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598 Chapter 11 APPLICATIONS IN ADAPTIVE FILTERING
to be in the range
1
0 < △ <
(11.9)
10NP x
where N is the length of the adaptive FIR filter and P x is the power in
the input signal, which can be approximated by
P x ≈ 1
1+M
M∑
n=0
x 2 (n) = r xx (0)
M +1
(11.10)
The mathematical justification of equations (11.9) and (11.10) and
the proof that the LMS algorithm leads to the solution for the optimum
filter coefficients is given in more advanced treatments of adaptive filters.
The interested reader may refer to the books by Haykin [8] and Proakis
and Manolakis [23].
11.1.1 MATLAB IMPLEMENTATION
The LMS algorithm (11.8) can easily be implemented in MATLAB.
Given the input sequence {x(n)}, the desired sequence {d(n)}, step size
△, and the desired length of the adaptive FIR filter N, we can use
(11.1), (11.2), and (11.8) to determine the adaptive filter coefficients
{h(n), 0 ≤ n ≤ N − 1} recursively. This is shown in the following function,
called lms.
function [h,y] = lms(x,d,delta,N)
% LMS Algorithm for Coefficient Adjustment
% ----------------------------------------
% [h,y] = lms(x,d,delta,N)
% h = estimated FIR filter
% y = output array y(n)
% x = input array x(n)
% d = desired array d(n), length must be same as x
% delta = step size
% N = length of the FIR filter
%
M = length(x); y = zeros(1,M);
h = zeros(1,N);
for n = N:M
x1 = x(n:-1:n-N+1);
y = h * x1’;
e = d(n) - y;
h = h + delta*e*x1;
end
In addition, the lms function provides the output {y(n)} of the adaptive
filter.
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