Adaptive Filters and Applications - Digital Signal Processing: Fundamentals and Applications - page 463

S i g n a l a n d n o i s e E r r o r s i g n a l

() () () ~ ()

dn sn nn () () ()

en dn yn sn

ADC

DAC

y ()

x ()

ADC Adaptive filter

Noise

LMS algorithm

FIGURE 10.7
Simplest noise canceller using a one-tap adaptive filter.

We first study the noise cancellation problem using a simple two-tap adaptive filter via Example
10.3 and assumed data. The purpose of doing so is to become familiar with the setup and operations of
the adaptive filter and LMS algorithm. The simulation for real adaptive noise cancellation follows.

EXAMPLE 10.3
Consider the DSP system for the noise cancellation application using an adaptive filter with two coefficients shown
in Figure 10.8 .
a. Set up the LMS algorithm for the adaptive filter.

b. Perform adaptive filtering to obtain outputs eðnÞ for n ¼ 0; 1; 2 given the following inputs and outputs:

xð0Þ¼1; xð1Þ¼1; xð2Þ¼1; dð0Þ¼2; dð1Þ¼1; dð2Þ¼2
The initial weights are wð0Þ¼wð1Þ¼0, and the convergence factor is set to be m ¼ 0:1.
Solution:
a. The adaptive LMS algorithm is set up as:
Initialization: wð0Þ¼0, wð1Þ¼0
Digital filtering: yðnÞ¼wð0ÞxðnÞþwð1Þxðn 1Þ

Signal and noise

dn sn nn () () ()

Output

e ()

Adaptive filter

Noise

x () yn w x nwx n () ()() ()( ) 0 1 1

y ()

FIGURE 10.8
Noise cancellation in Example 10.3.