Adaptive Filters and Applications - Digital Signal Processing: Fundamentals and Applications - page 462

Hence, we may make a selection of the convergence parameter as

9 : 3 10 10
N

1
N 2 30 z

m ¼ (10.15)

We further neglect time index for w n ðiÞ and use the notation wðiÞ¼w n ðiÞ , since only the current
updated coefficients are needed for next sample adaptation. We conclude the implementation of the
LMS algorithm with the following steps:

1. Initialize wð 0 Þ , wð 1 Þ , . wðN 1 Þ to arbitrary values.
2. Read dðnÞ , xðnÞ , and perform digital filtering:

yðnÞ¼wð 0 ÞxðnÞþwð 1 Þxðn 1 Þþ / þ wðN 1 Þxðn N þ 1 Þ

3. Compute the output error:

eðnÞ¼dðnÞyðnÞ

4. Update each filter coefficient using the LMS algorithm:

for i ¼ 0 ; / ; N 1
wðiÞ¼wðiÞþ 2 meðnÞxðn iÞ

We will apply the adaptive filter to solve real-world problems in the next section.

10.3 APPLICATIONS: NOISE CANCELLATION, SYSTEM MODELING, AND
LINE ENHANCEMENT
We now examine several applications of the LMS algorithm, such as noise cancellation, system
modeling, and line enhancement via application examples. First, we begin with the noise cancellation
problem to illustrate operations of the LMS adaptive FIR filter.

10.3.1 Noise Cancellation
The concept of noise cancellation was introduced in the previous section. Figure 10.7 shows the main
concept.
The DSP system consists of two ADC channels. The first microphone with ADC captures the noisy
speech, dðnÞ¼sðnÞþnðnÞ , which contains the clean speech sðnÞ and noise nðnÞ due to a noisy
environment, while the second microphone with ADC resides where it picks up only the correlated
noise and feeds the noise reference xðnÞ to the adaptive filter. The adaptive filter uses the LMS
algorithm to adjust its coefficients to produce the best estimate of noise yðnÞ z nðnÞ , which will be
subtracted from the corrupted signal dðnÞ¼sðnÞþnðnÞ . The output of the error signal
eðnÞ¼sðnÞþnðnÞyðnÞ z sðnÞ is expected to be the best estimate of the clean speech signal.
Through digital-to-analog conversion (DAC), the cleaned digital speech becomes analog voltage,
which drives the speaker.