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Quantization of Filter Coefficients 281
One can perform analysis similar to that for the movement of poles to
obtain maximum change in the magnitude or phase responses due to
changes in filter coefficients. However, such an analysis is very complicated
and may not add any new insight. Hence we will study these effects using
MATLAB. We provide the following two examples.
□ EXAMPLE 6.27 Compute and plot magnitude responses of filter structures given for the filter
in Example 6.26.
Solution
The filter is a bandpass filter with 10 tightly clustered poles implemented using
the direct and the cascade forms. For the direct-form structure, we compute the
magnitude response for infinite precision as well as for 16-bit quantization. For
the cascade-form structure, we use 16-bit and 11-bit representations.
r = 0.9; theta = (pi/180)*[-55:5:-35,35:5:55]’;
p = r*exp(j*theta); a = poly(p); b = 1;
w = [0:500]*pi/500; H = freqz(b*1e-4,a,w);
magH = abs(H); magHdb = 20*log10(magH);
% Direct form: quantized coefficients
N = 15; [ahat,L,B] = Qcoeff(a,N);
TITLE = sprintf(’%i-bit (1+%i+%i) Precision (DF)’,N+1,L,B);
Hhat = freqz(b*1e-4,ahat,w); magHhat = abs(Hhat);
% Cascade form: quantized coefficients: Same N
[b0,B0,A0] = dir2cas(b,a);
[BAhat1,L1,B1] = Qcoeff([B0,A0],N);
TITLE1 = sprintf(’%i-bit (1+%i+%i) Precision (CF)’,N+1,L1,B1);
Bhat1 = BAhat1(:,1:3); Ahat1 = BAhat1(:,4:6);
[bhat1,ahat1] = cas2dir(b0,Bhat1,Ahat1);
Hhat1 = freqz(b*1e-4,ahat1,w); magHhat1 = abs(Hhat1);
% Cascade form: quantized coefficients: Same B (N=L1+B)
N1 = L1+B; [BAhat2,L2,B2] = Qcoeff([B0,A0],N1);
TITLE2 = sprintf(’%i-bit (1+%i+%i) Precision (CF)’,N1+1,L2,B2);
Bhat2 = BAhat2(:,1:3); Ahat2 = BAhat2(:,4:6);
[bhat2,ahat2] = cas2dir(b0,Bhat2,Ahat2);
Hhat2 = freqz(b*1e-4,ahat2,w); magHhat2 = abs(Hhat2);
% Comparison of Magnitude Plots
Hf_1 = figure(’paperunits’,’inches’,’paperposition’,[0,0,6,4]);
subplot(2,2,1); plot(w/pi,magH,’g’,’linewidth’,2); axis([0,1,0,0.7]);
%xlabel(’Digital Frequency in \pi units’,’fontsize’,10);
ylabel(’Magnitude Response’,’fontsize’,10);
title(’Infinite Precision (DF)’,’fontsize’,10,’fontweight’,’bold’);
subplot(2,2,2); plot(w/pi,magHhat,’r’,’linewidth’,2); axis([0,1,0,0.7]);
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