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278 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
□ EXAMPLE 6.25 Consider the digital resonator in Example 6.24. Determine the change in the
pole locations using MATLAB.
Solution
The filter coefficients, a 1 = −0.9 and a 2 =0.81 can be quantized using
>> x = [-0.9,0.81]; [y,L,B] = Qcoeff(x,3)
y = -0.8750 0.7500
L = 0
B = 3
as expected. Now using the following MATLAB script, we can determine the
change in the location of the poles:
% Unquantized parameters
r = 0.9; theta = pi/3; a1 = -2*r*cos(theta); a2 = r*r;
p1 = r*exp(j*theta); p2 = p1’;
% Quantized parameters: N = 3;
[ahat,L,B] = Qcoeff([a1,a2],3); rhat = sqrt(ahat(2));
thetahat = acos(-ahat(1)/(2*rhat)); p1hat = rhat*exp(j*thetahat); p2 = p1’;
% Changes in pole locations
Dp1 = abs(p1-p1hat)
Dp1 = 0.0344
This is the same as before.
□
□ EXAMPLE 6.26 Consider the following IIR filter with 10 poles closely packed at a radius of
r =0.9 around angles ±45 ◦ with a separation of 5 ◦ . Due to large number of
poles, the denominator coefficients have values that require 6 bits for the integer
part. Using 9 bits for the fractional part for a total of 16-bit representation, we
compute and plot the new locations of poles:
r = 0.9; theta = (pi/180)*[-55:5:-35,35:5:55]’;
p = r*exp(j*theta); a = poly(p); b = 1;
% Direct form: quantized coefficients
N = 15; [ahat,L,B] = Qcoeff(a,N);
TITLE = sprintf(’%i-bit (1+%i+%i) Precision’,N+1,L,B);
% Comparison of Pole-Zero Plots
subplot(1,2,1); [HZ,HP,Hl] = zplane(1,a);
set(HZ,’color’,’g’,’linewidth’,1); set(HP,’color’,’g’,’linewidth’,1);
set(Hl,’color’,’w’); axis([-1.1,1.1,-1.1,1.1]);
title(’Infinite Precision’,’fontsize’,10,’fontweight’,’bold’);
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