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572 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
Following (10.43), (10.44), and (10.45), the scaling factor X max is given
by
1
X max = ∑ ∞
n=0 |h(n)| (10.58)
which is not easy to compute. However, lower and upper bounds on X max
are easy to obtain. From (10.52), the upper bound on the denominator of
(10.58) is given by
∞∑
∞∑
∞∑
n=0
|h(n)| = 1
sin θ
n=0
r n | sin[(n +1)θ]| ≤ 1
sin θ
or the lower bound on X max is given by
n=0
r n =
1
(1 − r) sin θ
(10.59)
X max ≥ (1 − r) sin θ (10.60)
The lower bound on the denominator of (10.58) is obtained by noting that
∣ |H(e jω ∞∑ ∣∣∣∣ ∞∑
)| =
h(n)e −jω ≤ |h(n)|
∣
n=0
n=0
Now from (10.51), the magnitude |H(e jω )| is given by
∣ |H(e jω )| =
1
∣∣∣
∣1 − 2r cos(θ)e −jω + r 2 e −j2ω
which has the maximum value at the resonant frequency ω = θ, which
can be easily obtained. Hence
∞∑
n=0
|h(n)| ≥ ∣ ∣ H(e jθ ) ∣ ∣ =
1
(1 − r) √ 1+r 2 − 2r cos(2θ)
or the upper bound on X max is given by
(10.61)
X max ≤ (1 − r) √ 1+r 2 − 2r cos(2θ) (10.62)
Substituting (10.60) and (10.62) in (10.56), the output SNR is upper and
lower bounded by
2 2(B+1) (1−r) 2 sin 2 θ ≤ SNR ≤ 2 2(B+1) (1−r) 2 (1+r 2 −2r cos 2θ) (10.63)
Substituting 1 − r = δ ≪ 1 and after some simplification, we obtain
(
2 2(B+1) δ 2 sin 2 θ ≤ SNR ≤ 4 2 2(B+1)) δ 2 sin 2 θ (10.64)
or the difference between the upper and lower SNR bounds is about 6 dB.
Once again the output SNR is directly proportional to B and δ. Furthermore,
it also depends on the angle θ. Some of these observations are
investigated in Example 10.12.
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