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548 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
We can also show that the variance of q(n), for both the truncation or
rounding, is given by
∞∑
∫ π
σq 2 = σe
2 |h(n)| 2 = σ2 e
|H(e jω )| 2 dω (10.15)
2π −π
−∞
The variance gain from the input to the output (also known as the normalized
output variance) isthe ratio
σq
2 ∞∑
= |h(n)| 2 = 1 ∫ π
|H(e jω )| 2 dω (10.16)
2π
σ 2 e
−∞
Forareal and stable filter, using the substitution z =e jω , the integral in
(10.16) can be further expressed as a complex contour integral
∫ π
|H(e jω )| 2 dω = 1 ∮
H(z)H(z −1 )z −1 dz (10.17)
2πj
−π
UC
where UC is the unit circle and can be computed using residues (or the
inverse Z-transform) as
∫ π
−π
|H(e jω )| 2 dω = ∑ [Residues of H(z)H(z −1 )z −1 inside UC](10.18a)
−π
= Z −1 [ H(z)H(z −1 ) ]∣ ∣
n=0
(10.18b)
10.1.7 MATLAB IMPLEMENTATION
Computation of the variance-gain for the A/D quantization noise can be
carried out in MATLAB using (10.16) and (10.18). For FIR filters, we can
perform exact calculations using the time-domain expression in (10.16).
In the case of IIR filters, exact calculations can only be done using (10.18)
in special cases, as we shall see (fortunately, this works for most practical
filters). The approximate computations can always be done using the
time-domain expression.
Let the FIR filter be given by the coefficients {b k } M−1
0 . Then using
the time-domain expression in (10.16), the variance-gain is given by
σq
2
σe
2
=
M−1
∑
k=0
|b k | 2 (10.19)
Let an IIR filter be given by the system function
∑ N−1
l=0
H(z) =
b lz −l
1+ ∑ N−1
k=1 a (10.20)
kz −k
with impulse response h(n). If we assume that the filter is real, causal, and
stable and has only simple poles, then using the partial fraction expansion,
we can write
N−1
∑ R k
H(z) =R 0 +
(10.21)
z − p k
k=1
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