Adaptive Filters and Applications - Digital Signal Processing: Fundamentals and Applications - page 470

of tones of 500, 1,500, and 2,500 Hz. The unknown system's frequency responses are shown in
Figure 10.12 .
The input waveform xðnÞ with three tones is shown as the first plot in Figure 10.13 . We can
predict that the output of the unknown system will contain a 1,500 Hz tone only, since the other two
tones are rejected by the unknown system. Now, let us look at adaptive filter results. We use an FIR
adaptive filter with the number of taps being 21, and a convergence factor set to 0.01. In the time
domain, the output waveforms of the unknown system dðnÞ and adaptive filter output yðnÞ are almost
identical after 70 samples when the LMS algorithm converges. The error signal eðnÞ is also plotted
to show the adaptive filter keeps tracking the unknown system's output with no difference after the
first 50 samples.
Figure 10.14 depicts the frequency domain comparisons. The first plot displays the frequency
components of the input signal, which clearly shows 500 Hz, 1,500 Hz, and 2,500 Hz. The second plot
shows the unknown system's output spectrum, which contains only a 1,500 Hz tone, while the third
plot displays the spectrum of the adaptive filter output. As we can see, in the frequency domain, the
adaptive filter tracks the characteristics of the unknown system. The MATLAB implementation is
given in Program 10.2.
Program 10.2. MATLAB program for adaptive system identification.

close all; clear all
%Design unknown system
fs ¼ 8000; T ¼ 1/fs; % Sampling rate and sampling period

0

-20

-40

-60

-80

0 500 1000 1500 2000 2500 3000 3500 4000

Frequency (Hertz)

200

100

0

-100

-200

0 500 1000 1500 2000 2500 3000 3500 4000

Frequency (Hertz)

FIGURE 10.12
The unknown system's frequency responses.