This paper discusses the optimization of an adaptive equalization system using the steepest gradient method. It presents the steepest gradient algorithm for minimizing a cost function with respect to adjustable filter parameters. Simulation results show the actual and estimated weights, true and estimated output signals, and estimation error over samples converging as the algorithm runs. The steepest gradient method provides an effective approach for removing limitations in the system and achieving weight equalization.

1. International Journal of Research in Advent Technology, Vol.2, No.6, June 2014
E-ISSN: 2321-9637
230
Optimization of a System with Weight Equalization
Using Steepest Gradient Method
Priya1, Yogesh Juneja2
Electronics and communication1, 2, PDM college of Engg1, 2
Email: priyavashist13@gmail.com1, yogeshjunejaer@gmail.com2
Abstract— This paper is concerned with Performance of Adaptive Equalizer Using Steepest Gradient
Algorithm. In this paper, the performance of adaptive equalizer with weight equalization using steepest
Gradient algorithm is evaluated. The method of steepest descent (Weight Equalization Using Steepest Gradient
Method) is a celebrated optimization procedure for minimizing the value of a cost function J(n) with respect to
a set of adjustable parameters W(n). The actual and estimated weight using steepest gradient method, depicts the
True and estimated output signal and demonstrates error estimation at every sample.
Index Terms: RLS, LMS, Optimization, Adaptive Algorithm.
1. INTRODUCTION
There are two main types of digital filtering: the
Finite Impulse Response (FIR) and the Infinite
Impulse Response (IIR). IIR can normally achieve
similar performance as FIR, with smaller amount of
coefficients and less computation. However, as the
complexity of the filter grows, the order of the IIR
filter increases a lot and the computational advantages
is less dominant. Also, IIR suffers from the instability
problem.
Adaptive Algorithm
The adaptive algorithm is a linear adaptive filter
algorithm, which in general consists of two basic
processes.
1. A filter process, which involves:
a. Computing the output of a linear
filter in response to an input signal.
b. Generating an estimation error by
comparing this output with a desired
response.
2. An adaptive process, which involves the
automatic adjustment of the parameters of
the filter in accordance with the estimation
error.
Factor affecting the performance of adaptive
Filters
In adaptive filtering a number of decisions have to be
made concerning the filter model and the adaptation
algorithm [1]. Factor affecting the performance of
adaptive Filter are
Filter type: This can be a finite impulse
response (FIR) filter, or an infinite impulse
response (IIR) filter. In this chapter we only
consider FIR filters, since they have good
stability and convergence properties and for
these reasons are the type often used in
practice.
Filter order: Often the correct number of
filter taps is unknown. The filter order is
either set using a priori knowledge of the
input and the desired signals, or it may be
obtained by monitoring the changes in the
error signal as a function of the increasing
filter order.
Adaptation algorithm: The two commonly
used adaptation algorithms are the recursive
least square (RLS) error and the least mean
square error (LMS) methods. The factors that
influence the choice of the adaptation
algorithm are the computational complexity,
the speed of convergence to optimal
operating conditions, the minimum error at
convergence, the numerical stability and the
robustness of the algorithm to initial
parameter states [2-3].
Optimisation criteria: In this section two
optimality criteria are used. One is based on
the minimization of the mean of squared
error (used in LMS, RLS) and the other is
based on constrained minimisation of the
norm of the incremental change in the filter
coefficients which results in normalised
LMS (NLMS) [4].
2. Adaptation Method
Fig. 1 shows the block diagram for the adaptive filter
method utilized in this work . Here ‘w’ represents the
coefficients of the FIR filter tap weight vector, ‘ x(n)’
is the input vector samples, ‘z-1’ is a delay of one
sample period, ‘y(n)’ is the adaptive filter output,

2. International Journal of Research in Advent Technology, Vol.2, No.6, June 2014
E-ISSN: 2321-9637
231
‘d(n)’ is the desired echoed signal and ‘e(n)’ is the
estimation error at time n.
The aim of an adaptive filter is to calculate the
difference between the desired signal and the adaptive
filter output, ‘e(n)’. This error signal is fed back into
the adaptive filter and its coefficients are changed
algorithmically in order to minimise a function of this
difference, known as the cost function. When the
adaptive filter output is equal to desired signal, the
error signal goes to zero [5].
The method used can be divided into two groups
based on their cost functions. First is known as Mean
Square Error (MSE) adaptive filters, they aim to
minimise a cost function equal to the expectation of
the square of the difference between the desired signal
d(n) and the actual output of the adaptive filter y(n).
ξ(n) = E[e2(n)] = E[(d(n) − y(n))2]
… (1)
Fig. 1 Block Diagram for the adaptive algorithm
Second is known as Recursive Least Squares (RLS)
adaptive filters and they aim to minimise a cost
function equal to the weighted sum of the squares of
the difference between the desired and the actual
output of the adaptive filter for different time
instances. The cost function is recursive in the sense
that unlike the MSE cost function, weighted previous
values of the estimation error are also considered. The
cost function is shown below in equation (2), the
parameter λ is in the range of 0λ1. It is known as
the forgetting factor as for λ1 it causes the previous
values to have an increasingly negligible effect on
updating of the filter tap weights. The value of 1/(1-
λ) is a measure of the memory of the algorithm, i.e. λ
=1. The cost function for RLS algorithm, ζ(n), is
stated in equation.
ζ (n)
= λn-k
… (2)
Where k=1, 2, 3,…, n, k=1 corresponds to the time at
which the RLS algorithm commences. Later it will be
seen that in practice not all previous values are
considered; rather only the previous N (corresponding
to the filter order) error signals are considered [6].
3. SIMULATION SET UP AND
RESULTS
Weight Equalization using steepest gradient
method
Fig. 2 shows the actual and estimated weight using
steepest gradient method. Fig. 3 depicts the True and
estimated output signal using steepest gradient
method. Fig. 4 demonstrates error estimation at every
sample using steepest gradient method.
Fig. 2 Actual and estimated weight

3. International Journal of Research in Advent Technology, Vol.2, No.6, June 2014
E-ISSN: 2321-9637
232
Fig. 3 True and estimated output signal
Fig. 4 Error estimation at every sample
4. CONCLUSION
Here, we focus on the performance we evaluate the
performance adaptive equalizer using steepest
gradient algorithm is presented. The weighted cost
function based adaptive algorithm for equalization. Is
used. Other adaptive algorithm such as RLS, NLMS,
BLMS can also be used for the channel equalization.
Equalization techniques compensate for the time
dispersion introduced by communication channels and
combat the resulting inter-symbol interference (ISI)
effect. The weigth equalization steepest descent
method provide the better approach to remove the
such limitations of the system.
REFERENCE
[1]. S. Qureshi, Adaptive equalization, IEEE
Communications Magazine, vol. 73, no. 9, pp.
1349-1387, Sept 1985.
[2]. S. Haykin, Adaptive Filter Theory, Prentice-
Hall, 3rd Ed., 1996.
[3]. Thomas Drumright, Adaptive Filtering, Spring
1998.
[4]. Weerackody, V. ; Kassam, S.A. ; Laker, K.R.,
“Convergence analysis of an algorithm for blind
equalization”, IEEE Transactions on
Communications, Vol. 39 , Issue: 6, Pag: 856 –
865, 1991.
[5]. Mathew, G. ; Farhang-Boroujeny, B. ; Wood,
R.W., Design of multilevel decision feedback
equalizers”, IEEE Transactions on Magnetics,
Vol. 33, Issue: 6, Pag: 4528 – 4542, 1997.
[6]. Bakhtiar Qutub Ali,” A New Blind Equalization
Scheme based on Principle of Minimal
Disturbance”, king fahd university of petroleum
and minerals, May 2004.