UploadFile_6417

596 Chapter 11 APPLICATIONS IN ADAPTIVE FILTERING
In this chapter we describe a basic algorithm, called the least-meansquare
(LMS) algorithm, toadaptively adjust the coefficients of an FIR
filter. The adaptive filter structure that will be implemented is the direct
form FIR filter structure with adjustable coefficients h(0),h(1),...,
h(N − 1), as illustrated in Figure 11.1. After we describe the LMS algorithm,
we apply it to several practical systems in which adaptive filters
are employed.
11.1 LMS ALGORITHM FOR COEFFICIENT ADJUSTMENT
Suppose we have an FIR filter with adjustable coefficients {h(k), 0 ≤ k ≤
N − 1}. Let {x(n)} denote the input sequence to the filter, and let the
corresponding output be {y(n)}, where
y(n) =
N−1
∑
k=0
h(k)x (n − k) , n =0,...,M (11.1)
Suppose that we also have a desired sequence {d(n)} with which we can
compare the FIR filter output. Then we can form the error sequence
{e(n)} by taking the difference between d(n) and y(n), that is,
e(n) =d(n) − y(n), n =0,...,M (11.2)
The coefficients of the FIR filter will be selected to minimize the sum of
squared errors. Thus we have
E =
=
[
M∑
M∑
e 2 (n) = d(n) −
n=0
M∑
n=0
n=0
N−1
∑
d 2 (n) − 2
where, by definition,
r dx (k) =
r xx (k) =
k=0
N−1
∑
k=0
h(k)r dx (k)+
h(k)x (n − k)] 2
(11.3)
N−1
∑
k=0
N−1
∑
l=0
h(k)h (l) r xx (k − l)
M∑
d(n)x (n − k) , 0 ≤ k ≤ N − 1 (11.4)
n=0
M∑
x(n)x (n + k) , 0 ≤ k ≤ N − 1 (11.5)
n=0
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.