UploadFile_6417

Round-off Effects in IIR Digital Filters 557
with zero initial conditions. This filter has two complex-conjugate poles and
hence is a bandpass filter. Let the input be x(n) =0.375δ(n). Analyze the limit
cycle behavior using a 3-bit quantizer.
Solution
In the filter implementation the coefficient products are quantized, which
results in
% Bandpass filter
a1 = 0.875; a2 = -0.75;
ŷ(n) =Q[0.875ŷ(n − 1)] −Q[0.75ŷ(n − 2)] + x(n) (10.35)
where ŷ(n) isthe quantized output. We simulate (10.35) in MATLAB using
both the rounding and truncation operations.
% Rounding operation in multipliers
yn1 = 0; yn2 = 0;
m = 0:20; y = [yn2,yn1,zeros(1,length(m))];
x = 0.375*impseq(m(1),m(1)-2,m(end));
for n = m+3
yn1 = y(n-1); yn2 = y(n-2);
y(n) = QFix(a1*yn1,3,’round’,’satur’)+QFix(a2*yn2,3,’round’,’satur’)+x(n);
end
subplot(’position’,[0.1,0.2,0.39,0.6]);
plot([-1,20],[0,0],’w’); axis([-1,20,-0.5,0.5]); hold on;
Hs_1 = stem([-2,-1,m],y,’filled’); set(Hs_1,’markersize’,3,’color’,[0,1,0]);
set(gca,’ytick’,[-0.5:0.25:0.5],’fontsize’,6); ylabel(’Amplitude’,’fontsize’,8);
title(’Rounding Operation’,’fontsize’,10);
xlabel(’Sample index n’,’fontsize’,8);
% Truncation operation in multipliers
yn1 = 0; yn2 = 0;
m = 0:20; y = [yn2,yn1,zeros(1,length(m))];
x = 0.375*impseq(m(1),m(1)-2,m(end));
for n = m+3
yn1 = y(n-1); yn2 = y(n-2);
y(n) = QFix(a1*yn1,3,’trunc’,’satur’)+QFix(a2*yn2,3,’trunc’,’satur’)+x(n);
end
subplot(’position’,[0.59,0.2,0.39,0.6]);
plot([-1,20],[0,0],’w’); axis([-1,20,-0.5,0.5]); hold on;
Hs_1 = stem([-2,-1,m],y,’filled’); set(Hs_1,’markersize’,3,’color’,[0,1,0]);
set(gca,’ytick’,[-0.5:0.25:0.5],’fontsize’,6); ylabel(’Amplitude’,’fontsize’,8);
title(’Truncation Operation’,’fontsize’,10);
xlabel(’Sample index n’,’fontsize’,8);
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.