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Round-off Effects in IIR Digital Filters 559
rounding operation. The details are in the MATLAB script:
M = 100; B = 3; A = 1-2^(-B);
a1 = A; a2 = -A; yn1 = -A; yn2 = A;
m = 0:M; y = [yn2,yn1,zeros(1,length(m))];
% Infinite precision
for n = m+3
yn1 = y(n-1); yn2 = y(n-2);
y(n) = a1*yn1 + a2*yn2;
end
subplot(’position’,[0.08,0.2,0.24,0.6]);
plot([-1,100],[0,0],’w’); axis([-1,80,-1,1]); hold on;
Hs_1 = stem([-2,-1,m],y,’filled’);set(Hs_1,’markersize’,3,’color’,[0,1,0]);
set(gca,’ytick’,[-1:0.25:1],’fontsize’,6); ylabel(’Amplitude’,’fontsize’,8);
title(’No Limit Cycles’,’fontsize’,10);
xlabel(’Sample index n’,’fontsize’,8);
% Granular limit cycle
for n = m+3
yn1 = y(n-1); yn2 = y(n-2);
y(n) = QFix(a1*yn1,B,’round’,’satur’)+QFix(a2*yn2,B,’round’,’satur’);
y(n) = QFix(y(n),B,’round’,’satur’);
end
subplot(’position’,[0.42,0.2,0.24,0.6]);
plot([-1,100],[0,0],’w’); axis([-1,80,-1,1]); hold on;
Hs_1 = stem([-2,-1,m],y,’filled’);set(Hs_1,’markersize’,3,’color’,[0,1,0]);
set(gca,’ytick’,[-1:0.25:1],’fontsize’,6); ylabel(’Amplitude’,’fontsize’,8);
title(’Granular Limit Cycles’,’fontsize’,10);
xlabel(’Sample index n’,’fontsize’,8);
% Overflow limit cycle
for n = m+3
yn1 = y(n-1); yn2 = y(n-2);
y(n) = a1*yn1 + a2*yn2;
y(n) = QFix(y(n),B,’round’,’twosc’);
end
subplot(’position’,[0.76,0.2,0.23,0.6]);
plot([-1,100],[0,0],’w’); axis([-1,80,-1,1]); hold on;
Hs_1 = stem([-2,-1,m],y,’filled’);set(Hs_1,’markersize’,3,’color’,[0,1,0]);
set(gca,’ytick’,[-1:0.25:1],’fontsize’,6); ylabel(’Amplitude’,’fontsize’,8);
title(’Overflow Limit Cycles’,’fontsize’,10);
xlabel(’Sample index n’,’fontsize’,8);
The resulting plots are shown in Figure 10.12. As expected, the infinite-precision
implementation has no limit cycles. The granular limit cycles are of smaller
amplitudes. Clearly, the overflow limit cycles have large amplitudes spanning
the −1 to1range of the quantizers.
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