In: ICONIP'96 Proceedings, pp.317-322
An On-line Learning Algorithm for Blind Equalization y
z
Howard Hua Yang , Eng Siong Chng
y
Lab. for Inf. Rep., FRP, RIKEN
2-1 Hirosawa, Wako-Shi, Saitama 351-01, JAPAN e-mail:
[email protected]
z
Inst. of Systems Science
National Univ. of S'pore SINGAPORE 119597 e-mail:
[email protected]
Abstract| An on-line algorithm for blind equalization of an FIR channel is proposed by minimizing the mutual information of the output. The algorithm is closely related to the blind separation algorithm based on independent component analysis. It is assumed that the channel impulse responses are unknown and the channel may be non-minimum phase. The algorithm is implemented on a linear neural network in which the weight matrix is updated by the proposed algorithm. The simulation results are used to demonstrate the e ectiveness of the algorithm. 1 Introduction Blind system identi cation and equalization are important in the areas such as data transmission, seismic deconvolution, and image de-blurring[6]. Some new developments in these areas are reviewed in [7] and [8] where two new methods for blind system identi cation and equalization are proposed. These methods are batch type algorithms and require matrix decompositions in the implementations. In addition, the system identi cation and equalization processes are two pipeline operations in these algorithms. The equalization can only be started after the channel is identi ed. Although the blind source separation seems to be an area di erent from the blind identi cation and equalization, in fact they are closely related. Recently, several blind separation algorithms have been proposed and analyzed [3, 4, 2, 1]. These algorithms can be applied for blind equalization by formulating the equalization problem appropriately. The application of blind separation algorithms leads to some on-line algorithms which carries the equalization directly without the channel identi cation process.
2 Channel Model and Problem Consider the following time invariant channel de ned in [7]: x(t) =
X
kT t
s(k)h(t 0 kT ) + n(t)
(1)
where fs(k)g is an input sequence, T the symbol interval, n(1) the additive noise, and h(1) the channel impulse response function. The problem of the blind identi cation and equalization is to identify the impulse response function and recover the input sequence. The model (1) becomes the following multi-channel FIR system when the sampling rate is M times faster than the baud rate: xm (k) =
L X l=0
hm (l)s(k 0 l) + nm (k); k = 1; 1 1 1 ; N;
(2)
m = 1; 11 1 ; M;
where xm (k ) is the output of the m-th channel, hm (l) the impulse response of the m-th channel, s(k) the common input to M channels, L the maximum order of these M channels, N the data length, and nm (k) a complex zero mean Gaussian noise.
A blind identi cation algorithm is given in [7] based on the second order statistics of the oversampled observations. The singular value decomposition (SVD) of the estimated data covariance matrix is used in this algorithm. Another blind identi cation algorithm is proposed in [8] by solving a linear equation which is derived by exploiting every channel output pair. The latter algorithm generally performs better than the former one and imposes a much weaker condition on the input sequence. Generally, both algorithms can achieve equalization within few hundreds baud intervals. However, both algorithms are batch type and require extensive matrix computations. The SVD is needed in [7] to decompose the data covariance matrix and a large matrix whose size depends on the data size is constructed to form the linear equations in [8]. Some diculties may arise in the implementation of the algorithms due to these matrix computations. In this paper, a new on-line algorithm is proposed for the blind equalization problem based on an algorithm for blind source separation. The new algorithm only requires matrix summations/multiplications to update the weight matrix for a linear neural network.
3 Blind Equalization Algorithm Based on ICA The independent component analysis (ICA) formulated in [5] is a general framework for developing blind separation algorithms. A linear mixture model is
x(t) = H s(t)
where H 2 Rn2n is an unknown non-singular mixing matrix, n the number of sources, s(t) = [s1 (t); 1 1 1 ; sn (t)]T the vector of sources and x(t) = [x1 (t); 1 1 1 ; xn (t)]T the vector of mixtures. To recover the original sources from the mixtures, we use the linear transform y(t) = W x(t). The following learning algorithm (LA) is used to update the weight matrix W :
W
I 0 f (y)yT gW
d = (t)f dt
(3)
where f (y ) = (f (y1 ); 1 1 1 ; f (yn ))T . This algorithm is derived in [2] by minimizing the mutual information of the outputs using natural gradient descent algorithm. There are many extensions[1] of this algorithm and several choices for the function f (1) as well. In this paper, we choose the simplest function f (y) = y3 among all options. To apply the above algorithm to the blind equalization problem, we write the system (2) in a vector form:
xk = H 1sk + nk
(4)
where xk = (x1 (k); 1 1 1 ; xM (k))T , sk = (s(k 0 L); s(k 0 L + 1); 1 1 1 ; s(k ))T , nk = (n1 (k ); 11 1 ; nM (k))T , H 1 = [ hL hL01 1 1 1 h0 ], hl = (h1(l); 1 1 1 ; hM (l))T , l = 0; 1; 1 1 1 ; L: De ne an m 2 (m + L) block matrix H m : 2
hL hL01 1 1 1 11 1 h0 0 hL hL01 1 1 1 1 1 1
H m = 664 1 1 1 1 1 1
11 1
1 11
0
11 1 0 h0 1 1 1 0 1 1 1 1 11 1 1 1 1 1 1 1 1 1 hL hL01 1 1 1 1 1 1 h0
0
0
3 7 7 5
and each block in H m is an M 2 1 vector. Stacking m observation vectors x1 ; 1 1 1 ; xm , we form a joint observation vector. From (4), we have 2
u1 = 64
x1
2
3
6 .. 7 . 5 = Hm 4
xm
s(1 0 L)
.. .
s(m)
2
3
7 6 5+4
n1
3
.. 7 . 5
(5)
nm
The next joint observation vector u2 is formed by removing x1 in u1 and appending xm to u1 . Generally, we stack m observation vectors xk ; 1 1 1 ; xm+k01 and obtain the k-th joint observation vector uk 2
uk = 64
xk .. .
xm+k01
3
7 5=
2
H m 64
s(k 0 L)
3
2
7 6 .. 5+4 . s(m + k 0 1)
nk .. .
nm+k01
3
7 5:
(6)
Note H m is of mM 2 (m + L). In this paper, we consider the blind equalization of the system (2) only for M = 2. For the system (1), we only consider the case in which the sampling rate is two times faster than the baud rate. Given M = 2, if we choose m = L, then H m becomes a square matrix. We treat the system (6) as a mixture model and apply the algorithm (3) to update the weight matrix W and recover the source vector by transforming the joint observation vector in the system (6). Since the elements of H m are unknown, we can not compute its inverse exactly. However, we can use the 1 blind separation algorithm such as (3) to obtain a scaled and permuted inverse DPH 0 m where D is a diagonal matrix with non-zero diagonal elements and P is a permutation matrix. With this scaled and permuted inverse matrix, we achieve the blind separation. We call this algorithm blind equalization by blind separation (BEBS). Denote
sk = (s(k 0 L); 1 1 1 ; s(L + k 0 1))T : The structure of the equalization system is illustrated in Figure 1.
- W 0 000 noise
sk
? - 6
- HL
- sck LA
Figure 1: A block diagram of the equalization system when m = L
4 Simulation Let T be the baud interval. Assume that the duration of an FIR channel is 12T and the sampling rate is twice faster than the baud rate. The fractional samples of the FIR channel output is the output of a multi-channel FIR system with the following 2-D vector impulse response:
h=
h1 where h2 h1 = [00:0013 0 0:0837 0:0138 0:3132 0:5167 0:3874 0:0843
and h2 = [0:0368
0 0:0703 0 0:0196 0:0414 0:0098 0 0:0279]
0 0:0514 0 0:0675 0:1520 0:4494 0:4931 0:2351 0 0:0241 0 0:0604 0:0216 0:0343 0 0:0156]. 0.6
0.5
0.5
0.4
0.4 0.3
h2
h1
0.3 0.2
0.2 0.1 0.1
0
0
−0.1 0
2
4
(a)
6
8
10
12
−0.1 0
2
4
6
8
10
12
(b)
Figure 2: The impulse responses of the two FIR channels with the common input. The impulse responses of the two channels are plotted in Figure 2. The same set of tap values are also used in [7] with a di erent setting. We test the BEBS algorithm to equalize the above channel. The
source symbols are drawn from an iid binary sequence. We choose M = 2 and m = L in the BEBS. Assume that the noise in the observation is a Gaussian noise with zero mean and variance 2 . The simulation result in Figure 3 is used as a benchmark for comparison. It is obtained by using the algorithm in [8] to estimate the channel responses rst, then use the following equation to estimate the source vector in (6) for equalization: c m )01 uk (^s(k 0 L); 1 1 1 ; s^(m + k 0 1))T = (H
c m is de ned from the estimated channel responses in the same way as H m is de ned. We call where H this algorithm a least-squares blind equalization (LSBE). The simulation results obtained by the LSBE and BEBS algorithms are shown in Figure 3-4 respectively. In each test, 2000 source symbols are used to equalize the channel. After the equalization, 20000 source symbols are transmitted to estimate the bit-error-rate for each algorithm. When = 0:0005, the biterror-rate for the BEBS algorithm is 0.025% while bit-error-rate for the LSBE is 29%. Therefore, the BEBS algorithm performs better than the LSBE algorithm when sampling rate is two times faster than the baud rate. 3
2
estimated source
1
0
−1
−2
−3 0
500
1000 1500 k (source index)
2000
2500
Figure 3: The equalization achieved by the LSBE algorithm when M = 2. 2
1.5
estimated source
1
0.5
0
−0.5
−1
−1.5
−2 0
200
400
600
800 1000 1200 k (source index)
1400
1600
1800
2000
Figure 4: The equalization achieved by the blind separation algorithm when M = 2.
5 Conclusion The on-line blind separation algorithm is applied for the equalization of an FIR channel with fractional sampling or a multi-channel system driven by a single input. When the sampling rate is two times the
baud rate, the blind separation algorithm achieves the equalization with much better quality than the least-squares blind equalization algorithm.
References [1] S. Amari, A. Cichocki, and H. H. Yang. Recurrent neural networks for blind separation of sources. In Proceedings 1995 International Symposium on Nonlinear Theory and Applications, volume I, pages 37{42, December 1995. [2] S. Amari, A. Cichocki, and H. H. Yang. A new learning algorithm for blind signal separation. In Advances in Neural Information Processing Systems, 8, eds. David S. Touretzky, Michael C. Mozer and Michael E. Hasselmo, MIT Press: Cambridge, MA. (to appear), 1996.
[3] A. J. Bell and T. J. Sejnowski. An information-maximisation approach to blind separation and blind deconvolution. Neural Computation, 7:1129{1159, 1995. [4] J.-F. Cardoso and Beate Laheld. Equivariant adaptive source separation. on Signal Processing, 1996. [5] P. Comon. Independent component analysis, a new concept? [6] S. Haykin.
Blind Deconvolution.
To appear in IEEE Trans.
Signal Processing,
36:287{314, 1994.
Prentice-Hall, Inc., 1994.
[7] L. Tong, G. Xu, and T. Kailath. Blind Identi cation and Equalization Based on Second-Order Statistics: A Time Domain Approach . IEEE Trans. on Information Theory, 40(2):340{349, March 1994. [8] G. Xu, H. Liu, L. Tong, and T. Kailath. A least-squares approach to blind channel identi cation. IEEE Trans. on Signal Processing, 43(12):2982{2993, December 1995.
