Adaptive Filters and Applications - Digital Signal Processing: Fundamentals and Applications - page 487

10.8. Given a quadratic MSE function for the Wiener filter

2

J ¼ 10 30 w þ 15 w

use the steepest descent method with an initial guess of w 0 ¼ 2 and a convergence factor
m ¼ 0 : 02 to find the optimal solution for w and determine J min by iterating three times.
10.9. Consider the following DSP system used for noise cancellation applications ( Figure 10.25 ) ,
in which dð 0 Þ¼ 3, dð 1 Þ¼ 2, dð 2 Þ¼ 1, xð 0 Þ¼ 3, xð 1 Þ¼ 1, xð 2 Þ¼ 2, and there is
an adaptive filter with two taps yðnÞ¼wð 0 ÞxðnÞþwð 1 Þxðn 1 Þ with initial values
wð 0 Þ¼ 0, wð 1 Þ¼ 1, and m ¼ 0 : 1,
a. Determine the LMS algorithm equations

yðnÞ¼
eðnÞ¼
wð 0 Þ¼
wð 1 Þ¼

b. Perform adaptive filtering for each n ¼ 0 ; 1 ; 2.
10.10. Given a DSP system with a sampling rate of 8,000 samples per second, implement an
adaptive filter with five taps for system modeling.

As shown in Figure 10.26 , assume the unknown system transfer function is

0 : 25 þ 0 : 25 z 1
1 0 : 5 z 1

HðzÞ¼

Determine the DSP equations

yðnÞ¼
eðnÞ¼
wðiÞ¼

d ()

Output

e ()

Input

x () y ()

Adaptive
FIR filter

FIGURE 10.25

Noise cancellation in Problem 10.9.