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600 Chapter 11 APPLICATIONS IN ADAPTIVE FILTERING
2. An unknown system module that may be selected is an IIR filter and
implemented by its difference equation. For example, we may select an
IIR filter specified by the second-order difference equation
d(n) =a 1 d (n − 1) + a 2 d (n − 2) + x(n)+b 1 x (n − 1) + b 2 x (n − 2)
(11.11)
where the parameters {a 1 ,a 2 } determine the positions of the poles and
{b 1 ,b 2 } determine the positions of the zeros of the filter. These parameters
are input variables to the program. This can be implemented by
the filter function.
3. An adaptive FIR filter module where the FIR filter has N tap coefficients
that are adjusted by means of the LMS algorithm. The length
N of the filter is an input variable to the program. This can be implemented
using the lms function given in the previous section.
The three modules are configured as shown in Figure 11.2. From this
project we can determine how closely the impulse response of the FIR
model approximates the impulse response of the unknown system after
the LMS algorithm has converged.
To monitor the convergence rate of the LMS algorithm, we may compute
a short-term average of the squared error e 2 (n) and plot it. That is,
we may compute
ASE(m) = 1 K
n+K
∑
k=n+1
e 2 (k) (11.12)
where m = n/K =1, 2,.... The averaging interval K may be selected
to be (approximately) K =10N. The effect of the choice of the step
size parameter △ on the convergence rate of the LMS algorithm may be
observed by monitoring the ASE(m).
Besides the main part of the program, you should also include, as an
aside, the computation of the impulse response of the unknown system,
which can be obtained by exciting the system with a unit sample sequence
δ(n). This actual impulse response can be compared with that of the FIR
model after convergence of the LMS algorithm. The two impulse responses
can be plotted for the purpose of comparison.
11.3 SUPPRESSION OF NARROWBAND INTERFERENCE
IN A WIDEBAND SIGNAL
Let us assume that we have a signal sequence {x(n)} that consists of
a desired wideband signal sequence, say {w(n)}, corrupted by an additive
narrowband interference sequence {s(n)}. The two sequences are
uncorrelated. This problem arises in digital communications and in signal
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