Robust Blind Beamforming Algorithm Using Joint ... - Semantic Scholar
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IEEE SENSORS JOURNAL, VOL. 7, NO. 1, JANUARY 2007
Robust Blind Beamforming Algorithm Using Joint Multiple Matrix Diagonalization Xiaozhou Huang, Hsiao-Chun Wu, Senior Member, IEEE, and Jose C. Principe, Fellow, IEEE
Abstract—The objective of the blind beamforming is to restore the unknown source signals simply based on the observations, without a priori knowledge of the source signals and the mixing matrix. In this paper, we propose a new joint multiple matrix diagonalization (JMMD) algorithm for the robust blind beamforming. This new JMMD algorithm is based on the iterative eigen decomposition of the fourth-order cumulant matrices. Therefore, it can avoid the problems of the stability and the misadjustment, which arise from the conventional steepest-descent approaches for the constant-modulus or cumulant optimization. Our Monte Carlo simulations show that our proposed algorithm significantly outperforms the ubiquitous joint approximate diagonalization of eigen-matrices algorithm, relying on the Givens rotations for the phase-shift keying source signals in terms of signal-to-interference-and-noise ratio for a wide variety of signal-to-noise ratios. Index Terms—Blind beamforming, cumulants, givens rotation, higher order statistics (HOS), joint approximate diagonalization of eigen-matrices (JADE), joint diagonalization, singular value decomposition (SVD).
I. INTRODUCTION EAMFORMING is the scheme for the reconstruction of the source signals from the acquired data of a sensor or antenna array. The traditional beamforming techniques such as the minimum variance distortionless response (MVDR) approach [1] and the linear constrained minimum variance (LCMV) method [1] are based on the a priori knowledge of the directional vector associated with the desired source signal and , thus, may be quite sensitive to the perturbation of this vector. The perturbation of the vector direction results from the unknown deformation of the antenna or sensor array, the drifting effect in the electronics or the multipath propagation. Consequently, those factors strictly limit the performance of MVDR and LCMV beamformers in practice.
B
Manuscript received March 24, 2006; revised May 26, 2006; accepted June 14, 2006. This work was supported in part by the Information Technology Research Award for National Priorities from the National Science Foundation (NSF-ECS 0426644), in part by the Research Initiation Grant from the Southeastern Center for Electrical Engineering Education, in part by the Research Enhancement Award from the Louisiana-NASA Space Consortium, in part by the NSF-Louisiana EPSCOR Pilot Fund, and in part by the Faculty Research Grant from the Louisiana State University. The associate editor coordinating the review of this paper and approving it for publication was Prof. Ralph Etienne-Cummings. X. Huang and H.-C. Wu are with the Communications and Signal Processing Laboratory, Department of Electrical and Computer Engineering, Louisiana State University, Baton Rouge, LA 70803 USA (e-mail: [email protected]; [email protected]). J. C. Principe is with the Computational Neuro-Engineering Laboratory, Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/JSEN.2006.886881
On the other hand, blind beamforming methods try to recover the source signals without this a priori information and rely only on the received signal waveforms. The pioneering blind beamforming techniques were based on the direction finding, for example, the scheme illustrated in [2]. The direction of each incoming signal is estimated; at the same time, a beamformer is constructed to recover one source signal from that direction [2]. This kind of technique require that the antenna array is sophisticatedly calibrated. If multiple sources and/or different delay paths are considered, the direction finding techniques are too complicated for any practical solution. Thus, the applicability of these techniques depends on the channel conditions. More recently, a group of striking blind beamforming techniques, the constant modulus algorithms (CMA), were introduced [3], [4]. The CMAs can separate the source signals using the fact that the source signals possess a constant amplitude, such as the phase-modulated signals in digital communications. A major CMA advantage is that these CMA based beamformers depend on neither channel properties nor array calibration. For the man-made signals, such as the signals encountered in digital communications, the corresponding properties are usually well known. These properties can lead to the robust CMAs. However, sometimes the explicit source properties are not known at the receiver. Hence, new blind beamforming schemes are of interest [5]–[9] to recover the source signals in the absence of the explicit source information. The second-order statistics approaches [6], [10], [11] can be applied to explore the temporal characteristics of the covariance matrices, such as nonstationary temporal covariance matrices or time-lagged cross-correlation matrices for blind beamforming. These second-order statistics approaches are established under the assumption of Gaussian noise, for instance, the AMUSE algorithm [12] and the SOBI algorithm [13]. The second-order statistics algorithms perform well only when the power spectral densities (PSDs) of the sources are different; otherwise, the higher order statistics (HOS) methods are required [9], [14]–[17], [22], [23]. In the relaxation of the assumption for Gaussian noise and different source PSDs required by the second-order statistics approaches, in this paper, we adopt the HOS or the cumulants for blind beamforming. It has been proved that the normalized cumulant optimization will lead to the unique solution of source recovery [18]. Therefore, the cumulants can be chosen as the appropriated objective functions for blind beamforming, whose primary goal is to separate the individual source signals using the antenna array [19]. Here, we propose a novel iterative blind beamforming method based on the diagonalization of the cumulant matrices. The main advantages of our new algorithm are that, 1) like all other HOS methods, it can lead to the robust performance for the sources of identical PSD, and 2) our algorithm
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significantly outperforms the ubiquitous joint approximate diagonalization of the eigen-matrices (JADE) algorithm in the high signal-to-noise ratio conditions. This paper is orginazed as follows. Section II formulates the mathematical model for blind beamforming and provides the definition for the performance measure, signal-to-interferenceand-noise ratio (SINR). Section III introduces a couple of classical methods involving the fourth-order moments [16] and the JADE algorithm [20], which will be used to benchmark our proposed algorithm. Our algorithm, joint multiple matrix diagonalization (JMMD) scheme will be described in Section IV. Monta Carlo simulation results are presented in Section V. Section VI concludes this paper. Notations: and denote a column vector and a matrix, redenotes the entry of a matrix ; , spectively; and denote the transpose, complex conjugate, and Hermitian adjoint operators; denotes the estimate of a parameter ; denotes the statistical expectation operator; represents the diagonal matrix which has the same diagonal elements as a square matrix ; the Frobenius norm of a matrix is ex, according to [21]. pressed as II. BLIND BEAMFORMING PROBLEM DESCRIPTION We assume that there are sensors (antennae) and unknown sources. The source signals are assumed to be statistically independent and have zero means. The sensors (antennae) receive the linear combination of the source signals in the presence of additive white Gaussian noise (AWGN). Therefore, the received signal vector is given by
For the future evaluation of the blind beamforming schemes, here, we define a measure quantifying the closeness of a matrix to a permutaion matrix . Such a measure, SINR is defined as
(4) If a matrix is a permutation matrix , the defined SINR in (4) is infinity. The larger the SINR, the closer a matrix to a permutation matrix. Therefore, we could use the defined SINR to meansure the closeness between a matrix to a permuation matrix [perfect reconstruction in (3)]. III. CONVENTIONAL HIGHER ORDER STATISTICS BLIND BEAMFORMING METHODS Blind beamforming methods using the HOS have been proposed by several researchers. Among them, two important tech[16] and the niques named the fourth-moment method JADE algorithm [20] are presented for comparison hereafter. As previously mentioned, in order to achieve the blind beamforming, the source signals are assumed to be statistically independent and zero mean. However, the independence assumption may not be valid for the received signals in the mixture. To reconstruct the independence of the sources, the first step in the blind beamforming is whitening. The whitening matrix is ob, which is the tained by invoking the eigen decomposition of , defined as correlation matrix of the received signal
(1) is
(5)
transis mitted signals, the AWGN vector, and is a constant matrix with . In this paper, the matrix is an unkown the size of deterministic matrix, which is called the mixing matrix. The AWGN vector is assumed to be zero-mean and statistically independent of the source signals. Consequently, the blind beamforming problem here becomes how to restore the unknown, statistically independent random sources from the available observations which are the linear combinations of these sources signals. In this paper, we always assume that the number of received signals is greater than or equal to the . If the noise is number of source signals, i.e., neglected, our goal is to build a blind beamformer of the size , such that
denote the largest eigenvalues of , and denote the corresponding eigenvectors. Thus, the whitening matrix is given by
where the vector consisting of
received signals, is the vector of
(2) or (3) where
is an arbitrary
permutation matrix, and
estimate of the source signal vector
, i.e.,
is the .
Let
(6) and the whitened signal
is obtained as (7)
According to (5)–(7), the correlation matrix is are uncorrelated with an identity matrix or the entries in is just the rotated each other. The whitened signal vector [18], [22], [23]. Therefore, vector of the source vector our goal is to determine or compensate the corresponding rotation variants. We describe how the existing blind beamforming schemes determine such rotation variants in the following. A. Fourth-Order Moment Algorithm A blind beamforming algorithm using the fourth-order moment was proposed in [16]. The fourth-order moment of the
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whitened signal
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is defined as
where
(8)
is the
indicator vector. Thus, we can construct a cumulant matrix as [20]
A singular value decomposition (SVD) is invoked on , i.e., ( is the left singular vector matrix) and the is formulated as blind beamformer .. .
.. .
.. .
(9)
B. Joint Approximate Diagonalization of Eigen-Matrices (JADE) Algorithm The JADE algorithm using the higher order cumulants was proposed in [20]. The fourth-order cumulants of the whitened , are given by [20] signals ,
Then, most significant eigenpairs assoand are the largest ciated with are carried out, where eigenvalues and the corresponding eigenvectors [20]. A unitary matrix is determined to jointly diagonalize the set of matrices . The matrix results from the vector using a classic “stacking–unstacking” device [20]. The ultimate blind beamfomer using the JADE algorithm is formulated as
(14) (10) where is denoted as for notational convenience. Since are independent of each other and have the source signals zero means, the following statistical property exists:
In the JADE algorithm, the joint diagonalization is implemented by extending the single-matrix Givens rotations for several matrices. It consists of the joint diagonalization criterion via successive Givens rotations, which is given by
(11) where is the Kronecker delta function and the kurtosis is defined as
max (15)
(12) and is denoted as for notational convenience. According to (11), the blind beamformer obtained when we maximize the objective function as
is
(13)
and is denoted as for notational where is the instantaneous output vector of the convenience. blind beamformer as defined in (2). The maximization of the objective function as given by (13) is equivalent to the minimization of the sum of the squared cross cumulants with distinct first and second indices [20]. In the JADE algorithm, after the received signals are whitened, the fourth-order cumulants defined by (10) are calculated. Define the cumulant submatrix as
where is the unitary matrix for the optimal JADE beamformer as given by (14) such that . The stop criterion associated with the optimization procedure in (15) has to be predetermined [20]. If the stop criterion or the error tolerance threshold is not properly chosen initially, it would take a long time for the Givens rotation procedures to converge. Therefore, in this paper, we propose a new iterative eigen-decomposition algorithm to carry out the joint diagonalization of the set of in [20], instead of using the Givens matrices rotations. IV. PROPOSED JOINT MULTIPLE MATRIX DIAGONALIZATION (JMMD) ALGORITHM A. Joint Multiple Matrix Diagonalization In this section, we describe a new robust blind beamforming scheme via our proposed joint multiple matrix diagonalization (JMMD) algorithm. The JMMD blind beamformer is constructed as
(16)
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where is the whitening matrix as given by (6) and is the “diagonalizing” matrix, which is not necessarily identical to . as described in SecGiven a set of matrices tion III-B[20], we propose a new cost function as
Equation (19) can be solved using an alternative search procedure, namely the eigen decomposition with a Gibbs sampler, such that (20) where
(17) , is the matrix for the optimal where JMMD beamformer given by (16). When our proposed cost is minimized, the magnitudes of function the off-diagonal elements are minimized aggregately. The is very difficult to get, since close-form solution of is highly nonlinear. According to (17), , or is achieved when is equivalent to the diagonal matrix , such that the off-diagonal elements in are all zeros. It means that
where . Thus, the cost function defined in (17) is equivalent to the following equation:
is the eigenvector associated with the smallest eigenvalue and . According to of is found, the (20), once the optimal diagonalizing matrix blind beamformer can be achieved using (16) where . In summary, our proposed JMMD blind beamforming algorithm can be stated as follows. Step 1) Invoke the eigen decomposition of the received as given by (5). signal’s correlation matrix Step 2) Construct the whitened signal vector, , where the whitening matrix is given by (6). Step 3) Randomly initiate a nonsingular matrix , set iteraand start the optimization procetion number dure for the JMMD blind beamformer. Step 4) Apply the eigen decomposition to search for the column vector , , in the matrix using (19), (20). Step 5) For iteration number , repeat Step 4). B. Computational Complexity Analysis In this section, we provide the computational complexity analysis for the JADE and the JMMD algorithms. Assume that there are sources. The number of complex multiplications involved in the JADE algorithm is while that of our for each iteration. However, the JMMD algorithm is numbers of iterations required for the JADE algorithm and the JMMD algorithm might be very different due to the individual stop criteria. V. SIMULATION
(18) where and , , are the column vectors. According to (17) and (18), the optimality can be related to
(19)
To test our proposed JMMD blind beamforming algorithm, we first consider independently identically distributed source signals consisting of QPSK symbols. The matrix in (1) is specified using a linear half-wavelength equispaced array of unit-gain sensors and the incident waves are assumed to have the plane wave fronts. The angles of arrival are randomly picked; 100 random matrices are generated over 100 Monte Carlo trials, accordingly. For each trial, 200 QPSK source symbols are of JMMD randomly generated and only one iteration algorithm is applied to compare with the JADE algorithm [20] and the beamforming algorithm using the fourth-order moment , respectively, matrices [16], which are denoted as JADE and in the figures. The beamforming performance measurement of SINR is given by (4). The SINR curves versus signal-to-noise and our JMMD algorithms are ratio (SNR) using JADE,
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Fig. 1. SINR comparison versus SNR for two QPSK sources, two antennae (p = q = 2; = 1).
Fig. 3. SINR comparison versus SNR for two QPSK sources, three antennae (p = 3; q = 2; = 1).
Fig. 2.
Fig. 4.
(p =
q
SINR comparison versus SNR for five QPSK sources, five antennae
= 1).
= 5;
depicted in Fig. 1 (two antennae, two sources), Fig. 2 (five antennae, five sources), Fig. 3 (three antennae, two sources), and Fig. 4 (five antennae, three sources). According to Figs. 1–4, our algorithms up to JMMD algorithm outperforms JADE and 15 dB, even though only one iteration of JMMD procedure is applied. Fig. 5 illustrates the beamforming performance variations for among different iteration numbers . According to Fig. 5, our JMMD algorithm is very robust even for only a few iterations. The number of iterations for the JADE algorithm can not be directly controlled since its stop criterion is based on the error tolerance, which depends on the diversified data statistics. For the JMMD algorithm, we may just set a small iteration number as an early stop criterion. Thus, the JMMD algorithm is more flexible to terminate than the JADE algorithm especially when many sources are considered. For the investigation of the high-order-modulation source signals, we carry out the simulations for 16QAM and 64QAM
(p =
q
SINR comparison versus SNR for five QPSK sources, five antennae
= 1).
= 5;
source symbols. Figs. 6 and 7 depict the SINRs versus SNR in , , comparison for the 16QAM source signals with , , respectively. and the 64QAM source signals with According to Figs. 6 and 7, for higher order-modulation source signals, our JMMD method would perform similarly to the JADE algorithm. VI. CONCLUSION In this paper, we formulate a new cost function and propose a novel robust joint multiple matrix diagonalization algorithm for the blind beamforming, which is based on the eigen decomposition instead of the Givens rotations in the exiting JADE technique. Consequently, our new blind beamforming algorithm involves a flexible stop criterion. In addition, it significantly outperforms the JADE algorithm for the QPSK source signals in the high signal-to-noise ratio conditions, while it achieves similar performances to the JADE algorithm for high-order-modulation source signals. Our new JMMD algorithm can be employed at
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REFERENCES
Fig. 5. SINR comparison versus SNR for five QPSK sources, five antennae (p = q = 5).
Fig. 6. SINR comparison versus SNR for three 16QAM sources, five antennae (p = 5; q = 3; = 10).
[1] H. L. Van Trees, Optimum Array Processing. New York: Wiley, 2002. [2] The Signal Processing Handbook. Boca Raton, FL: CRC, 1998. [3] R. Gooch and J. Lundell, “The CM array: An adaptive beamformer for constant modulus signals,” in Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing, April 1986, vol. 11, pp. 2523–2526. [4] A.-J. Van Der Veen and A. Paulraj, “An analytical constant modulus algorithm,” IEEE Trans. Signal Process., vol. 44, no. 5, pp. 1136–1155, May 1996. [5] J.-F. Cardoso and P. Comon, “Independent component analysis, a survey of some algebraic methods,” in Proc. IEEE Int. Symp. Circuits and Systems, May 1996, vol. 2, pp. 93–96. [6] J.-F. Cardoso, “Blind signal separation: Statistical principles,” Proc. IEEE, vol. 86, no. 10, pp. 2009–2025, Oct. 1998. [7] P. Comon, “Independent component analysis, a new concept?,” Signal Process., vol. 36, no. 3, pp. 287–314, Apr. 1994. [8] C. Jutten and J. Herault, “Blind separation of sources, Part 1: An adaptive algorithm based on neuromimetic architecture,” Signal Process., vol. 24, no. 1, pp. 1–10, July 1991. [9] D. T. Pham, “Blind separation of instantaneous mixture of sources via an independent component analysis,” IEEE Trans. Signal Process., vol. 44, no. 11, pp. 2768–2779, Nov. 1996. [10] S. C. Douglas and A. Cichocki, Tech. Rep. 1996. [11] F. M. Silva and L. B. Almeida, “On the stability of symmetric adaptive decorrelation,” in Proc. IEEE Int. Neural Networks, Jun./Jul. 1994, vol. 1, pp. 66–71. [12] L. Tong, V. C. Soon, Y. F. Huang, and R. Liu, “AMUSE: a new blind identification algorithm,” in Proc. IEEE Int. Symp. Circuits and Systems, May 1990, vol. 3, pp. 1784–1787. [13] A. Belouchrani, K. Abed-Meraim, J.-F. Cardoso, and E. Moulines, “A blind source separation technique using second-order statistics,” IEEE Trans. Signal Process., vol. 45, no. 2, pp. 434–444, Feb. 1997. [14] J.-F. Cardoso, “Source separation using higher order moments,” in Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing, May 1989, vol. 4, pp. 2109–2112. [15] D. T. Pham, “Blind separation of instantaneous mixture of sources based on order statistics,” IEEE Trans. Signal Process., vol. 48, no. 2, pp. 363–375, Feb. 2000. [16] L. Tong, R.-W. Liu, V. C. Soon, and Y.-F. Huang, “Indeterminacy and identifiability of blind identification,” IEEE Trans. Circuits Syst., vol. 38, no. 5, pp. 499–509, May 1991. [17] Y. Inouye and K. Hirano, “Cumulant-based blind identification of linear multi-input-multi-output systems driven by colored inputs,” IEEE Trans. Signal Process., vol. 45, no. 6, pp. 1543–1552, Jun. 1997. [18] H.-C. Wu and D. Xu, “Blind equalization of communication sequences based on optimization of cumulant criteria,” in Proc. IEEE Wireless Communications and Networking Conf., Mar. 2003, vol. 1, pp. 618–622. [19] A.-J. Van Der Veen, “Algebraic methods for deterministic blind beamforming,” Proc. IEEE, vol. 86, no. 10, pp. 1987–2008, Oct. 1998. [20] J.-F. Cardoso and A. Souloumiac, “Blind beamforming for non-Gaussian signals,” Proc. Inst. Elect. Eng. F, Radar Signal Process., vol. 140, no. 6, pp. 362–370, Dec. 1993. [21] G. Golub and C. Van Loan, Matrix Computations, 2nd ed. Baltimore, MD: Johns Hopkins Univ. Press, May 1991, John Hopkins Studies in Mathematical Sciences. [22] A. Yeredor, “Non-orthogonal joint diagonalization in the least-squares sense with application in blind source separation,” IEEE Trans. Signal Process., vol. 50, no. 7, pp. 1545–1553, Jul. 2002. [23] ——, “Approximate joint diagonalization using nonorthogonal matrices,” in Proc. IEEE Workshop ICA BSS, Jun. 2000, pp. 33–38.
Fig. 7. SINR comparison versus SNR for three 64QAM sources, five antennae (p = 5; q = 3; = 10).
Xiaozhou Huang received the B.S. and M.S. degrees in wireless communications from Northern Jiaotong University, Beijing, China, in 1993 and 1996, respectively, and the M.S. degree in physics science from the Physics Department and the M.S. degree in digital signal processing from the Department of Electrical and Computer Engineering, University of Massachusetts, North Dartmouth, in 1999 and 2002, respectively. She is currently pursuing the Ph.D. degree in the Department of Electrical and Computer Engineering, Louisiana State University,
the wireless base station which has the powerful computing resource and requires high-performance beamformers.
Baton Rouge. She was with the Beijing University of Post and the TelecommunicationNORTEL Research Laboratory from 1996 to 1998. Her research interests are in the areas of digital communication and communication theory.
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Hsiao-Chun Wu (S’98–M’00–SM’05) received the B.S. degree in electrical engineering from the National Cheng Kung University, Taiwan, R.O.C., in 1990, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Florida, Gainesville, in 1993 and 1999, respectively. From March 1999 to January 2001, he was with the Motorola Personal Communications Sector Research Laboratories as a Senior Electrical Engineer. Since January 2001, he has been with the Faculty of the Department of Electrical and Computer Engineering, Louisiana State University, Baton Rouge. His research interests include optimization, estimation, wireless communications, and signal processing. Dr. Wu currently serves as an Associate Editor of IEEE TRANSACTIONS ON BROADCASTING. He has also served on the technical program committees of numerous IEEE conferences in signal processing, communications, computers, and networking areas.
IEEE SENSORS JOURNAL, VOL. 7, NO. 1, JANUARY 2007
Jose C. Principe (M’83–SM’90–F’00) received the B.S. degree from the University of Porto, Portugal, and the M.S. and Ph.D. degrees from the University of Florida, Gainesville, all in electrical engineering, and the Laurea Honoris Causa degree from the Universita Mediterranea, Reggio Calabria, Italy. He has been a Distinguished Professor of Electrical and Biomedical Engineering at the University of Florida since 2002. He joined the University of Florida in 1987, after an eight-year appointment as Professor at the University of Aveiro, Portugal. He holds five patents and has seven patents pending. He was the supervisory committee Chair of 47 Ph.D. and 61 M.S. students, and he is the author of more than 400 refereed publications (three books, four edited books, 14 book chapters, 116 journal papers, and 276 conference proceedings). His interests lie in nonlinear, non-Gaussian optimal signal processing and modeling and biomedical engineering. In 1991, he created the Computational NeuroEngineering Laboratory to synergistically focus on research in biological information processing models. Dr. Principe is a past President of the International Neural Network Society and Editor-in-Chief of the Transactions of Biomedical Engineering since 2001, as well as a former member of the Advisory Science Board of the FDA.
IEEE SENSORS JOURNAL, VOL. 7, NO. 1, JANUARY 2007
Robust Blind Beamforming Algorithm Using Joint Multiple Matrix Diagonalization Xiaozhou Huang, Hsiao-Chun Wu, Senior Member, IEEE, and Jose C. Principe, Fellow, IEEE
Abstract—The objective of the blind beamforming is to restore the unknown source signals simply based on the observations, without a priori knowledge of the source signals and the mixing matrix. In this paper, we propose a new joint multiple matrix diagonalization (JMMD) algorithm for the robust blind beamforming. This new JMMD algorithm is based on the iterative eigen decomposition of the fourth-order cumulant matrices. Therefore, it can avoid the problems of the stability and the misadjustment, which arise from the conventional steepest-descent approaches for the constant-modulus or cumulant optimization. Our Monte Carlo simulations show that our proposed algorithm significantly outperforms the ubiquitous joint approximate diagonalization of eigen-matrices algorithm, relying on the Givens rotations for the phase-shift keying source signals in terms of signal-to-interference-and-noise ratio for a wide variety of signal-to-noise ratios. Index Terms—Blind beamforming, cumulants, givens rotation, higher order statistics (HOS), joint approximate diagonalization of eigen-matrices (JADE), joint diagonalization, singular value decomposition (SVD).
I. INTRODUCTION EAMFORMING is the scheme for the reconstruction of the source signals from the acquired data of a sensor or antenna array. The traditional beamforming techniques such as the minimum variance distortionless response (MVDR) approach [1] and the linear constrained minimum variance (LCMV) method [1] are based on the a priori knowledge of the directional vector associated with the desired source signal and , thus, may be quite sensitive to the perturbation of this vector. The perturbation of the vector direction results from the unknown deformation of the antenna or sensor array, the drifting effect in the electronics or the multipath propagation. Consequently, those factors strictly limit the performance of MVDR and LCMV beamformers in practice.
B
Manuscript received March 24, 2006; revised May 26, 2006; accepted June 14, 2006. This work was supported in part by the Information Technology Research Award for National Priorities from the National Science Foundation (NSF-ECS 0426644), in part by the Research Initiation Grant from the Southeastern Center for Electrical Engineering Education, in part by the Research Enhancement Award from the Louisiana-NASA Space Consortium, in part by the NSF-Louisiana EPSCOR Pilot Fund, and in part by the Faculty Research Grant from the Louisiana State University. The associate editor coordinating the review of this paper and approving it for publication was Prof. Ralph Etienne-Cummings. X. Huang and H.-C. Wu are with the Communications and Signal Processing Laboratory, Department of Electrical and Computer Engineering, Louisiana State University, Baton Rouge, LA 70803 USA (e-mail: [email protected]; [email protected]). J. C. Principe is with the Computational Neuro-Engineering Laboratory, Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/JSEN.2006.886881
On the other hand, blind beamforming methods try to recover the source signals without this a priori information and rely only on the received signal waveforms. The pioneering blind beamforming techniques were based on the direction finding, for example, the scheme illustrated in [2]. The direction of each incoming signal is estimated; at the same time, a beamformer is constructed to recover one source signal from that direction [2]. This kind of technique require that the antenna array is sophisticatedly calibrated. If multiple sources and/or different delay paths are considered, the direction finding techniques are too complicated for any practical solution. Thus, the applicability of these techniques depends on the channel conditions. More recently, a group of striking blind beamforming techniques, the constant modulus algorithms (CMA), were introduced [3], [4]. The CMAs can separate the source signals using the fact that the source signals possess a constant amplitude, such as the phase-modulated signals in digital communications. A major CMA advantage is that these CMA based beamformers depend on neither channel properties nor array calibration. For the man-made signals, such as the signals encountered in digital communications, the corresponding properties are usually well known. These properties can lead to the robust CMAs. However, sometimes the explicit source properties are not known at the receiver. Hence, new blind beamforming schemes are of interest [5]–[9] to recover the source signals in the absence of the explicit source information. The second-order statistics approaches [6], [10], [11] can be applied to explore the temporal characteristics of the covariance matrices, such as nonstationary temporal covariance matrices or time-lagged cross-correlation matrices for blind beamforming. These second-order statistics approaches are established under the assumption of Gaussian noise, for instance, the AMUSE algorithm [12] and the SOBI algorithm [13]. The second-order statistics algorithms perform well only when the power spectral densities (PSDs) of the sources are different; otherwise, the higher order statistics (HOS) methods are required [9], [14]–[17], [22], [23]. In the relaxation of the assumption for Gaussian noise and different source PSDs required by the second-order statistics approaches, in this paper, we adopt the HOS or the cumulants for blind beamforming. It has been proved that the normalized cumulant optimization will lead to the unique solution of source recovery [18]. Therefore, the cumulants can be chosen as the appropriated objective functions for blind beamforming, whose primary goal is to separate the individual source signals using the antenna array [19]. Here, we propose a novel iterative blind beamforming method based on the diagonalization of the cumulant matrices. The main advantages of our new algorithm are that, 1) like all other HOS methods, it can lead to the robust performance for the sources of identical PSD, and 2) our algorithm
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significantly outperforms the ubiquitous joint approximate diagonalization of the eigen-matrices (JADE) algorithm in the high signal-to-noise ratio conditions. This paper is orginazed as follows. Section II formulates the mathematical model for blind beamforming and provides the definition for the performance measure, signal-to-interferenceand-noise ratio (SINR). Section III introduces a couple of classical methods involving the fourth-order moments [16] and the JADE algorithm [20], which will be used to benchmark our proposed algorithm. Our algorithm, joint multiple matrix diagonalization (JMMD) scheme will be described in Section IV. Monta Carlo simulation results are presented in Section V. Section VI concludes this paper. Notations: and denote a column vector and a matrix, redenotes the entry of a matrix ; , spectively; and denote the transpose, complex conjugate, and Hermitian adjoint operators; denotes the estimate of a parameter ; denotes the statistical expectation operator; represents the diagonal matrix which has the same diagonal elements as a square matrix ; the Frobenius norm of a matrix is ex, according to [21]. pressed as II. BLIND BEAMFORMING PROBLEM DESCRIPTION We assume that there are sensors (antennae) and unknown sources. The source signals are assumed to be statistically independent and have zero means. The sensors (antennae) receive the linear combination of the source signals in the presence of additive white Gaussian noise (AWGN). Therefore, the received signal vector is given by
For the future evaluation of the blind beamforming schemes, here, we define a measure quantifying the closeness of a matrix to a permutaion matrix . Such a measure, SINR is defined as
(4) If a matrix is a permutation matrix , the defined SINR in (4) is infinity. The larger the SINR, the closer a matrix to a permutation matrix. Therefore, we could use the defined SINR to meansure the closeness between a matrix to a permuation matrix [perfect reconstruction in (3)]. III. CONVENTIONAL HIGHER ORDER STATISTICS BLIND BEAMFORMING METHODS Blind beamforming methods using the HOS have been proposed by several researchers. Among them, two important tech[16] and the niques named the fourth-moment method JADE algorithm [20] are presented for comparison hereafter. As previously mentioned, in order to achieve the blind beamforming, the source signals are assumed to be statistically independent and zero mean. However, the independence assumption may not be valid for the received signals in the mixture. To reconstruct the independence of the sources, the first step in the blind beamforming is whitening. The whitening matrix is ob, which is the tained by invoking the eigen decomposition of , defined as correlation matrix of the received signal
(1) is
(5)
transis mitted signals, the AWGN vector, and is a constant matrix with . In this paper, the matrix is an unkown the size of deterministic matrix, which is called the mixing matrix. The AWGN vector is assumed to be zero-mean and statistically independent of the source signals. Consequently, the blind beamforming problem here becomes how to restore the unknown, statistically independent random sources from the available observations which are the linear combinations of these sources signals. In this paper, we always assume that the number of received signals is greater than or equal to the . If the noise is number of source signals, i.e., neglected, our goal is to build a blind beamformer of the size , such that
denote the largest eigenvalues of , and denote the corresponding eigenvectors. Thus, the whitening matrix is given by
where the vector consisting of
received signals, is the vector of
(2) or (3) where
is an arbitrary
permutation matrix, and
estimate of the source signal vector
, i.e.,
is the .
Let
(6) and the whitened signal
is obtained as (7)
According to (5)–(7), the correlation matrix is are uncorrelated with an identity matrix or the entries in is just the rotated each other. The whitened signal vector [18], [22], [23]. Therefore, vector of the source vector our goal is to determine or compensate the corresponding rotation variants. We describe how the existing blind beamforming schemes determine such rotation variants in the following. A. Fourth-Order Moment Algorithm A blind beamforming algorithm using the fourth-order moment was proposed in [16]. The fourth-order moment of the
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whitened signal
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is defined as
where
(8)
is the
indicator vector. Thus, we can construct a cumulant matrix as [20]
A singular value decomposition (SVD) is invoked on , i.e., ( is the left singular vector matrix) and the is formulated as blind beamformer .. .
.. .
.. .
(9)
B. Joint Approximate Diagonalization of Eigen-Matrices (JADE) Algorithm The JADE algorithm using the higher order cumulants was proposed in [20]. The fourth-order cumulants of the whitened , are given by [20] signals ,
Then, most significant eigenpairs assoand are the largest ciated with are carried out, where eigenvalues and the corresponding eigenvectors [20]. A unitary matrix is determined to jointly diagonalize the set of matrices . The matrix results from the vector using a classic “stacking–unstacking” device [20]. The ultimate blind beamfomer using the JADE algorithm is formulated as
(14) (10) where is denoted as for notational convenience. Since are independent of each other and have the source signals zero means, the following statistical property exists:
In the JADE algorithm, the joint diagonalization is implemented by extending the single-matrix Givens rotations for several matrices. It consists of the joint diagonalization criterion via successive Givens rotations, which is given by
(11) where is the Kronecker delta function and the kurtosis is defined as
max (15)
(12) and is denoted as for notational convenience. According to (11), the blind beamformer obtained when we maximize the objective function as
is
(13)
and is denoted as for notational where is the instantaneous output vector of the convenience. blind beamformer as defined in (2). The maximization of the objective function as given by (13) is equivalent to the minimization of the sum of the squared cross cumulants with distinct first and second indices [20]. In the JADE algorithm, after the received signals are whitened, the fourth-order cumulants defined by (10) are calculated. Define the cumulant submatrix as
where is the unitary matrix for the optimal JADE beamformer as given by (14) such that . The stop criterion associated with the optimization procedure in (15) has to be predetermined [20]. If the stop criterion or the error tolerance threshold is not properly chosen initially, it would take a long time for the Givens rotation procedures to converge. Therefore, in this paper, we propose a new iterative eigen-decomposition algorithm to carry out the joint diagonalization of the set of in [20], instead of using the Givens matrices rotations. IV. PROPOSED JOINT MULTIPLE MATRIX DIAGONALIZATION (JMMD) ALGORITHM A. Joint Multiple Matrix Diagonalization In this section, we describe a new robust blind beamforming scheme via our proposed joint multiple matrix diagonalization (JMMD) algorithm. The JMMD blind beamformer is constructed as
(16)
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where is the whitening matrix as given by (6) and is the “diagonalizing” matrix, which is not necessarily identical to . as described in SecGiven a set of matrices tion III-B[20], we propose a new cost function as
Equation (19) can be solved using an alternative search procedure, namely the eigen decomposition with a Gibbs sampler, such that (20) where
(17) , is the matrix for the optimal where JMMD beamformer given by (16). When our proposed cost is minimized, the magnitudes of function the off-diagonal elements are minimized aggregately. The is very difficult to get, since close-form solution of is highly nonlinear. According to (17), , or is achieved when is equivalent to the diagonal matrix , such that the off-diagonal elements in are all zeros. It means that
where . Thus, the cost function defined in (17) is equivalent to the following equation:
is the eigenvector associated with the smallest eigenvalue and . According to of is found, the (20), once the optimal diagonalizing matrix blind beamformer can be achieved using (16) where . In summary, our proposed JMMD blind beamforming algorithm can be stated as follows. Step 1) Invoke the eigen decomposition of the received as given by (5). signal’s correlation matrix Step 2) Construct the whitened signal vector, , where the whitening matrix is given by (6). Step 3) Randomly initiate a nonsingular matrix , set iteraand start the optimization procetion number dure for the JMMD blind beamformer. Step 4) Apply the eigen decomposition to search for the column vector , , in the matrix using (19), (20). Step 5) For iteration number , repeat Step 4). B. Computational Complexity Analysis In this section, we provide the computational complexity analysis for the JADE and the JMMD algorithms. Assume that there are sources. The number of complex multiplications involved in the JADE algorithm is while that of our for each iteration. However, the JMMD algorithm is numbers of iterations required for the JADE algorithm and the JMMD algorithm might be very different due to the individual stop criteria. V. SIMULATION
(18) where and , , are the column vectors. According to (17) and (18), the optimality can be related to
(19)
To test our proposed JMMD blind beamforming algorithm, we first consider independently identically distributed source signals consisting of QPSK symbols. The matrix in (1) is specified using a linear half-wavelength equispaced array of unit-gain sensors and the incident waves are assumed to have the plane wave fronts. The angles of arrival are randomly picked; 100 random matrices are generated over 100 Monte Carlo trials, accordingly. For each trial, 200 QPSK source symbols are of JMMD randomly generated and only one iteration algorithm is applied to compare with the JADE algorithm [20] and the beamforming algorithm using the fourth-order moment , respectively, matrices [16], which are denoted as JADE and in the figures. The beamforming performance measurement of SINR is given by (4). The SINR curves versus signal-to-noise and our JMMD algorithms are ratio (SNR) using JADE,
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Fig. 1. SINR comparison versus SNR for two QPSK sources, two antennae (p = q = 2; = 1).
Fig. 3. SINR comparison versus SNR for two QPSK sources, three antennae (p = 3; q = 2; = 1).
Fig. 2.
Fig. 4.
(p =
q
SINR comparison versus SNR for five QPSK sources, five antennae
= 1).
= 5;
depicted in Fig. 1 (two antennae, two sources), Fig. 2 (five antennae, five sources), Fig. 3 (three antennae, two sources), and Fig. 4 (five antennae, three sources). According to Figs. 1–4, our algorithms up to JMMD algorithm outperforms JADE and 15 dB, even though only one iteration of JMMD procedure is applied. Fig. 5 illustrates the beamforming performance variations for among different iteration numbers . According to Fig. 5, our JMMD algorithm is very robust even for only a few iterations. The number of iterations for the JADE algorithm can not be directly controlled since its stop criterion is based on the error tolerance, which depends on the diversified data statistics. For the JMMD algorithm, we may just set a small iteration number as an early stop criterion. Thus, the JMMD algorithm is more flexible to terminate than the JADE algorithm especially when many sources are considered. For the investigation of the high-order-modulation source signals, we carry out the simulations for 16QAM and 64QAM
(p =
q
SINR comparison versus SNR for five QPSK sources, five antennae
= 1).
= 5;
source symbols. Figs. 6 and 7 depict the SINRs versus SNR in , , comparison for the 16QAM source signals with , , respectively. and the 64QAM source signals with According to Figs. 6 and 7, for higher order-modulation source signals, our JMMD method would perform similarly to the JADE algorithm. VI. CONCLUSION In this paper, we formulate a new cost function and propose a novel robust joint multiple matrix diagonalization algorithm for the blind beamforming, which is based on the eigen decomposition instead of the Givens rotations in the exiting JADE technique. Consequently, our new blind beamforming algorithm involves a flexible stop criterion. In addition, it significantly outperforms the JADE algorithm for the QPSK source signals in the high signal-to-noise ratio conditions, while it achieves similar performances to the JADE algorithm for high-order-modulation source signals. Our new JMMD algorithm can be employed at
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REFERENCES
Fig. 5. SINR comparison versus SNR for five QPSK sources, five antennae (p = q = 5).
Fig. 6. SINR comparison versus SNR for three 16QAM sources, five antennae (p = 5; q = 3; = 10).
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Fig. 7. SINR comparison versus SNR for three 64QAM sources, five antennae (p = 5; q = 3; = 10).
Xiaozhou Huang received the B.S. and M.S. degrees in wireless communications from Northern Jiaotong University, Beijing, China, in 1993 and 1996, respectively, and the M.S. degree in physics science from the Physics Department and the M.S. degree in digital signal processing from the Department of Electrical and Computer Engineering, University of Massachusetts, North Dartmouth, in 1999 and 2002, respectively. She is currently pursuing the Ph.D. degree in the Department of Electrical and Computer Engineering, Louisiana State University,
the wireless base station which has the powerful computing resource and requires high-performance beamformers.
Baton Rouge. She was with the Beijing University of Post and the TelecommunicationNORTEL Research Laboratory from 1996 to 1998. Her research interests are in the areas of digital communication and communication theory.
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Hsiao-Chun Wu (S’98–M’00–SM’05) received the B.S. degree in electrical engineering from the National Cheng Kung University, Taiwan, R.O.C., in 1990, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Florida, Gainesville, in 1993 and 1999, respectively. From March 1999 to January 2001, he was with the Motorola Personal Communications Sector Research Laboratories as a Senior Electrical Engineer. Since January 2001, he has been with the Faculty of the Department of Electrical and Computer Engineering, Louisiana State University, Baton Rouge. His research interests include optimization, estimation, wireless communications, and signal processing. Dr. Wu currently serves as an Associate Editor of IEEE TRANSACTIONS ON BROADCASTING. He has also served on the technical program committees of numerous IEEE conferences in signal processing, communications, computers, and networking areas.
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Jose C. Principe (M’83–SM’90–F’00) received the B.S. degree from the University of Porto, Portugal, and the M.S. and Ph.D. degrees from the University of Florida, Gainesville, all in electrical engineering, and the Laurea Honoris Causa degree from the Universita Mediterranea, Reggio Calabria, Italy. He has been a Distinguished Professor of Electrical and Biomedical Engineering at the University of Florida since 2002. He joined the University of Florida in 1987, after an eight-year appointment as Professor at the University of Aveiro, Portugal. He holds five patents and has seven patents pending. He was the supervisory committee Chair of 47 Ph.D. and 61 M.S. students, and he is the author of more than 400 refereed publications (three books, four edited books, 14 book chapters, 116 journal papers, and 276 conference proceedings). His interests lie in nonlinear, non-Gaussian optimal signal processing and modeling and biomedical engineering. In 1991, he created the Computational NeuroEngineering Laboratory to synergistically focus on research in biological information processing models. Dr. Principe is a past President of the International Neural Network Society and Editor-in-Chief of the Transactions of Biomedical Engineering since 2001, as well as a former member of the Advisory Science Board of the FDA.
