to the overall computational burden of unitary MODE.

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4. RELATIONSHIP TO FB-MODE Taking into account (8), the fact that Q is a unitary matrix, and some properties of the trace operator, we obtain from (23) that H ˆ WU ˆH fU(B) ⫽ tr5PBQQHU S 2 S QQ 6

(25)

ˆ WU ˆH ⫽ tr5PBU S 2 S 6 ⫽ fFB(B).

(26)

The function fFB(B) in (26) can be recognized as the FB-MODE cost function derived by Stoica and Jansson [7]. Several interesting conclusions can be drawn from (25)–(26). First of all, we observe that the performance of unitary MODE does not depend on a particular choice of the matrix Q, although (7) ˆ in (6) can be obtained from seems to be a very good choice because the matrix C the matrix (5) with a very low computational cost. Furthermore, the expressions (25)–(26) lead to a reconsideration of the usefulness of FB-MODE. In their work [7], Stoica and Jansson proved that the conventional (forward-only) MODE technique performs in coherent source scenarios asymptotically better than FB-MODE, and therefore, they did not recommend using the FB approach in conjunction with MODE. However, our results demonstrate that FB-MODE is a useful method (if carefully implemented) because it enables reducing the computational complexity of the eigendecomposition step approximately by a factor of 4 relative to conventional MODE.

5. SIMULATIONS In the simulations, we assumed a ULA of 10 omnidirectional sensors with half-wavelength interelement spacing, N ⫽ 100 snapshots, and two equally powered narrowband sources with the DOAs ␪1 ⫽ 10° and ␪2 ⫽ 12° relative to broadside. A total of 1000 independent simulation runs were performed to obtain each simulated point. In the first example, we assumed uncorrelated sources. Figure 1 displays the DOA estimation root-mean-square errors (RMSEs) of conventional and unitary MODE versus the SNR. In this figure, the stochastic Crame´r–Rao bound (CRB) [8] is also shown. In the second example, fully coherent sources with zero phase difference at the first sensor have been assumed. Figure 2 shows the DOA estimation RMSEs of both techniques and the stochastic CRB versus the SNR. From Fig. 1, we observe that for uncorrelated sources, the unitary MODE algorithm performs better in the threshold domain than the conventional MODE technique. The asymptotic performances of both techniques are similar [7]. However, in the scenario with coherent sources, the situation changes (see Fig. 2); the conventional MODE method performs better than unitary MODE in the threshold domain. Similarly to the previous case, the asymptotic perfor-

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FIG. 1. The RMSE of the conventional and unitary MODE algorithms versus the SNR in the first example with uncorrelated sources.

mances of conventional MODE and FB-MODE at high SNR are nearly identical [7]. Hence, our simulations show that the performance loss of FB-MODE in coherent scenarios [7] is compensated by performance improvements for uncorrelated sources.

6. REAL DATA EXAMPLE To compare the conventional and unitary MODE approaches further, the experimental ultrasonic data recorded by the University of Wyoming Source Tracking Array Testbed (UW STAT) [9] have been used. These narrowband six-element array records are available on the World Wide Web at http:// wwweng.uwyo.edu/electrical/array.html and have been used as benchmark data for testing direction finding methods in [9–10]. The parameters of these data are: carrier frequency 40 kHz, signal bandwidth 200 Hz, and interelement spacing 2.1␭. The data set number 2 with one stationary source and one moving (constant velocity) source has been used. A rectangular sliding window with N ⫽ 150 snapshots was used to estimate the source trajectories. The results for conventional and unitary MODE are displayed in Figs. 3a and b, respectively.

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FIG. 2. The RMSE of the conventional and unitary MODE algorithms versus the SNR in the second example with coherent sources.

The true source trajectories are shown as well. From these plots, we see that both algorithms have serious problems when the sources become closely spaced. However, it can be observed that in our real data example, unitary MODE has better threshold performance than conventional MODE. In particular, the empirical RMSE (ERMSE),

ERMSE ⫽

冑

1

M

2

兺 兺 (␪ˆ (k) ⫺ ␪ (k)) ,

2M k⫽1

i

i

2

(27)

i⫽1

was equal to 6.723° and 5.821° for conventional and unitary MODE, respectively. In (27), M is the total number of estimated points, ␪ˆ i(k) corresponds to the estimate of the ith DOA at the kth time point, whereas ␪i(k) is the corresponding true DOA (source trajectory).

7. CONCLUSION A new unitary (real-valued) formulation of the popular MODE technique has been considered. Our unitary MODE algorithm has a reduced computational complexity because it is based on the eigen-decomposition of a real-valued

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FIG. 3.

The results of real data processing: (a) conventional MODE, (b) unitary MODE.

covariance matrix. Its exact equivalence to the forward–backward MODE (FB-MODE) estimator has been proven. This property sheds a new light on the usefulness of FB-MODE. In particular, the performance loss of FB-MODE relative to conventional MODE in coherent scenarios [7] can be viewed as a natural price for the computational savings achieved via real-valued computations. Of course, to obtain these savings, the implementation should be organized using the real-valued matrix (6) and the cost function (23), rather than through the direct application of the MODE approach to the forward– backward covariance matrix (4). Additionally, our simulations and real data example have shown that the aforementioned performance loss is compensated by the superior threshold performance of unitary MODE (FB-MODE) in scenarios with uncorrelated sources.

REFERENCES 1. Huarng, K. C., and Yeh, C. C. A unitary transformation method for angle of arrival estimation. IEEE Trans. Acoust. Speech Signal Process. ASSP-39 (1991), 975–977. 2. Zoltowski, M. D., Kautz, G. M., and Silverstein, S. D. Beamspace root-MUSIC. IEEE Trans. Signal Process. SP-41 (1993), 344–364. 3. Linebarger, D. A., DeGroat, R. D., and Dowling, E. M. Efficient direction-finding methods employing forward-backward averaging. IEEE Trans. Signal Processing. SP-42 (1994), pp. 2136–2145.

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4. Haardt, M., and Nossek, J. A. Unitary ESPRIT: How to obtain increased estimation accuracy with a reduced computational burden. IEEE Trans. Signal Processing. SP-43 (1995), pp. 1232–1242. 5. Stoica, P., and Sharman, K. C. Novel eigenanalysis method for direction estimation. IEE Proceedings. F-137 (1990), pp. 19–26. 6. Stoica, P., and Sharman, K.C. Maximum likelihood methods for direction-of-arrival estimation. IEEE Trans. Acoust., Speech, Signal Processing. ASSP-38 (1990), pp. 1132–1143. 7. Stoica, P., and Jansson, M. On forward-backward MODE for array signal processing. Digital Signal Processing—A Review Journal. 7 (1997), pp. 239–252. 8. Stoica, P., and Nehorai, A. Performance study of conditional and unconditional direction-ofarrival estimation. IEEE Transactions on Acoust., Speech, and Signal Processing. ASSP-38 (1990), pp. 1783–1795. 9. Pierre, J. W., Scott, E.D., and Hays, M. P. A sensor array testbed for source tracking algorithms. In Proc. ICASSP’97, Munich, 1997, pp.3769–3772. 10. Gershman, A, B., and Bo¨hme, J. F. A pseudo-noise approach to direction finding. Signal Processing. 71 (1998), pp. 1–13.

ALEX GERSHMAN received the Diploma and Ph.D. degree in radiophysics from the Nizhny Novgorod State University, Russia, in 1984 and 1990, respectively. His research interests are in the area of signal processing and include statistical signal and array processing, adaptive beamforming, parameter estimation, spectral analysis, and their applications to underwater acoustics, communications, seismology, and radar. He has published more than 90 journal and conference papers on these topics. During 1984–1989, he was with the Radiotechnical and Radiophysical Institutes, Nizhny Novgorod, Russia. Since 1989, he was with the Institute of Applied Physics of Russian Academy of Science, Nizhny Novgorod, Russia, recently as a senior research scientist. From the summer of 1994 until the beginning of 1995, he held a visiting research fellow position at the Swiss Federal Institute of Technology, Lausanne, Switzerland. Since 1995 until 1997, he held a guest scientist position at the Ruhr University, Bochum, Germany. Since 1997, he has been a member of staff at the Department of Electrical Engineering of the Ruhr University. In April 1999, he joins the Faculty of Engineering at the McMaster University, Hamilton, Canada, as an associate professor. He is a recipient of the 1993 URSI Young Scientist Award, the 1994 Distinguished Young Scientist Governmental Fellowship (Russia), the 1994 Swiss Academy of Engineering Science (SATW) Fellowship, and the 1995–1996 Alexander von Humboldt Fellowship. He is a senior member of the IEEE and is listed in Who’s Who in the World. PETRE STOICA received the M.Sc. and D.Sc. degrees, both in automatic control, from the Bucharest Polytechnic Institute, Romania, in 1972 and 1979, respectively. In 1993, he was awarded an honorary doctorate by Uppsala University, Sweden. His scientific interests include system identification, time series analysis and prediction, statistical signal and array processing, spectral analysis, and radar signal processing. He has published some 350 papers in journals and conference records on these topics. He has also published eight book chapters and seven books of which the most recent one is Introduction to Spectral Analysis, Prentice-Hall, 1997 (coauthored with R. Moses). He is co-recipient (with A. Nehorai) of the 1989 IEEE ASSP Senior Award for contributions to array processing, and recipient of the 1996 Technical Achievement Award of the IEEE Signal Processing Society for fundamental contributions to statistical signal processing, as well as several other awards and prizes. He is listed in Who’s Who in the World. He is also on the editorial boards of four journals in the field: Signal Processing, Circuits, Systems and Signals Processing, Multidimensional Systems and Signal Processing, and Journal of Forecasting. He was associated with the Department of Automatic Control and Computers at the Polytechnic Institute of Bucharest, lately as Professor of System Identification and Signal Processing. He spent 1992, 1993, and the first half of 1994 with the Systems and Control Group at Uppsala University in Sweden as a guest professor. In the second half of 1994 he held a 150th Anniversary Visiting Professorship with the Applied Electronics Department at Chalmers University of Technology, Gothenburg, Sweden. Currently he is affiliated with the Systems and Control Group at Uppsala University, where he holds the position of Professor of System Modeling. He is a corresponding member of the Romanian Academy, a fellow of the IEEE, and a fellow of the Royal Statistical Society.

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