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558 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
0.5
Rounding Operation
0.5
Truncation Operation
Amplitude
0.25
0
−0.25
Amplitude
0.25
0
−0.25
−0.5
0 5 10 15 20
Sample Index n
−0.5
FIGURE 10.11 Granular limit cycles in Example 10.7
0 5 10 15 20
Sample Index n
The resulting plots are shown in Figure 10.11. The round-off limit cycles have a
period of six samples and amplitude of 0.25, which agrees with (10.33). Unlike
in the case of 1st-order filters, the limit cycles for the 2nd-order exist even when
truncation is used in the quantizer.
□
10.2.3 OVERFLOW LIMIT CYCLES
This type of limit cycle is also a zero-input behavior that gives an oscillatory
output. It is due to overflow in the addition even if we ignore
multiplication or product quantization in the filter implementation. This
is a more serious limit cycle because the oscillations can cover the entire
dynamic range of the quantizer. It can be avoided in practice by using
the saturation characteristics instead of overflow in the quantizer. In the
following example, we simulate both granular and overflow limit cycles in
a second-order filter, in addition to infinite precision implementation.
□ EXAMPLE 10.8 To obtain overflow in addition we will consider the second-order filter with large
coefficient values and initial conditions (magnitudewise) excited by a zero input:
y(n) =0.875y(n − 1) − 0.875y(n − 1); y(−1) = −0.875, y(−2) =0.875
(10.36)
The overflow in the addition is obtained by placing the quantizer after the
additions as
ŷ(n) =Q[0.875ŷ(n − 1) − 0.875ŷ(n − 1)]; ŷ(−1) = −0.875, ŷ(−2) = 0.875
(10.37)
where ŷ(n) isthe quantized output. We first simulate the infinite-precision operation
of (10.36) and compare its output with the granular limit-cycle implementation
in (10.35) and with the overflow limit-cycle in (10.37). We use the
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