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Direction-of-Arrival Estimation Based on the Joint Diagonalization Structure of Multiple Fourth-Order Cumulant Matrices Wen-Jun Zeng, Student Member, IEEE, Xi-Lin Li, and Xian-Da Zhang, Senior Member, IEEE
Abstract—A new algorithm for direction-of-arrival (DOA) estimation of non-Gaussian sources is proposed. Based on the joint diagonalization structure of multiple fourth-order cumulant matrices, a novel cost function is designed and a new spatial spectrum for direction finding at hand is derived. Unlike the subspace-based techniques, it is not necessary to determine the number of sources in advance for the proposed algorithm. Moreover, the proposed method is insensitive to the spatially correlated noise. Simulation results are provided to demonstrate the performance of the proposed approach. Index Terms—Correlated noise, direction-of-arrival, high-order cumulants, joint diagonalization.
I. INTRODUCTION
D
IRECTION-OF-ARRIVAL (DOA) estimation is a basic technique widely used in various sensor arrays for localizing radiating sources. Subspace-based methods have found prominence in the problem of direction finding. A vast number of subspace-based DOA estimators have been proposed, e.g., MUSIC, ESPRIT, and subspace fitting [1]. Most of these require the sensor noise to be spatially white or the noise covariance matrix to be known. However, the assumption about the spatial whiteness of sensor noise is often violated and the noise covariance is usually unavailable in practice, which results in performance degradation of these DOA estimators [2]. Although a variety of techniques have been developed for eliminating the effects of unknown noise fields, most of them are based on the assumptions that the noises (or signals) satisfy some particular parametric models [3], or the array is composed of well-separated subarrays (referred to as sparse array) [4]. Clearly, such assumptions may severely restrict the applications of these methods. Another drawback of subspace-based methods is that they require the number of sources to be known or to be exactly estimated in advance. The information theoretic criteria such as AIC and MDL [5] are the most important methods for determining the number of sources. Nevertheless, the rate of correctly
detecting the number of sources is rather low, even for the white noise with moderate signal-to-noise ratio (SNR) [6]. The detection of the number of sources with high correct rate in colored noise fields is a more difficult task [6]. In numerous applications (e.g., digital communications), the signals are non-Gaussian. Non-Gaussian signals contain valuable statistical information in their high-order statistics (HOS). The main advantage of using high-order cumulants is that the colored noise can be suppressed, and it is not necessary to know or to estimate the noise covariance as long as the noise is normally distributed. In [7] and [8], two fourth-order cumulantbased MUSIC algorithms for direction finding are proposed. However, it still needs to determine the number of sources exactly before estimating the DOAs. In this letter, the fairly common case where the signals are non-Gaussian is considered. We propose a new direction finding algorithm to overcome the aforementioned shortcomings. By exploiting the joint diagonalization structure of a set of fourthorder cumulant matrices, a new spatial spectrum is derived and the DOAs can be estimated from it subsequently. Compared with the existing methods, the proposed algorithm has three advantages. Firstly, it is insensitive to spatially correlated noise. Secondly, it has no assumptions on the model of the noises or signals, or the sparsity of the sensor arrays. Thirdly, unlike the subspace-based methods, it is not necessary to determine the number of sources before computing the spatial spectrum. II. DATA MODEL Consider far-field, narrowband sources emitting plane waves impinging on a uniform linear array (ULA) of sensors with inter-sensor spacing . In subspace-based DOA estimators, must be strictly less than . In the proposed approach, we only need to assume . In other words, can be equal to for the proposed method. The complex baseband signal received by the th sensor is expressed as
(1) Manuscript received October 13, 2008; revised November 06, 2008. Current version published February 11, 2009. This work was supported by the National Natural Science Foundation of China under Grant 60675002 and funded by Basic Research Foundation of Tsinghua National Laboratory for Information Science and Technology (TNList). The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Richard J. Kozick. The authors are with the Department of Automation, Tsinghua University, Beijing 100084, China. (e-mail:
[email protected];
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2008.2010816
where is the th source, is the imaginary unit, is the wavelength of the signal, is the DOA of the th source, and denotes the additive noise. By arranging the output of the sensors in a vector with the superscript denoting transpose, the matrix formulation of (1) can be written as
1070-9908/$25.00 © 2009 IEEE
(2)
ZENG et al.: DIRECTION-OF-ARRIVAL ESTIMATION BASED ON THE JOINT DIAGONALIZATION STRUCTURE
where
is the source vector, is the noise vector, and is the array manifold matrix given by (3)
where
formed to achieve this [10]. Moreover, it is rather complicated for JADE to construct cumulant matrices having the desired joint diagonalization structure. Herein we propose a new cumulant matrix construction method, which leads to non-orthogonal joint diagonalization structure and can avoid pre-whitening of the received data. Define a cumulant matrix whose th entry is
is the steering vector
(8) (4)
Denoting ,
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as the th entry of matrix can also be represented as
, i.e.,
(5) Herein we give some assumptions about the properties of the sources and the noises: A1) the sources are zero-mean and stationary, mutually independently with each other, and non-Gaussian; A2) the noises are zero-mean, Gaussian, and can be spatially correlated; A3) the noises are statistically independent with the sources.
, one can construct such cumuSince lant matrices in all. However, by noting the fact , which means that and contain the same useful statistic information (only differ by a complex conjugate), it is not necessary to adopt all the cumulant matrices. Therefore, we can set , and in this way only cumulant matrices containing different statistic information are used for DOA estimation. Then by observing (7), it is clear that matrix can be rewritten in matrix form compactly as (9) where the superscript
denotes conjugate transpose and
(10)
III. DOA ESTIMATION BASED ON MULTIPLE FOURTH-ORDER CUMULANT MATRICES is
a
diagonal
matrix.
A. Cumulant Matrices In this letter, we adopt the fourth-order cumulants of the observed data for DOA estimation. The definition of the fourthorder cumulants of , , , and is given by
By defining for convenience, (9) can be
equivalently written as (11) cumulant Equation (9) and (11) mean that all the matrices have the joint diagonalization structure and span the same range space of , i.e.,
(6)
(12)
where denotes for short, denotes expectation, and the superscript denotes complex conjugate. According to (5) and using the additivity of the cumulants in the addition of independent variables and the multilinearity of cumulants [9], we can derive the expression of shown as
Therefore, we can utilize these multiple cumulant matrices to identify the range space of the array manifold matrix and estimate the DOA parameters. In [7], two DOA estimation methods based on single cumulant matrix are given. Compared with the single matrix-based method, exploiting multiple cumulant matrices can improve the performance greatly, especially in the case where the noise is spatially correlated.
(7) Note that the cumulants of the noise vanish (i.e., ) since the noise has been assumed to be normally distributed. According to the definition of (6), a set of fourth-order statistics can be obtained. The joint approximate diagonalization of eigen-matrix (JADE) algorithm [10] uses the statistics to blindly identify the array manifold. However, the JADE requires the array manifold matrix to be squared and orthogonal; therefore, dimensionality reduction and pre-whitening must be per-
B. Novel Cost Function Derivation For the th source, let us define a vector which is orthogonal to the space spanned by the steering vectors except for , i.e., (13) In other words, we have
.
(14)
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IEEE SIGNAL PROCESSING LETTERS, VOL. 16, NO. 3, MARCH 2009
Since we have assumed , there must exist such nonzero vector satisfying (14). Substituting(14) into (11) leads to
For fixed
and , the optimal
is given by (23)
which leads to (15) where is a scalar. The geometry meanings of (15) is that there exists a vector which makes and are collinear when equals one of the true DOAs, i.e., (16) (16) holds true, it is natural to Since for all propose the new cost function for searching for the azimuth
subject to
(17)
denotes the Euclidian norm, is the steering where vector with parameter to be optimized, is a vector, and is a vector whose entries are . Namely, can be expressed as
(24) Here subscript denotes the Moore–Penrose pseudoinverse. For fourth-order cumulant matrices exactly satisfying the structure in (9), the rank of is . In this case, can be singular due to and the pseudoinverse must be used. In practice, the estimates of cumulant matrices are always noisy, and . Hence, has full rank and the pseudoinverse can be replaced with inverse. By substituting into (22), the cost function can be simplified to (25) By observing (25), it is clear that for fixed , the optimal is given by the unit norm eigenvector of corresponding to its maximum eigenvalue. Thus, the simplified cost function of only is (26)
(18) , therefore Note that a trivial solution of (17) is constraint is used to avoid this trivial solution.
denotes the maximum eigenvalue of a matrix. where Therefore, we can introduce a new “spatial spectrum” (27)
C. Optimization Method for Minimizing the Cost Function Since both and are nuisance parameters to be optimized, it is difficult to use the cost function in (17) to search for the DOAs directly. Thus, the nuisance parameters should be reduced to obtain a simple cost function of interesting parameter only. Expanding (17) leads to
Thus, one can detect and estimate the DOAs by searching for the . It is noteworthy that there is no need to determaxima of mine the number of sources in advance in the proposed method. After plotting the spatial spectrum, the number of sources can be determined by counting the number of peaks in . IV. SIMULATION RESULTS
(19) Define as
matrix
and
matrix
(20) (21) By exploiting the facts that , , and , the cost function in (19) can be rewritten as (22)
sensors with We consider a ULA consisting of inter-sensor spacing . The DOAs of two independent quadrature phase-shift keying (QPSK) sources are and . The noise used in this simulation is zero mean, normally distributed, and spatially colored. The correlated noise model adopted in many literatures such as [3] and [11] is used in our simulation. The th entry of the noise covariance matrix is , where is adjusted to give the desired SNR defined by SNR . Herein we compare the performance of the proposed algorithm with two cumulant-based MUSIC algorithms. One algorithm adopts a single fourth-order cumulant matrix referred to as contracted quadricovariance[7]. The other algorithm is based on the matrix composed of the cumulants of the tensor product (Kronecker product) of the received signal [8]. Hence, we refer it to as Kronecker HOS MUSIC. For the proposed algorithm, fourth-order cumulant matrices are used. The number of snapshots is set to 500 in the following simulations. 1) Experiment 1: In the first example, we compare the three algorithms with closely-spaced sources and show the advantage of not needing to determine the number of sources. We set , , and SNR dB. The number of source
ZENG et al.: DIRECTION-OF-ARRIVAL ESTIMATION BASED ON THE JOINT DIAGONALIZATION STRUCTURE
Fig. 1. Plots of ten typical spectrum estimates. The vertical red lines show the true DOAs.
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Fig. 3. RMSEs of the DOA estimations versus SNR.
V. CONCLUSION A novel direction finding algorithm is designed in unknown correlated noise fields based on the joint diagonalization structure of a set of fourth-order cumulant matrices. An interesting advantage of the proposed algorithm is that it is not necessary to determine the number of sources before computing the spatial spectrum. Such an advantage is highly desirable for practical applications, where the detection of the number of signals is always difficult. Simulation results confirmed the high performance of the proposed algorithm.
Fig. 2. Probability of success versus SNR.
is determined by the AIC criterion for the two cumulant-based MUSIC algorithms. Our new algorithm does not need an estimate of the number of sources. Fig. 1 displays ten typical spatial spectrum estimates. Since the two sources are very close, the two cumulant-based MUSIC algorithms failed to distinguish them and the number of sources is always erroneously estimated as one. However, the spatial spectrum estimates of the proposed algorithm always have two distinct peaks, which suggests that the number of sources is . 2) Experiment 2: In the second example, 400 Monte Carlo trials are performed to evaluate the performance of the DOA estimation. We use two performance indices: one is the root mean squared errors (RMSEs) of the estimated DOA, and the other is the probability of success trails. A trial is successful if it successes in distinguishing two sources, i.e., the corresponding spatial spectrum has two peaks. We always assume that the number of sources is known for the cumulant-based MUSIC algorithms since it is difficult to detect the number of sources correctly. Fig. 2 shows the probability of success versus SNR (from 0 to 15 dB) of the three methods. Fig. 3 illustrates the RMSEs of the estimated DOAs of source 1 and source 2 versus SNR, respectively. As we can see, the probability of success is higher (especially in the case where the SNR is low), and the DOA estimations are much more accurate by using the proposed algorithm rather than the two cumulant-based MUSIC algorithms.
REFERENCES [1] H. Krim and M. Viberg, “Two decades of array signal processing research: The parametric approach,” IEEE Signal Process. Mag., vol. 13, no. 4, pp. 67–94, Jul. 1996. [2] F. Li and R. J. Vaccaro, “Performance degradation of DOA estimators due to unknown noise fields,” IEEE Trans. Signal Process., vol. 40, no. 3, pp. 686–690, Mar. 1992. [3] M. Viberg, P. Stoica, and B. Ottersten, “Maximum likelihood array processing in spatially correlated noise fields using parameterized signals,” IEEE Trans. Signal Process., vol. 45, no. 4, pp. 996–1004, Apr. 1997. [4] S. A. Vorobyov, A. B. Gershman, and K. M. Wong, “Maximum likelihood direction-of-arrival estimation in unknown noise fields using sparse sensor arrays,” IEEE Trans. Signal Process., vol. 53, no. 1, pp. 34–42, Jan. 2005. [5] M. Wax and T. Kailath, “Detection of signals by information theoretic criteria,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-33, no. 3, pp. 387–392, Apr. 1985. [6] X.-L. Li, “Studies on joint diagonalization and phase recovery in blind source separation,” Ph.D. dissertation, Tsinghua Univ., Beijng, China, 2007. [7] J.-F. Cardoso and E. Moulines, “Asymptotic performance analysis of direction-finding algorithms based on fourth-order cumulants,” IEEE Trans. Signal Process., vol. 43, no. 1, pp. 214–224, Jan. 1995. [8] B. Porat and B. Friedlander, “Direction finding algorithms based on high-order statistics,” IEEE Trans. Signal Process., vol. 39, no. 9, pp. 2016–2023, Sep. 1991. [9] J. M. Mendel, “Tutorial on higher-order statistics (spectra) in signal processing and system theory: Theoretical results and some applications,” Proc. IEEE, vol. 79, no. 3, pp. 278–305, Mar. 1991. [10] J.-F. Cardoso and A. Souloumiac, “Blind beamforming for non-Gaussian signals,” Proc. Inst. Elect. Eng. F, vol. 140, no. 6, pp. 362–370, Dec. 1993. [11] A. Belouchrani, M. G. Amin, and K. Abed-Meraim, “Direction finding in correlated noise fields based on joint block-diagonalization of spatiotemporal correlation matrices,” IEEE Signal Process. Lett., vol. 4, no. 9, pp. 266–268, Sep. 1997.
