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Quantization of Filter Coefficients 283
Infinite Precision (DF)
16–bit (1+6+9) Precision (DF)
Magnitude Response
0.6
0.5
0.4
0.3
0.2
0.1
Magnitude Response
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.2 0.4 0.6 0.8 1
0
0 0.2 0.4 0.6 0.8 1
16–bit (1+1+14) Precision (CF)
11–bit (1+1+9) Precision (CF)
Magnitude Response
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.2 0.4 0.6 0.8 1
Digital Frequency in π Units
Magnitude plots for direct- and cascade-form structures in Exam-
FIGURE 6.33
ple 6.27
Magnitude Response
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.2 0.4 0.6 0.8 1
Digital Frequency in π Units
Solution
Unlike the previous example, some of the filter coefficient values (specifically
those of the autoregressive part) are greater than one and hence require bits for
the integer part. This assignment is done for all coefficients since in practice,
the same bit-pattern is used for the filter representation. These and other steps
are given in the following MATLAB script.
% The following 3 lines produce filter coefficients shown in Table 6.1.
wp = [0.35,0.65]; ws = [0.25,0.75]; Rp = 1; As = 50;
[N, wn] = ellipord(wp, ws, Rp, As);
[b,a] = ellip(N,Rp,As,wn);
w = [0:500]*pi/500; H = freqz(b,a,w); magH = abs(H);
magHdb = 20*log10(magH);
% 16-bit word-length quantization
N1 = 15; [bahat,L1,B1] = QCoeff([b;a],N1);
TITLE1 = sprintf(’%i-bits (1+%i+%i)’,N1+1,L1,B1);
bhat1 = bahat(1,:); ahat1 = bahat(2,:);
Hhat1 = freqz(bhat1,ahat1,w); magHhat1 = abs(Hhat1);
magHhat1db = 20*log10(magHhat1); zhat1 = roots(bhat1);
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