# An Improved Method for Decision-Directed Blind Equalization Algorithm

**Topics:**Computational complexity theory, Blind equalization, Hyperbolic function

**Pages:**16 (1708 words)

**Published:**March 6, 2014

BLIND EQUALIZATION ALGORITHM

Seongmin Kim, Wangrok Oh and Whanwoo Kim

Department of Electronics Engineering

Chungnam National University, Daejeon, Republic of Korea

wwkim@cnu.ac.kr

ABSTRACT

s ( n)

u ( n)

ˆ(

s n)

y ( n)

The decision-directed blind equalization algorithm is often

used due to its simplicity and good convergence property

when the eye pattern is open. However, in a channel

where the eye pattern is closed, the decision-directed

algorithm is not guaranteed to converge. Hence, a

modified decision-directed algorithm using a hyperbolic

tangent function for zero-memory nonlinear function has

been proposed and applied to avoid this problem by Filho

et al. But application of this algorithm includes the

calculation of hyperbolic tangent function and its

derivative or a look-up table which may need a large

amount of memory due to channel variations. To reduce

the computational and/or hardware complexity of Filho’s

algorithm, in this paper, an improved method for the

decision-directed algorithm is proposed. It is shown that

the proposed scheme, when it is combined with decisiondirected algorithm, reduces the computational complexity drastically while it retains the convergence and steadystate performance of the Filho’s algorithm.

Figure 1. Block diagram of a receiver system using

Bussgang blind equalization algorithm.

Index Terms— Blind equalization, decision-directed,

zero-memory nonlinear function.

where W ( n ) = [ w 0 ( n ) … w L −1 ( n ) ] is the tap weight

1. INTRODUCTION

e( n)

η [ ⋅ ]

Bussgang algorithm can be described by the

following equations:

W ( n + 1) = W (n) + μ ( n)e( n) U ( n)

(1)

e(n) = η ⎡ y ( n )⎤ − y ( n )

⎣

⎦

(2)

T

y (n) = U (n) W(n)

(3)

T

vector of the equalizer at time n , (here, w i ( n) is the i -th equalizer coefficient), L is the equalizer length, μ (n) is the step size; U ( n ) = [u ( n ) u ( n − 1) … u ( n − L + 1) ] is the equalizer input vector of L sample measurements at the

channel output, y ( n ) is the equalizer output, and η [⋅] is a T

Blind equalization algorithms are typically classified into

two categories, depending on whether the adaptation is

performed by a linear or a nonlinear filter. Bussgang

algorithms exploit in the implicit sense the higher order

cumulants that are associated with nonlinear filter [1].

This paper deals with a Bussgang algorithm and its

modified versions that are adequate for real time

processing and simple realization. Bussgang algorithm

can be obtained with the minimization of a nonlinear cost

function which may be achieved by a LMS (least mean

square) algorithm, as in [1], by means of a stochastic

gradient steepest descent algorithm, as in [2] and [3], or

any suitable known optimization method. Especially,

Bussgang-type algorithm with a derivative of zeromemory nonlinear (ZNL) function using a hyperbolic tangent function is described in [2]. We consider this

algorithm to improve the computational complexity in this

paper. The block diagram of a receiver system using

Bussgang blind equalization algorithm is illustrated in

Figure 1.

zero-memory nonlinear (ZNL) function.

Each Bussgang blind equalization algorithm is

determined by choosing the corresponding ZNL function

η [⋅] , which is used to generate the error signal e(n) . When this nonlinearity is chosen as the decision device

D(⋅), the decision-directed (DD) algorithm is obtained.

The DD algorithm is often used in association with

another blind equalization algorithm, due to its good

convergence property and its simplicity when the eye

pattern is open [2]. However, in a channel where the eye

pattern is closed, the DD algorithm is not guaranteed to

converge. Hence, a modified decision-directed (MDD)

algorithm using a hyperbolic tangent function for the ZNL

function has been proposed and applied to avoid this

problem by Filho et al [2]. But...

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