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Analysis of A/D Quantization Noise 549
where R 0 is the constant term and R k ’s are the residues at the pole
locations p k . This expansion can be computed using the residue function.
Note that both poles and the corresponding residues are either real-valued
or occur in complex-conjugate pairs. Then using (10.18a), we can show
that (see [17] and also Problem P10.3)
σq
2
σe
2
= R 2 0 +
N−1
∑
k=1
N−1
∑
l=1
R k Rl
∗
1 − p k p ∗ l
(10.22)
The variance-gain expression in (10.22) is applicable for most practical
filters since rarely do they have multiple poles. The approximate value of
the variance-gain for IIR filters is given by
σq
2
σe
2
≃
K−1
∑
k=0
|h(n)| 2 , K ≫ 1 (10.23)
where K is chosen so that the impulse response values (magnitudewise)
are almost zero beyond K samples. The following MATLAB function,
VarGain, computes variance-gain using (10.19) or (10.22).
function Gv = VarGain(b,a)
% Computation of variance-gain for the output noise process
% of digital filter described by b(z)/a(z)
% Gv = VarGain(b,a)
a0 = a(1); a = a/a0; b = b/a0; M = length(b); N = length(a);
if N == 1
% FIR Filter
Gv = sum(b.*b);
return
else
% IIR Filter
[R,p,P] = residue(b,a);
if length(P) > 1
error(’*** Variance Gain Not computable ***’);
elseif length(P) == 1
Gv = P*P;
else
Gv = 0;
end
Rnum = R*R’; pden = 1-p*p’;
H = Rnum./pden; Gv = Gv + real(sum(H(:)));
end
It should be noted that the actual output noise variance is obtained by
multiplying the A/D quantization noise variance by the variance-gain.
□ EXAMPLE 10.4 Consider an 8-order IIR filter with poles at p k = r e j2πk/8 , k =0,...,7. If r is
close to 1, then the filter has 4 narrowband peaks. Determine the variance-gain
for this filter when r =0.9 and r =0.99.
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