IMPROVING THE PERFORMANCE OF EQUALIZATION

AND A FAST START-UP TECHNIQUE FOR

COMMUNICATION SYSTEMS

Hua Ye, Wanlei Zhou

School of Information Technology

Deakin University

221 Burwood HWY, Burwood.

VIC. 3125. Australia.

Keywords: least-mean-square (LMS), discrete cosine transform ( DCT ), recursive least squares ( RLS ), filtered-X

LMS, H

2

.

Abstract: This paper explores the potential of several popular equalization techniques and proposes new

approaches to overcome their disadvantages. Such as the conventional least-mean-square (LMS )

algorithm, the recursive least-squares ( RLS ) algorithm, the filtered-X LMS algorithm and their

development. An H

2

optimal initialization has been proposed to overcome the slow convergence

problem while keeping the simplicity algorithms. The effectiveness of the methods proposed in

this paper has been verified.

1 INTRODUCTION

The least-mean-square (LMS) based adaptive

algorithm have been successfully applied in

many communication equalization practices. The

importance of the LMS algorithm is largely due

to two unique attributes(R.D.Gitlin et al., 1992):

• Simplicity of implementation

• Model-independent and therefore robust

performance

The main limitation of the LMS algorithm is its

relatively slow rate of convergence. Two

principal factors affect the convergence

behaviour of the LMS algorithm: the step-size

parameter

µ

, and the eigenvalues of the

correlation matrix R of the tap-input vector.

The recursive least-square (RLS) algorithm is

derived as a natural extension of the method of

least square algorithm. The derivation was based

on a lemma in matrix algebra known as the

matrix inversion lemma. (E.A. Lee et al., 1994). (S.

Haykin, 1996).

The fundamental difference between the RLS

algorithm and the LMS algorithm can be stated

as follows: The step-size parameter

µ

in the

LMS algorithm is replaced in RLS algorithm by

)(

1

n

−

φ

,

that is, the inverse of the correlation

matrix of the input vector U(n). This

modification has a profound impact on the

convergence behavior of the RLS algorithm in a

stationary environment, as summarized here

(David S. Bayard, 1997)-(Steven L. Gay, 1993):

1. The rate of convergence of the RLS

algorithm is typically an order of magnitude

faster than that of the LMS algorithm.

2. The rate of convergence of the RLS algorithm

is invariant to the eigenvalue spread ( i.e.,

condition number ) of the ensemble-averaged

correlation matrix R of the input vector U(n).

1. The excess mean-squared error

)(xJ

ex

of the RLS algorithm converges to zero as the

number of iterations, n, approaches infinity.

The computational load of the conventional RLS

algorithm is prohibited in real time applications.

The recursive least-squares ( RLS ) algorithm is

characterized by a fast rate of convergence that

is relatively insensitive to the eigenvalue spread

129

Hua Y. and Zhou W. (2004).

IMPROVING THE PERFORMANCE OF EQUALIZATION AND A FAST START-UP TECHNIQUE FOR COMMUNICATION SYSTEMS.

In Proceedings of the First International Conference on E-Business and Telecommunication Networks, pages 129-135

DOI: 10.5220/0001382101290135

Copyright

c

SciTePress

of the underlying correlation matrix of the input

data, and a negligible misadjustment, although

its computational complexity is increased(J.M.

Cioffi , 1984)-(BOUCHARD M., 2000).

Though the LMS algorithm does not actually

converge to the least--mean--square solution

that optimal

2

H model matching solution

achieves, they are very close if the adaptive

step size

µ

is small enough. Interestingly,

not much effort is needed to find the filter

^

P

(z) as the filtered-X LMS algorithm still

converges so long as the estimate of the

channel P (z) has less than

0

90± phase

shift and unlimited amplitude distortion. The

∞

H robust performance analysis of the LMS

algorithm conducted by Hassibi, et al. reveals

that sum of the squared errors is always upper

bounded by the combined effects of the initial

weight uncertainty (

^

W (0) --

0

W ) and

the noise

υ

( i ). This evidence strongly

supports that the

H

2

optimal initialisation

presented in this thesis can confine the error to

a low level right from the beginning and hence

improve the convergence rate dramatically.

A big benefit of this approach is that it

makes the adaptive process a virtual fine-tuning

process if a reasonable initialization is

obtained, which avoids experiencing a

possibly long adaptation process in transit to

the fine-tuning period. The advantage will

be more clearly illustrated by a high

eigenvalue spread case. Extensive simulation

experiment has shown that, in many cases,

the adaptive process starts from an

acceptable performance, and it does not need

any remedy like Discrete Cosine Transform

( DCT ) or Discrete Fourier Transform ( DFT )

even in the case with a very high input signal

eigenvalue spread where the conventional

LMS algorithm may fail and traditionally a

remedy like DCT and DFT technique is

required.

The conventional filtered-X LMS is modified

and introduced for the purpose of equalization.

Generic integration of the filtered-X structure,

LMS algorithm, RLS algorithm and

optimal

H

2

initialization is conducted to

meet all paramount criteria of simplicity,

robust and fast convergence for equalization of

high-speed, distorted communication channels.

Finally, various techniques proposed in this

thesis are tested using a popular communication

channel example, under both slight non-

stationary and sever non-stationary conditions.

Comparisons are made with other

conventional methods. Significant performance

improvement has been observed by Mont Carlo.

The effectiveness of the methods proposed in

this thesis has been verified.

2 EVALUATION

We present the experiment results of three adaptive

equalization algorithms:

least-mean-square (LMS) algorithm, discrete

cosine transform-least mean square ( DCT-LMS )

algorithm, and recursive least square ( RLS )

algorithm. Based on the experiments, we obtained

that the convergence rate of LMS is slow; the

convergence rate of RLS is great faster while the

computational price is expensive; the performance

of that two parameters of DCT-LMS are between

the previous two algorithms, but still not good

enough. Therefore we will propose an algorithm

based on H

2

in a coming paper to solve the

problems.

It is well known that high data rate

transmission through dispersive communication

channels is limited by the inter-symbol

interference (ISI). Equalization is an effective

way to reduce the effects of ISI by cancelling

the channel distortion. However, dynamic,

random and time-varying characteristics of

communication channels make this task very

challenging. High speed of data transmission

demands a low computational burden. Hence,

simplicity and robust performance play a

crucial role in equalizer design. Due to its

good robust performance and computational

simplicity, least-mean-square (LMS) based

algorithms have received a wide attention and

been adopted in most applications (E.A. Lee et

al., 1994), but one major disadvantage of the

LMS algorithm is its very slow convergence

rate, especially in high condition number case .

To solve this problem, a variety of improved

algorithm have been proposed in the literature.

Although their actual implementations and

properties may be different but the underlying

principle remains the same: trying to

orthogonalize as much as possible the input

ICETE 2004 - GLOBAL COMMUNICATION INFORMATION SYSTEMS AND SERVICES

130

autocorrelation matrix and to follow a steepest-

descent path on the transformed error function.

Therefore, we extend the least square algorithm

to a recursive algorithm for the design of

adaptive transversal filter. An important feature

of the RLS algorithm is that it utilizes

information contained in the input data,

extending back to the instant of time when

the algorithm is initiated. The resulting rate of

convergence is therefore typically an order of

magnitude faster than the simple LMS algorithm.

This improvement in performance, however, is

achieved at the expense of a large increasing in

computational complexity.

The RLS algorithm implements recursively an

exact least squares solution (Steven L. Gay,

1993). At each time, RLS estimates the

autocorrelation matrix of the inputs and cross

correlation between inputs and desired outputs

based on all past data, and updates the weight

vector using the so-called matrix inversion

lemma. The DFT/LMS and DCT/LMS

algorithms are composed of three simple stages

(Steven L. Gay, 1993). First, the tap-delayed

inputs are preprocessed by a discrete Fourier or

cosine transform. The transformed signals are

then normalized by the square root of their

power. The resulting equal power signals are

input to an adaptive linear combiner whose

weights are adjusted using the LMS algorithm.

With these two algorithms, the orthogonalizing

step is data independent; only the power

normalization step is data dependent. Because

of the simplicity of their components, these

algorithms retain the robustness and

computational low cost while improving its

convergence speed.

Although the structure of the filtered-X LMS

adaptive equalization scheme is a little bit

different from that of the basic LMS adaptive

equalization scheme, the control adjustment

process is the same: adjusting the FIR model

of the equalizer to minimize the least mean

square error

k

e .

3 A FAST START-UP

TECHNIQUE

Though the LMS algorithm does not actually

converge to the least--mean--square solution

that optimal

2

H model matching solution

achieves, they are very close if the adaptive

step size

µ

is small enough. Interestingly, not

much effort is needed to find the filter

^

P (z)

as the filtered-X LMS algorithm still

converges so long as the estimate of the

channel P (z) has less than

0

90± phase

shift and unlimited amplitude distortion. The

∞

H robust performance analysis of the LMS

algorithm conducted by Hassibi, et al. reveals

that sum of the squared errors is always upper

bounded by the combined effects of the initial

weight uncertainty (

^

W (0) --

0

W ) and

the noise

υ

( i ). This evidence strongly

supports that the

H

2

optimal initialisation

presented in this thesis can confine the error to

a low level right from the beginning and hence

improve the convergence rate dramatically.

A big benefit of this approach is that it

makes the adaptive process a virtual fine-tuning

process if a reasonable initialization is

obtained, which avoids experiencing a

possibly long adaptation process in transit to

the fine-tuning period. The advantage will

be more clearly illustrated by a high

eigenvalue spread case. Extensive simulation

experiment has shown that, in many cases,

the adaptive process starts from an

acceptable performance, and it does not need

any remedy like Discrete Cosine Transform

( DCT ) or Discrete Fourier Transform ( DFT )

even in the case with a very high input signal

eigenvalue spread where the conventional

LMS algorithm may fail and traditionally a

remedy like DCT and DFT technique is

required.

The conventional filtered-X LMS is modified

and introduced for the purpose of equalization.

Generic integration of the filtered-X structure,

LMS algorithm, RLS algorithm and

optimal

H

2

initialization is conducted to

meet all paramount criteria of simplicity,

robust and fast convergence for equalization of

high-speed, distorted communication channels.

Finally, various techniques proposed in this

thesis are tested using a popular communication

channel example, under both slight non-

stationary and sever non-stationary conditions.

Comparisons are made with other

conventional methods. Significant performance

improvement has been observed by Mont Carlo.

IMPROVING THE PERFORMANCE OF EQUALIZATION AND A FAST START-UP TECHNIQUE FOR

COMMUNICATION SYSTEMS

131

The effectiveness of the methods proposed in

this thesis has been verified.

This experiment has verified a well known fact

that the conventional adaptive LMS algorithm

can track slight non-stationary environments

such as slowly varying parameters. Now, a

more severe non-stationary situation is tested by

abruptly increasing the channel impulse

response coefficients by 35% of its nominal

value. Fig. 1 shows the simulation result.

The conventional adaptive LMS algorithm

begins to diverge while the filtered-X LMS

algorithm with or without optimal initialization

still maintains a good robust performance. The

conventional RLS algorithm still has an

acceptable performance, which matches the

observation that when the time variation of the

channel is not small, the RLS algorithm will

have a tracking advantage over the LMS

algorithm . The filtered-X RLS algorithm has a

better robust performance. The robust

performance enhancement by the introduction of

the filtered-X structure is obvious and

significant.

From a computational point of view, optimal

initialization needs an additional effort to solve

an

2

H optimal model matching or

2

H

filtering problem. Since this procedure is a

non-iterative solution and can be done off-line,

it does not increase the computational burden

in online operation. The only extra online

computational burden concerned comes from the

extra filter that is involved in every adaptive

step. However, that structure increases only a

computation of one simple algebraic

convolution. This poses no serious problem in

computation at all.

Figure 1: Learning curves of the various adaptive

algorithms experiencing a abrupt increase of impulse

response of the channel by 35%

4 INTEGRATION METHOD

Therefore, the

2

H optimal solution is actually

the limit of the best solution for the filtered-X

LMS adaptive equalization algorithm. Similar to

the case of the basic LMS adaptive

equalization scheme, the filtered-X LMS

adaptive equalization scheme can not, in

general, achieve this limit. However, its optimal

solution is still expected to be close to that

point if the adaptive size is small. Therefore,

the optimal

2

H initialization method proposed

still applies here. Why not simply use the

2

H

optimal model matching filter as the final

equalizer? This is because of the presence of

the model uncertainty and other unexpected

disturbances. The optimal solution obtained in

offline computation may not be optimal when

the filter is implemented in the real world

system because of model uncertainty and other

unexpected disturbance. For the optimal

initialization, a poorly identified system model

may give rise to a low quality model matching

solution.

However, due to robustness of filtered-X LMS

adaptive equalization scheme, this solution may

still be well within the convergence region. By

extensive simulations and experiments, it is

observed that method proposed here can also

cope with wide eigenvalue spread of the input

without having to use Discrete Cosine

Transformation (DCT) that was conventionally

required. This is an advantage in real-time

operation environment where computation

burden is a critical factor.

The focus of this work is on improving the

equalization performance of the powerful LMS

and RLS adaptive algorithms while minimizing

the increase of the related computational

complexity. Since these algorithms are very

popular in real world applications, the

attemption is significant.

Channel equalization is an effective signal

processing technique that compensates for

channel-induced signal impairment and the

resultant inter-symbol interference (ISI) in

communications system. Many sophisticated

techniques have been proposed for equalization,

most of successful real world applications are

still dominated by techniques that are related to

several popular algorithms, such as the adaptive

LMS algorithm, the filtered-X LMS algorithm

ICETE 2004 - GLOBAL COMMUNICATION INFORMATION SYSTEMS AND SERVICES

132

and the RLS algorithm. For high-speed

commercial communication systems, simplicity,

robust and fast convergence rate are critical

criteria for the design of a good equalizer. The

adaptive LMS algorithm, the filtered-X LMS

algorithm, and the RLS algorithm meet some

of these criteria. Unfortunately, none of them,

alone, satisfies all these criteria. Therefore,

research on exploring the potential of these

techniques while overcoming their disadvantages

is important and necessary, which is exactly

what has been conducted in this paper.

(1). The practical importance of LMS algorithm

is largely due to simplicity of implementation

and its robust performance and its main

limitation is relatively slow rate of convergence.

The RLS algorithm is characterized by a fast

rate of convergence that is relatively insensitive

to the eigenvalue spread of the underlying

correlation matrix of the input data, and a

negligible misadjustment. Although it is

computational complexity.

(2). The conventional filtered-X LMS is

modified and introduced for the purpose of

equalization. The famous filtered-X LMS

algorithm has found very successful applications

in the field of active noise and vibration

control. It has inherited the elegant simplicity

of the conventional LMS algorithm, and is very

robust. For approach in analyzing the

performance of the filtered-X LMS algorithm, a

heuristic method based on linear time-invariant

operator theory has been provided to analyze

the robust performance of the filtered-X

structure. It indicates that the extra filter could

enhance the stability margin of the

corresponding non filtered-X structure. In this

thesis, a generic integration of the filtered-X

structure, LMS algorithm, RLS algorithm and

optimal

2

H

initialization has been conducted

to meet all paramount criteria of simplicity,

robust and fast convergence for equalization of

high-speed communication channels.

(3). To overcome the slow convergence

problem while keeping the simplicity of the

LMS based algorithms, an

2

H optimal

initialization is proposed. Though the LMS

algorithm does not actually converge to the

least-mean-square solution that optimal

2

H

model matching solution achieves, they are

very close if the adaptive step size

µ

is small

enough. Interestingly, not much effort is needed

to find the filter

(

)

zP

ˆ

as the filtered-X LMS

algorithm still converges so long as the

estimate of the channel P(z) has less than

0

90± phase shift and unlimited amplitude

distortion [21]. The

∞

H

robust performance

analysis of the LMS algorithm conducted by

Hassibi, et al. reveals that the sum of the

squared errors is always upper bounded by the

combined effects of the initial weight

uncertainty

(

)

(

)

0

0

ˆ

WW −

and the noise v(i).

This evidence strongly supports that the

2

H

optimal initialization presented in this thesis can

confine the error to a low level right from the

beginning and hence improved the convergence

rate dramatically.

A big benefit of this approach is that it makes

the adaptive process a virtual fine-tuning

process if a reasonable initialization is obtained,

which avoids experiencing a possibly long

adaptation process in transit to fine-tuning

period. The advantage will be more clearly

illustrated by a high eigenvalue spread case. As

it is well known that the conventional LMS

converges very slowly or even fails to

converge with a no matter how small adaptive

step size due to high input signal eigenvalue

spread.

2

H optimal model matching solution is

independent of this input signal eigenvalue

spread, and hence could avoid this trouble.

Moreover, this idea can be combined with

other speed-up techniques such as Discrete

Cosine Transform (DCT) and Discrete Fourier

Transform (DFT) as well as various adaptive

algorithms. Extensive simulation experiment has

shown that, in many cases, the adaptive process

starts from an acceptable performance, elated

ven in the case with a very high input signal

eigenvalue spread.

Another approach proposed here is that it

generally does not require detailed knowledge

of the external signal which is a great

advantage in practice. Since there exist many

powerful tools solving

2

H

filtering problem,

including explicit solution, the method proposed

in this thesis is very promising.

(4). A popular communication channel example

is used to test the proposed techniques, under

both slight non-stationary and severe non-

stationary conditions. The level of channel

distortion is deliberately raised to a level that

IMPROVING THE PERFORMANCE OF EQUALIZATION AND A FAST START-UP TECHNIQUE FOR

COMMUNICATION SYSTEMS

133

is much higher than any published result with

system condition number as high as nearly 390.

Furthermore, it is assumed that each tap weight

of the channel undergoes an independent

stationary stochastic process with each

parameter fluctuating around its nominal value

with a uniform probability distribution over the

interval

()

kkkk

hhhh %8,%8 +− , in

addition to the white noise disturbance of

variance 0.001 at the channel output. Mont

Carlo simulation experiment of 1000

independent trials is conducted to obtain an

ensemble-averaged learning curve. All adaptive

algorithms have shown a good robust

performance against the time varying, random

Gaussian impulse response coefficient

fluctuations specified above. The filtered-X

LMS with the optimal

2

H initialization has

been shown to have the fastest convergence

rate and best performance.

(5). A more severe non-stationary situation was

tested by abruptly increasing the channel

impulse response coefficient by 35% of its

nominal value. The conventional adaptive LMS

algorithm begins to diverge while the filtered-X

LMS algorithm with or without optimal

initialization still maintains a good robust

performance. The conventional RLS algorithm

has an acceptable performance, which matches

the observation that when the time variation of

the channel is not small, the RLS algorithm

will have a tracking advantage over the LMS

algorithm (R.D.Gitlin et al., 1992). The filtered-X

RLS algorithm has a better robust performance.

The performance improvement by using the

proposed techniques is significant and hence,

the effectiveness of the new method has been

verified.

5 CONCLUSIONS

The contributions of this paper are: we are compared

the LMS with DCT-LMS and RLS for adaptive

equalizer first, then we will be conducted on how to

speed up the convergence rate of LMS based

algorithm while keeping the increased in-line

computational burden as low as passible, we will

overcome the slow convergence problem while

keeping the simplicity of the LMS based algorithm,

and the H

2

Optimal initialization has been applied in

adaptive equalizer for communication systems.

There still exists many open problems. For

instance, the analysis of the stability margin of

the filtered-X LMS was conducted in a heuristic

manner. Can we extend this to a general case

such as a discrete time MIMO case? What about

the filtered-X RLS algorithm? Can we apply the

ideas to other adaptive equalization techniques

such as decision-feedback equalization, etc.? What

happens if we use the

∞

H optimal

initialization instead of the

2

H optimal

initialization? Another very active area of

equalization is wireless communication where the

phenomenon of fast multiple-path fading

(Rayleigh fading) is very challenging. As

indicated in the simulation, rapid and not so

small channel variations can cause the

conventional LMS algorithm to diverge. It will

be interesting and challenging, therefore, to

apply the new techniques presented here to those

areas in the future.

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COMMUNICATION SYSTEMS

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