package such as MATLAB that uses double-precision floating-point format to repre-
sent filter coefficients. Let b
H
l
and a
H
m
denote the quantized values corresponding to b
l
and a
m
, respectively. The I/O equation that can be actually implemented is given by
Equation (3.5.10), and its transfer function is expressed as
H
H
z
LÀ1
l0
b
H
l
z
Àl
1
M
m1
a
H
m
z
Àm
:
6:6:10
Similar to the concept of input quantization discussed in Section 3.5, the nonlinear
operation of coefficient quantization can be modeled as a linear process that introduces
a quantization noise expressed as
b
H
l
Qb
l
b
l
el
6:6:11
and
a
H
m
Qa
m
a
m
em,
6:6:12
where the coefficient quantization errors el and em can be assumed to be a random
noise that has zero mean and variance as defined in (3.5.6).
If the wordlength is not large enough, some undesirable effects occur. For ex-
ample, the frequency characteristics such as magnitude and phase responses of H
H
z
may be different from those of Hz. In addition, for high-order filters whose poles
are closely clustered in the z-plane, small changes in the denominator coefficients can
cause large shifts in the location of the poles. If the poles of H(z) are close to the
unit circle, the pole(s) of H
H
z may move outside the unit circle after coefficient
quantization, resulting in an unstable implementation. These undesired effects are
more serious when higher-order filters are implemented using the direct-form I and II
realizations discussed in Section 6.4. Therefore the cascade and parallel realizations are
preferred in practical DSP implementations with each H
k
z in a first-or second-order
section.
Example 6.15: Given the IIR filter with transfer function
Hz
1
1 À 0:9z
À1
0:2z
À2
,
the poles are located at z 0:4 and z 0:5. This filter can be realized in the
cascade form as
Hz H
1
zH
2
z,
where H
1
z 1=1 À 0:4z
À1
and H
2
z 1=1 À 0:5z
À1
:
Assuming that this IIR filter is implemented in a 4-bit (a sign bit plus 3 data
bits) DSP hardware, 0.9, 0.2, 0.4, and 0.5 are quantized to 0.875, 0.125, 0.375, and
0.5, respectively. Therefore the direct-form realization is described as
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DESIGN AND IMPLEMENTATION OF IIR FILTERS