H
H
z
1
1 À 0:875z
À1
0:125z
À2
and the cascade realization is expressed as
H
HH
z
1
1 À 0:375z
À1
Á
1
1 À 0:5z
À1
:
The pole locations of the direct-form H
H
z are z 0:18 and z 0:695, and the
pole locations of the cascade form H
HH
z are z 0:375 and z 0:5. Therefore the
poles of cascade realization are closer to the desired Hz.
In practice, one must always check the stability of the filter with the quantized
coefficients. The problem of coefficient quantization may be studied by examining
pole locations in the z-plane. For a second-order IIR filter given in (6.4.1), we can
place the poles near z Æ1 with much less accuracy than elsewhere in the z-plane. Since
the second-order IIR filters are the building blocks of the cascade and parallel forms, we
can conclude that narrowband lowpass (or highpass) filters will be most sensitive to
coefficient quantization because their poles close to z 1 or (z À1). In summary, the
cascade form is recommended for the implementation of high-order narrowband IIR
filters that have closely clustered poles.
As discussed in Chapter 3, the effect of the input quantization noise on the output can
be computed as
s
2
y;e
s
2
e
2pj
z
À1
HzHz
À1
dz,
6:6:13
where s
2
e
2
À2B
=3 is defined by (3.5.6). The integration around the unit circle jzj 1 in
the counterclockwise direction can be evaluated using the residue method introduced in
Chapter 4 for the inverse z-transform.
Example 6.16: Consider an IIR filter expressed as
Hz
1
1 À az
À1
, jaj < 1,
and the input signal x(n) is an 8-bit data. The noise power due to input quantiza-
tion is s
2
e
2
À16
=3. Since
R
za
z À a
1
z À a1 À az
za
1
1 À a
2
,
the noise power at the output of the filter is calculated as
s
2
y;e
2
À16
31 À a
2
:
IMPLEMENTATION CONSIDERATIONS
277