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280 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
16–bit (1+1+14) Precision
11–bit (1+1+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 6.32 Pole-zero plots for cascade-form structure in Example 6.26
% 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’,N1+1,L2,B2);
Bhat2 = BAhat2(:,1:3); Ahat2 = BAhat2(:,4:6);
[bhat2,ahat2] = cas2dir(b0,Bhat2,Ahat2);
subplot(1,2,2); [HZhat2,HPhat2,Hlhat2] = zplane(bhat2,ahat2);
set(HZhat2,’color’,’r’,’linewidth’,1); set(HPhat2,’color’,’r’,’linewidth’,1);
set(Hlhat2,’color’,’w’);title(TITLE2,’fontsize’,10,’fontweight’,’bold’);
axis([-1.1,1.1,-1.1,1.1]);
The results are shown in Figure 6.32. We observe that not only for 16-bit representation
but also for 11-bit representation, the resulting filter is essentially
the same as the original one and is stable. Clearly, the cascade form structure
has better finite word-length properties than the direct form structure. □
6.8.3 EFFECTS ON FREQUENCY RESPONSE
The frequency response of the IIR filter in (6.50) is given by
∑ M
H(e jω k=0
)=
b k e −jωk
1+ ∑ N
k=1 a (6.76)
k e −jωk
When the coefficients {a k } and {b k } are quantized to {â k } and {ˆb k },
respectively, the new frequency response is given by
∑ M
Ĥ(e jω k=0
)=
ˆb k e −jωk
1+ ∑ (6.77)
N
k=1 âk e −jωk
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