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556 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
Amplitude
1
0.75
0.5
0.25
0
−0.25
−0.5
−0.75
−1
α = −0.5, Rounding
0 5 10
Sample Index n
Amplitude
1
0.75
0.5
0.25
0
−0.25
−0.5
−0.75
−1
α = 0.5, Rounding
0 5 10
Sample Index n
Amplitude
1
0.75
0.5
0.25
0
−0.25
−0.5
−0.75
−1
FIGURE 10.10 Granular limit cycles in Example 10.6
α = −0.5, Truncation
0 5 10
Sample Index n
Hs_1 = stem([-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(’\alpha = -0.5, Truncation’,’fontsize’,10);
xlabel(’Sample index n’,’fontsize’,8);
The resulting plots are shown in Figure 10.10. The output signal in the left
plot agrees with that in Example 10.5 and has an asymptotic period of two
samples. The middle plot for α =0.5 (lowpass filter) shows that the limit cycle
has a period of one sample with amplitude of 1 . Finally, the right plot shows
8
that the limit cycles vanish for the truncation operation. This behavior for the
truncation operation is also exhibited for lowpass filters.
□
In the case of 2nd-order and higher-order digital filters, granular limit
cycles not only exist but also are of various types. These cycles in 2ndorder
filters can be analyzed, and dead-band as well as frequency of oscillations
can be estimated. For example, if the recursive all-pole filter is
implemented with rounding quantizers in the multipliers as
ŷ(n) =Q[a 1 ŷ(n − 1)] + Q[a 2 ŷ(n − 2)] + x(n) (10.32)
where ŷ(n) isthe quantized output, then using the analysis similar to that
of the 1-order case, the dead-band region is given by
ŷ(n − 2) ≤
∆
2(1 −|a 2 |)
(10.33)
with a 1 determining the frequency of oscillations. For more details see
Proakis, and Manolakis [23]. We provide the following example to illustrate
granular limit cycles in 2nd-order filters using 3-bit quantizers.
□ EXAMPLE 10.7 Consider the 2nd-order recursive filter
y(n) =0.875y(n − 1) − 0.75y(n − 2) + x(n) (10.34)
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