Fifth International Symposium on Signal Processing and its Applications, . ISSPA ‘99, Brisbane, Australia, 22-25 August, 1999 Organised by the Signal Processing Research Centre, QUT, Brisbane, Australia

Robust Adaptive Equalization Using the Filtered-X LMS Algorithm J. Hu and H.R. Wu* *School of Computer Science & Software Eng., Monash University, Clayton 3168, AUSTRALIA, E-mail: [email protected] Abstract k -

The filtered-X LMS algorithm is currently the most popular method for adaptive filter design when there exists a transfer function in the error path. We introduce and modify this concept in the application of adaptive equalization for communication channels. Computer experiment was conducted on the equalization of a time-varying communication channel using both conventional LMS and modified filtered-X LMS algorithms. Mont Carlo simulation results demonstrate that the filtered-X has a better robust performance than that of the conventional LMS.

1 Introduction It is well known that wireless digital communications systems suffer from many channel impairments. To reduce inter-symbol interference (ISI), an equalizer can be provided to reshape each received pulse into something more like its transmitted form [l]. The LMS-based algorithms are very important techniques in adaptive equalization. The power of the LMS algorithm lies in its simplicity. It does not require any knowledge of external signals and only instant data is needed in adaptation [2]. This is very important in high speed transmission systems where timing dominates. In fact, the simplicity of the LMS algorithm has made it the standard against which other adaptive filtering algorithms are benchmarked [2]. In real-world transmission systems, the communication channels are time varying and always fluctuate around their nominal parameters in an unpredictable way. Therefore, the robust performance of the equalizers are essential. The

Figure 1: Block diagram of adaptive equalizer.

filtered-X LMS algorithm has been found to be robust in both against plant variation and noise disturbance [3][4]. It is currently the most popular method for adapting a filter when there exists a transfer function in the error path, especially in the field of active noise and vibration control [5][6]. In this paper, we are going to apply this concept to robust adaptive equalizer design for a communication channel. Under this investigation, each impulse response parameter of the channel undergoes an independent stochastic fluctuation around its nominal value. Mont Carlo simulation has been conducted. The experiment shows that the modified filtered-X LMS algorithm has reduced the mean-squared error by 15% compared with the conventional LMS adaptive equalization method. To the best of our knowledge, no similar attempt using the filtered-X LMS algorithm has been reported in the field of adaptive equalization.

2 Filtered-X LMS Algorithm For comparison, conventional LMS equalization and the filtered-X LMS scheme [3] are depicted in Figs. 1 and 2 respectively. In Fig.2, P(z) is an FIR model of the channel obtained by either

41 9

Fifth International Symposium on Signal Processing and its Applications, ISSPA '99, Brisbane, Australia, 22-25 August, 1999 Organised by the Signal Processing Research Centre, QUT, Brisbane, Australia

Figure 3: Block diagram of a transmission system.

Figure 2: Adaptive inverse model control using the filtered-X LMS algorithm.

on-line or off-line identification. ylk is the additive noise of the plant. rk is the reference command input, and ek is the error between the desired response and the output of the system. The LMS algorithm is used to adjust the controller that is an FIR model in the channel. Two major differences are observed in these two structures. One is that the filtered-X LMS algorithm places the controller in front of the plant instead of after the plant as in the case of the conventional LMS adaptive inverse modelling. The other is that one of the input to the LMS algorithm in the filtered-X LMS algorithm comes from the output of the FIR model of the plant channel instead of the output of the disturbed true plant as in the case of the conventional LMS inverse modelling. These two characteristics have offered the filtered-X LMS algorithm a strong robust capacity and hence become the most popular method for a filter when there exists a transfer function in the error path, especially in the field of active noise and vibration control [5][6].

Figure 4: Block diagram of the modified filtered-X LMS algorithm in adaptive equalization.

However, refemng to Fig.2, a similar problem still exists when trying to apply the filtered-X LMS algorithm to adaptive equalization. The adaptive controller would be physically separated by the transmission channel if the filtered-X LMS algorithm is directly applied in the adaptive equalization. There is no feasible way to feed the error signal at the receiver back to the transmitting end as that is required by the filtered-X LMS algorithm. To overcome this difficulty, we swap back the adaptive controller and the channel plant. Thus we have lost one characteristic of the filtered-X LMS algorithm, but the filter channel p ( z ) still remains. Therefore, a modified filtered-X LMS algorithm is proposed for adaptive equalization as shown in Fig. 4.As explained above, the output y k of the decision block can be used in real systems. Hence, this modified filtered-X LMS algorithm is applicable. In this paper, similar to the case of [2], the true reference input signal is assumed available. Identification is allowed to obtain a impulse response of the channel as a nominal value. A more general case using the output of the decision block is being conducted and will be reported later. Now consider the following communication system [2]. The impulse response of the channel is described by the raised cosine:

3 Application of the Filtered-X LMS Algorithm to Adaptive Equalization A general linear equalizer receiver is shown in Fig.3 [7]. Refemng to the conventional LMS adaptive equalization shown in Fig.1, a natural question arises: How can we, at the receiver end, get the reference input signal rk at the transmitting end that is physically separated by the transmission medium? A common practice is to use the output yk of the decision block instead of the true reference input signal rk [1][7]. Normally, the high accurate rate of the output of the decision block makes no essential difference between using y k and the true reference rk [1][7].

hk =

420

{ 0,$[l+cos(g(k-2))],

k = 1,2,3 otherwise(1)

Fifth International Symposium on Signal Processing and its Applications, ISSPA ‘99, Brisbane, Australia, 22-25 August, 1999 Organised by the Signal Processing Research Centre, QUT, Brisbane, Australia

where the parameter W controls the amount of amplitude distortion produced by the channel, with the distortion increasing with W . Equivalently, the parameter W controls the eigenvalue spread x(R) of the correlation matrix of the tap inputs in the equalizer, with the eigenvalue spread increasing with W. The random input sequence {rk} applied to the channel is in polar form, with rk = f l , so the sequence {rk} has zero mean. The additive output noise sequence nk has zero mean and variance 0: = 0.001. The equalizer has M = 11 taps. Since the channel has an impulse response h k that is symmetric about time k = 2, it follows that the optimum tap weights Wop of the equalizer are likewise symmetric about time n = 5. Accordingly, the channel input {rk} is delayed by 2 5 = 7 samples to provide the desired response for the equalizer. A medium size distortion W = 3.1 is chosen, with an eigenvalue spread x(R) = = 11.1238 of the correlation matrix R of the 11 tap*inputs in the equalizer. For the case where the step-size parameter p is small compared to 2 a condition for convergence of the LMS algorithm in the mean square is given by [2]

tap weight of the channel undergoes an independent stationary stochastic process after the system has reached the steady-state, with each parameter fluctuating around its nominal value with a uniS form probability distribution on the interval (hk 12%hk,hk 12%hk). An approximation to the ensemble-averaged learning curve of the adaptive equalizer is obtained by Mont Carlo simulation experiment, with a fixed signal-to-noise ratio (SNR) of 30dB. lo4 samples are used in each simulation trial. The results of this experiment using the conventional LMS algorithm and the modified filtered-X LMS algorithm are shown in Figs.5-6. The equalization performance has been significantly affected by the random time varying disturbance on the channel parameters starting at the 200th sample. Both algorithms still maintain satisfactory equalization performance during the whole lo4 sample period with the modified-X LMS algorithm performing better. Statistics data are drawn from the Mont Carlo experiment. The modified filtered-X LMS algorithm has reduced the mean squared error by 15% compared with the conventional LMS algorithm. By suddenly increasing the impulse response parameters to 35% of their nominal values, the simulation result is shown in Fig.7. It is observed that the result of the conventional LMS algorithm is unacceptable while the result of the modified filtered-X LMS algorithm is still satisfactory. This experiment has clearly shown that the modified-X LMS algorithm has better robust performance than that of the conventional LMS algorithm.

Therefore, the choice of p = 0.075 assures the convergence of the adaptive equalizer in stationary environment. To mimic a more practical situation, the plant is assumed unknown. A simple identification is conducted to obtain the F!IR filter a(z) for the modified filtered-X LMS algorithm. The true impulse response parameters of the channel and the identified FIR model are listed in Table 1. To observe the robust performance of the systems, perturbation will be added after the system has reached its steady-state, ie., after 200 samples of adaptation. Assume each

4 Conclusion In this paper, the concept of the popular filtered-X LMS algorithm is introduced and modified for the adaptive equalization. Mont Carlo simulation was conducted on a communication channel that was subjected to stochastic time-varying disturbance on its nominal impulse response parameters. The experiment has shown that the modified filteredX LMS has much better robust performance than that of the conventional LMS algorithm. Since in any real transmission systems, the output of the decision block should be used instead of the true reference signal, a wrong decision can increase the error and hence tend to affect the decision

Table 1: Impulse Responses of the True Channel and . the Identified Channel Description

P(z) ..

I P(z) 1

I

I

ho hl 0 I 0.2798 0.001 I 0.277

I1

11

h2 h3 1 0.2798 0.9921 I 0.28 I

+

I

+

w,

421

Fifth International Symposium on Signal Processing and its Applications, ISSPA '99, Brisbane, Australia, 22-25 August, 1999 Organised by the Signal Processing Research Centre, QUT, Brisbane, Australia

and then circulate again to escalate the problem, if the system is not robust enough. The modified filtered-X LMS algorithm has inherited virtually all of the advantages of the conventional LMS algorithm in terms of its implementation simplicity while offers a better robust performance. It has virtually the same amount of computation except the additional computation of a simple convolution. This is a great advantage in real time implementation. How to incorporate decision block and compare with other adaptive equalization methods, etc. are research topics still under investigation.

1.1

e $10''

8 . . .g

.i.

...................

....

10"

Figure 5: Learning curves of the conventional LMS and the modified filtered-X LMS algor i t h m s for adaptive equalization, starting from beginning

References [l] R.D. Gitlin, J.F. Hayes, and S.B. Weinstein, Data communication principles, Plenum Press, New York, 1992.

E

[2] S. Haykin, Adaptive filter theory, Second Edition Edition, Prentice Hall Information and System Sciences Series, 1991.

t

. . . . . . . . . . . . . . . :. . . i

. i . . :

. . . . . . . . . . .j . . . .

1

110''

i 1 E

[3] S. Shaffer, Adaptive inverse-model control, Ph.D. Dissertation, Stanford University, August 1982.

10-

I

NunOvdh-

[4] B. Widrow, and E. Walach, Adaptive Inverse Control,Prentice Hall, 1996.

Figure 6: Learning curves of the conventional LMS and the modified filtered-X LMS algor i t h m s for adaptive equalization, last 500

[5] T. Kosaka, S.J. Elliott, and C.C. Boucher, A novel frequency domain filtered-X LMS algorithm for active noise reduction, IEEE International Conference on Acoustics, Speech, and Signal Processing, 1997, vol.1, pp.403-406.

sample section to-,

,

,

,

,

,

.,

. . . . . . .

.

,

..

..

,

,

.; . . . . . . . .

[6] E.A. Wan, Adjoint LMS: An efficient alternative to the Filtered-X LMS and multiple error LMS algorithms. IEEE International Conference on Acoustics, Speech, and Signal Processing, 1996, ICASSP-96, vo1.3, pp.1842-1845. [7] E.A. Lee and D.G. Messerschmitt, Digital comunicafion, Second Edition, Kluwer Academic Publishers, 1994.

. . . . . . . . . .. .. . . . . . . I

...

....i..'... .

Figure 7: Squared error versus k curves of the conventional LMS and the modified filtered-X LMS algorithms for adaptive equalization

422