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Quantization of Filter Coefficients 279
Infinite Precision
16–bit (1+6+9) Precision
1
1
Imaginary Part
0.5
0
−0.5
10
Imaginary Part
0.5
0
−0.5
10
−1
−1
−1 −0.5 0 0.5 1
−1 −0.5 0 0.5 1
Real Part
Real Part
FIGURE 6.31 Pole-zero plots for direct-form structure in Example 6.26
subplot(1,2,2); [HZhat,HPhat,Hlhat] = zplane(1,ahat);
set(HZhat,’color’,’r’,’linewidth’,1); set(HPhat,’color’,’r’,’linewidth’,1);
set(Hlhat,’color’,’w’); title(TITLE,’fontsize’,10,’fontweight’,’bold’);
axis([-1.1,1.1,-1.1,1.1]);
Figure 6.31 shows the pole-zero plots for filters with both infinite and 16-
bit precision coefficients. Clearly, with 16-bit word length, the resulting filter
is completely different from the original one and is unstable. To investigate finite
word-length effect on the cascade-form structure, we first converted the
direct-form coefficients into the cascade-form coefficients using the dir2cas
function, quantized the resulting set of coefficients, and then converted back
to the direct-form for pole-zero plotting. We show results for two different word
lengths. In the first case, we used the same 16-bit word length. Since the cascade
coefficients have smaller integer parts that require only one integer bit,
the number of fractional bits is 14. In the second case we used 9 fractional bits
(same as those in the direct form) for a total word length of 11 bits.
% 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’,N+1,L1,B1);
Bhat1 = BAhat1(:,1:3); Ahat1 = BAhat1(:,4:6);
[bhat1,ahat1] = cas2dir(b0,Bhat1,Ahat1);
subplot(1,2,1); [HZhat1,HPhat1,Hlhat1] = zplane(bhat1,ahat1);
set(HZhat1,’color’,’g’,’linewidth’,1); set(HPhat1,’color’,’g’,’linewidth’,1);
set(Hlhat1,’color’,’w’); axis([-1.1,1.1,-1.1,1.1]);
title(TITLE1,’fontsize’,10,’fontweight’,’bold’);
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