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580 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
x(n)
z −1 z −1 z −1
h(0)
h(1)
h(2)
h(M − 1) h(M − 2)
y(n)
x(n)
h(0)
(a)
z −1 z −1 z −1
h(1) h(2)
h(M − 1) h(M − 2)
Q Q Q Q Q
e 0 (n)
e 1 (n) e 2 (n) e M−1 (n) e M−1 (n)
x(n)
(b)
z −1 z −1 z −1
h(0)
h(1)
h(2)
h(M − 1) h(M − 2)
(c)
Q
e(n)
y(n) ˆ
= y(n) + q(n)
FIGURE 10.26 Direct-form FIR filter: (a) structure, (b) round-off noise model
with quantizers after each multiplier, (c) round-off noise mode with one quantizer
after the final sum
and is similar to that for the IIR filter in (10.42)–(10.44). The upperbound
on y(n) isobtained as
∣
∣
∣∣
∑
|y(n)| = ∣∑
h(k) x(n − k) ≤ Xmax |h(n)| (10.83)
where X max is the upper-bound on x(n). To guarantee that |y(n)| ≤1,
we need the scaling factor X max on x(n) as
X max ≤
1
∑ |h(n)|
(10.84)
which is the most conservative scaling factor. There are other scaling
factors, depending on the applications—for example, the narrowband
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