BLIND SOURCE SEPARATION FOR BLAST P.Sansrimaliachai, D. B. Ward and A . G. Constantinides Imperial College of Science, Technology and Medicine, Department of Electrical and Electronic Engineering Exhibition Road, London SW7 ZBT, UK e-mail:
[email protected]
Abstract: We present a new blind source separation (BSS) approach for Bell Labs Layered Space-time (BLAST) communication system. The proposed algorithm is derived from the multimodulus algorithm (MMA) and employs the Gram-Schmidt orthogonalization procedure. The basic idea is t o adjust the real and imaginary parts of the equalizer matrix separately and then project the updated parameters to the orthogonality constraints which ensure the independence among the equalizer outputs. The simulation results show that the proposed algorithm has better performance cornpared to the niultiuser kurtosis algorithm (MUK) with comparable computational complexities. 1. INTRODUCTION
Theoretical and experimental results have shown that the Bell Labs Layered space-time (BLAST) architecture [I],[2], [3], [4], a wireless communications technique, has a high transmission capacity. This technique splits a single data stream into multiple substreams and simultaneously transmits all substreams by using an array of antennas. At the receiver, an array of antennas and the BLAST algorithm are used to extract the transmitted data. The algorithm assumes that the channel characteristic is known to the receiver via a training sequence, but unknown to the transmitter. There are, however: practical situations where training can be expensive or impossible. Blind techniques are therefore considered. Blind source separation (BSS) of instantaneous mixtures is a technique of interest. Some techniques based on higher order statistics [SI, [GI7 [7], [8] have been proposed for BSS. These techniques are constant modulus (CM)-based and kurtois-based BSS approaches. To prevent the same source signal to be extracted at different outputs, the CM-based BSS algorithm proposed in [SI employs the cross-correlation between different outputs. When two outputs extract the same source signal, this term will take a nonzero value which is then added to the update equation to penalize this situation. In the kurtosisbased BSS algorithm proposed in [7]; the Gram-Schmidt orthogonalization approach is used to prevent this situation. In this paper we propose a new BSS technique for the BLAST architecture based on the inultiinodulus algorithm (MMA) [9], [lo]; [ll]and a constraint that is used in [7]. The basic idea of the proposed algorithm is to adjust the real and imaginary parts of the equalizer matrix separately and then project the updated parameters to the orthogonality constraints which ensure the independence among the equalizer outputs. The paper is organised as follows. We briefly re-
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02002 IEEE
view the BLAST system in section 2. In this section, we show the connection between the BLAST system and the problem of BSS. The proposed algorithm which is called multiple-input multiple-output MMA (MIMO MMA) is derived in section 3. Computer simulations are discussed in section 4: and conclusions are given in section 5. 2.
SYSTEM MODEL
The BLAST system is shown in figure 1 where N represents the number of the receive antennas and M represents the number of the transmit antennas. Denoting a(k),n(k) and H the M x 1 transmitted signal vector, the N x 1 noise vector at time instant k and the N x M channel matrix, respectively, the corresponding N x 1 received signal vector is given as
r ( k ) = Ha(k)
+ n(k)
(1)
It is assumed that each component of a(k) has the same statistical properties and is drawn from the same constellation, i.e. Q 4 M [Z]. The total transmitted power is constrained to P regardless of M. The same amount of power is therefore fed to each transmit antenna. The channel matrix H can be modelled by a matrix having independent identically distributed (iid), complex, zero-mean, unit-variance entries. We assume that H is unitary. If it is not unitary, the received signals are then prewhitened.
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Fig. 1. The BLAST System.
The noise at the receiver is assumed to be a complex additive white Gaussian noise. Each component of the noise vector is statistically independent and of identical power a t the antennas output. Equation (1) can be viewed as the problem of BSS whose goal is to recover the transmitted signal a(k) using only the received signal r(k). Basically, BSS involves multiple independent sources of information and multiple receivers, given that the number of sources is equal to the number of receivers. Provided that the source signals are independent and the channel matrix H is nonsingular, it is possible to find an equalizer matrix W which results in y(k) = WTr(k)= G T a ( k )
I‘
JT
;,
, I I
(2)
where y(k) is an M-vector of the output signal; ( . ) T denotes transpose operation and G ( k ) = H*W(k) is the global system matrix. The matrix W is feasible to separate the sources, except for a possible permutation of y and an arbitrary scaling of each source signal. Equation ( 2 ) can be written as y ( k ) = DPa(k)
,
Fig. 2. The meaning of MIMO MMA. By using the stochastic gradient method arid dropping the expectation E{.}: the update equation for the equalizer matrix W is obtained as
W ( k + 1)= W(k) - p r * ( k ) e ( k )
(8)
(3)
where D is a nonsingular diagonal matrix and P is a permutation matrix.
where p is afixed step-size and e ( k ) = e z ( k ) + j e F ( k ) is the error signal vector given as
3. THE MIMO MMA
The proposed algorithm is the extension of the MMA [9], [lo], [ll]to the multiple-input multiple-output ) yr,%(k)the (MIMO) system. Denoting y ~ , + ( kand real and imaginary parts of the zLIL equalizer ouput y l ( k ) , the criterion of the MIMO MMA can be formulated as M JArlA40
A4A/A(k)
= x { J R , ‘ ( k ) + J l , ~ ( k ) } (4) L=l
where J R , , ( ~= ) E { ( I Y R , , ( ~) ~R~,R)’) ’ J I , & (= ~ )E{(lyr,+(k)12-
RZJ)’}
The operation 0 denotes point-by-point multiplication. It can be seen from equations (9) and (10) that a t the perfect equalization point where e R = er = 0, the MIMO MMA attempts to drive the real and imaginary parts of the equalizer output to lie on the points of values The MIMO MMA therefore forces the equalizer output to form a constellation that corresponds to the source constellation with a modulo T f 2 phase rotation. In order to ensure that each equalizer output is obtained from different sources, the criterion for the MIMO MMA is modified as
+a.
(5) (6)
The operation E{.} denotes statistical expecta, tion. For the QAM constellation. the real. R ~ , Rand imaginary, R ~ , Jconstants , are identical. These constants represent real and imaginary parameters for piecewise linear contours which the measure of dispersion of y ( k ) is made around. The constants determined by the real and imaginary parts of the input signal a(k) = aR(k) .lar(k) are defined as
min subject to
(11)
JMrMo M M A ( ~ )
GHG = 1
~
~
~
4
where ( . ) H denotes complex conjugate transpose operation. To satisfy the constraint given in equation (ll), the channel matrix H is assumed to be unitary in order that the constraint can be reduced to
+
+
WH(k 1)W(k
+ 1) = I M X M
(12)
It was suggested that we must choose W defined by Figure 2 illustrates the meaning of the cost function given in (4). In this figure, the piecewise linear contours mentioned previously are represented by the dotted lines which together form a square.
as [71
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w = [WI IV2 . . . WM]
Fig. 3. Mean square error (MSE) comparison between the MIMO MMA and the MUK for 100 simulations (SNR = 15 dB, ~ M U K= ~ M I M OMMA = 0.002, N=M=2).
Fig. 4. Mean square error (MSE) comparison between the MIMO MM,4 and the MUK for 1000 simulations (SNR = 15 dB, ~ M U K= ~ M I M OMMA = 0.002, N = M = 2). denoised received signal covariance matrix as
R,, = HHH = USV,
+
W p ( k 1) =
lQp(k
+ 1) - Q
IlIQp(k + 1)- QII
,p=
(14)
where U and V are unitary matrix, and S is a diagonal matrix with real entries. The matrix US$ equals to H up to a unitary ambiguity matiix. We therefore prewhiten the recived signal as
2, . . . , M
where
i. = ( U S + ) + r For proof and derivation of (13), please refer to [7]. When the matrix H is not unitary, the received signal is prewhitened before performing the algorithm. 4. SIMULATIONS
Computer simulation results are presented in this section. We first assume that N = M = 2. Each transmitted signal is drawn from 4-&AM constellation and undergoes a 2 x 2 unitary channel matrix chosen randomly. The length of the transmitted signal vector is chosen to be 2000. We compare the proposed MIMO MMA and the multiuser kurtosis algorithm (MUK) [7] by running 100 siniulations with WO= 12x2,~ M U K= ~ M I M OM M , ~= 0.002 and signal-to-noise ratio (SNR) = 15 dB. After the algorithms converge, the output signal is processed to remove the phase ambiguities. The results are shown in figure 3. It can be seen that the mean square error (MSE) of the MIMO MMA is less than that of the MUK. We then expect that the MIMO MMA could result in a lower bit error rate (BER). This is shown later in this paper. To realize the BLAST system, a 2 x 2 complex channel matrix is chosen randomly from a complex Gaussian distribution of zero mean and unit variance. Since the channel is not unitary, we prewhiten the received signal by using a simple singular value decomposition (SVD) algoirthm. Denoting RrTthe
(15)
where (.)+ denotes matrix pseudo-inverse. For 1000 simulations, the averaged results are given in figure 4. It can be seen from this figure that the proposed algorithm has a better performance than the MUK in terms of MSE. We observe further in figure 5 that the MIMO MMA also results in a lower BER. We consider the case where N > M , i.e. N = 4. The channel matrix is chosen to be a complex matrix.
t
I
Fig. 5. Bit error rate (BER) comparison between the MIMO MMA and the MUK for 1000 simulations ( ~ M U K= ~ M I M OM M A = 0.002, N = M = 2).
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[3] G. D. Golden, C. J . Foschini, R. 4.Valenzuela, P. W. Wolniansky, “Detection algorithm and initial laboratory results using V-BLAST spacetime communication architecture”, Electronics Letters, vol. 35, no. 1, January 1999. [4] G. J . Foschini, G. D. Golden, R. A. Valenzuela, P.W. Wolniansky, “Simplified Processing for High Spectral Efficiency Wireless Communication Employing Multi-Element Arrays”, IEEE Journal on Selected Areas i n Communications, vol. 17, no. 11, pp. 1841-1852, 1999. 10.2
0
’
2W
’
’
6W
900
BW
1002 ltelallOnS
1200
14W
16W
1800
20.20
[5] S. Haykin, “Unsupervised adaptive filtering, volume 1: blind source separation”, John Wiley & Sons, Inc., 2000.
Fig. 6 . Mean square error (MSE) comparison between the MIMO MMA and the MUK for 1000 simulations (SNR = 15 dB, ~ M U K= ~ M I M OM M A = 0.002, N = 4, M = 2).
[GI Y. Li, K.J. Ray Liu, “Adaptive blind source separation and equalization for multipleinput/multiple-output systems”, IEEE Transaction on Information Theory, vo1.44, pp.28642876, 1998.
The received signal is prewhitened by the matrix that contains the first M columns of US; before being separated. Figure G shows the performance comparison between the MIMO MMA and the MUK when N = 4 and M = 2. The results are again show that MIMO MMA exhibits a better performance than the MUK. When we compare figures 4 and 6 , the benefit of using N > M can be obviously seen. MSE of N = 4 is less than that of N = 2.
[7] C. B. Papadias, “Globally convergent blind source separation based on a multiuser kurtosis maximization criterion”, IEEE Transactions o n Signal Processing, vol. 48, no. 12, pp. 35083519: 2000.
5.
CONCLUSION
In this paper we presented a new BSS for BLAST. The proposed algorithm is based on the MMA and the Gram-Schmidt orthogonalization procedure. This algorithm can be viewed as the generalization of the MMA extended to the MIMO system. We investigated the performance of the MIMO MMA by comparing to that of the MUK. Simulation results have shown that the proposed algorithm has better performance compared to the MUK with comparable computational complexities.
[8] L. Castedo, C. J. Escudero and A. Dapena; “A blind signal separation method for multiuser communications”, IEEE Transactions on Si,ynal Processing, \Jol. 45, No. 5, pp. 1343-1348, 1997. [9] K. N. Oh and Y. 0. Chin, “Modified constant modulus algoirthm: blind equalization and carrier phase recovery algorithm”, Proc. ICC’95, Seattle; WA, pp. 498-502, 1995.
[lo] J .
Yang, J. J. Werner and G. A. Dumont, “The multimodulus blind equalization algorithm”, Proceedings 13th International Conference on Di,yital Signal Processing, Vol. 1, pp. 127-130, 1997.
[ll] J . C. Lin and L. S. Lee, “A modified blind equalization technique based on a constant modulus algorithm”, IEEE International Conference on Communications, vol.1, pp. 344-348, 1998.
REFERENCES G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas”, Bell Labs Technical Journal, Vol. 1, No. 2, pp.41-59, 199G. P. W. Wolniansky, G. J. Foschini, G. D. Golden, R. A. Valenzuela, “V-BLAST: An architecture for realizing very high data rates over the Rich-scattering wireless channel”, invited paper, Proc. ISSSE-98, Pisa, Italy, Sept. 29, 1998.
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