Digital Signal Processing Reference
In-Depth Information
FIGURE 7.1.
Basic adaptive filter structure.
desirable to design the filter to be self-learning so that it can adapt itself to the sit-
uation at hand.
The coefficients of an adaptive filter are adjusted to compensate for changes in
input signal, output signal, or system parameters. Instead of being rigid, an adaptive
system can learn the signal characteristics and track slow changes. An adaptive filter
can be very useful when there is uncertainty about the characteristics of a signal or
when these characteristics change.
Conceptually, the adaptive scheme is fairly simple. Most of the adaptive schemes
can be described by the structure shown in Figure 7.1. This is a basic adaptive filter
structure in which the adaptive filter's output
y
is compared with a desired signal
d
to yield an error signal
e
, which is fed back to the adaptive filter. The error signal is
input to the adaptive algorithm, which adjusts the variable filter to satisfy some pre-
determined criteria or rules. The desired signal is usually the most difficult one to
obtain. One of the first questions that probably comes to mind is: Why are we trying
to generate the desired signal at
y
if we already know it? Surprisingly, in many appli-
cations the desired signal does exist somewhere in the system or is known a priori.
The challenge in applying adaptive techniques is to figure out where to get the
desired signal, what to make the output
y
, and what to make the error
e.
The coefficients of the adaptive filter are adjusted, or optimized, using an LMS
algorithm based on the error signal. Here we discuss only the LMS searching algo-
rithm with a linear combiner (FIR filter), although there are several strategies for
performing adaptive filtering. The output of the adaptive filter in Figure 7.1 is
N
-
Â
0
1
()
=
()
-
(
)
yn
w nxn k
k
(7.1)
k
=
where
w
k
(
n
) represent
N
weights or coefficients for a specific time
n
. The convolu-
tion equation (7.1) was implemented in Chapter 4 in conjunction with FIR filter-
ing. It is common practice to use the terminology of weights
w
for the coefficients
associated with topics in adaptive filtering and neural networks.
A performance measure is needed to determine how good the filter is. This
measure is based on the error signal,
()
=
()
-
()
en
dn
yn
(7.2)
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