Adaptive Filters and Applications - Digital Signal Processing: Fundamentals and Applications - page 495

Residue error signal: eðnÞ¼dðnÞ filtering yðnÞ using coefficients sðnÞ¼dðnÞyðnÞ
sðnÞ , where the symbol “*” denotes the filter convolution.
Implement the ANC system and plot the residue sensor signal to verify the effectiveness. The
primary noise at cancelling point dðnÞ , and filtered reference signal uðnÞ can be generated in
MATLAB as follows:
d ¼ filter([0.2 0.25 0.2],1,x); % Simulate physical media
u ¼ filter([0.2 0.2],1,x);
The residue error signal eðnÞ should be generated sample by sample and embedded into the
adaptive algorithm, that is,
e(n) ¼ d(n)-(s(1)*y(n)+s(2)*y(n-1)); % Simulate the residue error
where s(1) ¼ 0.2 and s(2) ¼ 0.2. Details of active control systems can be found in the textbook
by Kuo and Morgan (1996).

10.28. Frequency tracking:

An adaptive filter can be applied for real-time frequency tracking (estimation). In this appli-
cation, a special second notch IIR filter structure, as shown in Figure 10.33 , is preferred for
simplicity.

The notch filter transfer function

1 2cos q z 1

þ z 2

HðzÞ¼

1 2 r cos ðqÞz 1

2

z 2 þ r

has only one adaptive parameter q . It has two zeros on the unit circle resulting in an infinite-
depth notch. The parameter r controls the notch bandwidth. It requires 0 << r < 1 for
achieving a narrowband notch. When r is close to 1, the 3-dB notch filter bandwidth can be
approximated as BW z 2 ð 1 rÞ (see Chapter 8). The input sinusoid whose frequency f
needs to be estimated and tracked is given below:

xðnÞ¼A cos ð 2 pfn=f s þ aÞ
where A and a are the amplitude and phase angle. The filter output is expressed as
y n ¼ x n 2cos q n x n 1 þ x n 2 þ 2 r cos q n y n 1 r

y n 2

2

xn ()

yn ()

Second-order adaptive IIR
notch filter

FIGURE 10.33
A frequency tracking system.