An Adaptive Linear Equalizer With Optimum Filter Length and ...

An Adaptive Linear Equalizer With Optimum Filter Length
and Decision Delay
Yu Gong
School of Systems Engineering
University of Reading,
Reading RG6 6AY, UK
Colin F. N. Cowan and Jian Chen
The ECIT Institute
Queen’s University of Belfast
Belfast BT3 9DT, UK.
Abstract
The power of an adaptive equalizer is maximized when the structural parameters including
the tap-length and decision delay can be optimally chosen. Although the method for adjusting
either the tap-length or decision delay has been proposed, adjusting both simultaneously
becomes much more involved as they interact with each other. In this paper, this problem is
solved by putting a linear prewhitener before the equalizer, with which the equivalent channel
becomes maximum-phase. This implies that the optimum decision delay can be simply fixed
at the tap-length minus one, while the tap-length can then be chosen using a similar approach
as that proposed in our previous work.
1 Introduction
In a classic design of the linear adaptive equalizer, the structure parameters, including the filter
length and decision delay, are usually fixed to some compromise values, implying that often the
equalizer is too long and sometimes inadequate for the severity of conditions, and furthermore, the
resource may not be best utilized unless the fixed decision delay happens to be (which rarely does)
the optimum value that minimizes the MMSE.
In our previous work, we have proposed algorithms to adapt either the filter length [GC05b]
or the decision delay [GC05a] by fixing one of them. While the length adaptation was successfully
implemented, however, the proposed delay adaptation suffers from slow convergence under
some scenarios. Up to now, the objective of the structure adaptation is only partially achieved as
eventually the length and delay need to be adjusted simultaneously. In a recent paper [GC06], we
proposed a new method to optimally chose the filter length and decision delay simultaneously for
the decision feedback equalizer (DFE). Ironically, although the linear equalizer appears to have a
simpler structure than the DFE, it bears more difficult to adjust the tap-length and decision delay.
Thus the approaches developed for the DFE cannot be directly applied.
In this paper, we solve this problem by adding a linear pre-whitener prior to the equalizer, with
which the channel becomes maximum-phase. We can then circumvent the difficulty of the delay
adaptation by fixing the delay at the filter length minus one and only apply the length adaptation
to the equalizer. Although this brings up a new problem of choosing an appropriate length for the
pre-whitener itself, it can be relatively easily tackled. Assuming a moving average model of the
channel, a feedback pre-whitener needs only to be as long as the channel memory, while the channel
memory can be obtained by traditional order selection criteria (e.g. [RR97, Aka74]). Although

adding a pre-whitener slightly increases the complexity, the cost is justified since a pre-whitener is
often used to improve the convergence of the equalizer as well.
2 Linear Prewhitener
Assuming the channel is causal with finite length of N h , we obtain the channel output as:
y(n) = Hs(n) + n(n), (1)
where H is the channel matrix, s(n) is the transmission vector and n(n) is the noise vector. Without
losing generality, we assume E[s 2 (n)] = 1.
It is known [Qur85] that the optimum MMSE linear equalizer is given by
W (z) =
H ∗ (1/z ∗ )
H(z)H ∗ (1/z ∗ ) + σ 2 , (2)
where H(z) is the channel transfer function and σ 2 is the noise power. The denominator of (2) can
be decomposed as
H(z)H ∗ (1/z ∗ ) + σ 2 = F (z)F ∗ (1/z ∗ ), (3)
where F (z) and F (1/z ∗ ) contain only the zero inside and outside the unit circle respectively. Then
(2) can be written as:
W (z) = A(z) · B(z), (4)
where
A(z) = 1
F (z) , B(z) = H∗ (1/z ∗ )
F ∗ (1/z ∗ )
Eq. (4) - (5) implies that when a linear prewhitener, A(z), is placed before the equalizer B(z),
the equivalent channel becomes “maximum-phase”, i.e. all of the channel zeros are outside the
unit circle. Thus the optimum decision-delay is simply N − 1, where N is the tap-length. It is
clear from (5) that the linear prewhitener can be either a feedback filter with length N h or an
infinitely long feedforward filter. Although the feedforward filter is more stable, from the structure
adaptation point of view, it is preferable to use the feedback filter since its length is just N h which
can be obtained by, for example, channel order estimation [Aka74]. Since the prewhitener is in
fact a self-adapted linear predictor with no training symbols required, its length can also be on-line
adapted using the approach proposed in [GC05b]. From this point of view, both the feedback and
feedforward prewhitener can be applied. To focus on the main point, in this paper, we only consider
the feedback prewhitener and assume the channel order is a priori known. We note that the idea
of putting a prewhitener before the equalizer was also reported in [LML98] but not for structural
adaptation, which is the main goal of this paper.
3 Variable Length Equalizer
In this section, we describe how the tap-length of the equalizer after the prewhitener is chosen.
Theoretically a maximum-phase channel requires an infinitely long transversal equalizer. Too long
a filter, however, not only complicates the design with little performance benefit, but also introduces
more adaptation noise. To make it worse, since the time lag is fixed at N − 1, a very long taplength
also means a very long time lag which then renders the equalizer impractical, if not useless.
From the performance point of view, we need to chose an appropriate filter length to balance the
performance and the complexity.
First, we define the optimum tap-length. Recall that the equivalent channel is maximum phase,
so the optimum delay ∆ is fixed at N − 1. This implies that when the filter length is very large, the
(5)

first several coefficients of the filter are very small. Thus the difference between the final MMSE
and the so-called “segment MMSE” can be used as a measure to indicate whether the equalizer is
long enough or not. The “segment MMSE” is defined as
ξ {N−K} E|e{N(n)−K} 2 (n)| (6)
where e {N−K} (n) s(n − ∆) − w T K+1:N · y K+1:N(n), K is a positive integer much smaller than N,
and w K+1:N and y K+1:N (n) consist of the last N − K coefficients of the equalizer w and inputvector
y(n) respectively. K is set to avoid local minima corresponding to zero channel taps. Then
similar to that in [GC05b], the optimum tap-length, N ′ opt, can be defined as the minimum N that
satisfies:
ξ {N−K} − ξ {N} < E ′ , (7)
where E ′ is a smaller positive constant.
As in [GC05b], we define n f (n) as the pseudo-fractional tap-length which can take fractional
values. Then the length adaptation rule becomes:
[
]
n f (n + 1) = (n f (n) − α) − γ · e{N(n)} 2 (n) − e {N(n)−K} 2 (n) , (8)
where both α and γ are small positive numbers. Specifically, α is an additive leaky factor which is
used to prevent the length adapting to unnecessarily large values, and γ is the step-size parameter for
the above adaption rule. To ensure stability, we must have α ≪ γ. Initially we have n f (0) = N(0).
The “true” tap-length N(n) is determined according to:
{
⌊nf (n)⌋, |N(n) − n
N(n + 1) =
f (n)| M
N(n), otherwise
(9)
where ⌊.⌋ rounds the embraced value to the nearest integer. If N(n) is to be increased, then M
number of zeros are padded at the head of the equalizer. Otherwise if N is to be decreased, the
first M coefficients of the equalizer are taken out. It can verify that, if α/γ = E ′ , (8) converges to
within a range of (N
f, ′ opt − M, N f, ′ opt
+ M) in the mean.
4 Simulations
In the simulation below, we assume the channel vector is given by h = [0.1 0.3 1 0.3 0.1] T , the
SNR= 20dB, and a feedback prewhitener with length of 5 is placed before the equalizer. Fig. 1
plots the MMSE vs. decision-delay for different tap-lengths. For clarification, only the curves for
tap-length at 3, 6, 9 and 11 are shown. Two observations can be made: First, for every tap-length,
the optimum decision-delay is N − 1, which verifies our previous analysis that a prewhitener makes
the channel maximum-phase; Second, as long as the tap-length is equal to or larger than 6, the
minimum MMSE can always be reached with ∆ = N − 1. Thus it is not necessary to have any
linear equalizer longer than 6 in this example.
Fig. 2 plots the length learning curves of the linear equalizer, where it is clearly shown that the
tap-length adapts to around the optimum value, i.e. 6, regardless whether the initial tap-length is
3 or 20.
References
[Aka74] H. Akaike. A new look at the statistical model identification. IEEE Trans Automat.
Contr, AC-19:716 – 723, Dec. 1974.

10 0 N=11
N=3
N=6
N=9
MSE
10 −1
10 −2
0 2 4 6 8 10 12 14 16 18
Decision Delay ∆
Figure 1: The MMSE performance of the linear equalizer with a feedback prewhitener.
20
18
N(0) = 3
N(0) = 20
16
14
Tap length N
12
10
8
6
4
2
0 500 1000 1500 2000 2500 3000
Number of symbols
Figure 2: The learning curve of the length adaptation.
[GC05a] Y. Gong and C. F. N. Cowan. Equalization with adaptive time lag. IEE Proceedings -
Communications, 152(5):661 – 667, Oct. 2005.
[GC05b] Y. Gong and C. F. N. Cowan. An LMS style variable tap-length algorithm for structure
adaptation. IEEE Trans. on Signal Processing, 53(7):2400 – 2407, July 2005.
[GC06]
Y. Gong and C. F. N. Cowan. A self-constructed adaptive decision feedback equalizer.
IEEE Signal Processing Letters, 13(3):169 – 172, March 2006.
[LML98] J. Labat, O. Macchi, and C. Laot. Adaptive decision feedback equalization: can you skip
the training period. IEEE Trans. on Communications, 46(7):921 – 930, July 1998.
[Qur85] S. U. H. Qureshi. Adaptive equalization. Proceedings of the IEEE, 73(9):1349 – 1387,
Sept. 1985.
[RR97]
I. A. Rezek and S. J. Roberts. Parametric model order estimation: a brief review. In
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Analysis of Biomedical Signals, pages 3/1–3/6, London, UK, April 1997.