Digital Signal Processing Reference
In-Depth Information
One can easily show that this second-order system is stable as long as |a
1
| \ 2
and |a
2
| \ 1. To graphically appraise the sensitivity of the filter implementation,
one can do the following:
(a) Assume Direct Form—I realization: Plot all the possible locations of the
complex stable poles when the coefficients a
1
and a
2
are represented in sign-
magnitude format with 5-bit word-length.
(b) Realize the second-order IIR filter using coupled form and repeat the z-plane
plot as in (a).
This second-order system is stable as long as |a
1
| \ 2 and |a
2
| \ 1. There are a
number of possibilities for implementing this filter. The first way to implement it is
with a Direct Form—I realization, with this realization being illustrated in Fig.
4.5
.
An alternative way to implement the filter is with the so-called coupled form
realization. This form uses the real and imaginary parts of the filters pole locations
explicitly in the transfer function expression. The real part of the pole pair is
rsin(h), the imaginary part of the pole pair is rsin(h) and the coupled-form
expression is:
r sin
ð
h
Þ
1
2r cos
ð
h
Þ
z
1
þ
r
2
z
2
:
H
ð
z
Þ¼
ð
4
:
19
Þ
The coupled-form implementation structure is shown in Fig.
4.6
. To implement
this structure, the coefficients rcos(h) and rsin(h) must be quantized. A plot of all
possible stable pole positions is shown in Fig.
4.7
, assuming that 5-bit quantization
is used. As can be seen in the figure, the pole positions are distributed uniformly
within the unit circle. On the other hand, Fig.
4.8
shows the set of all possible pole
locations when one uses 5-bit quantization in a direct form realization. The pole
locations are seen to be non-uniformly distributed, i.e., there is a bias in the
distribution of pole locations when one uses direct form filter implementations. For
this reason, coupled- form representations give rise to more reliable implemen-
tations than direct form realizations. However, there is a price to be paid for this
advantage, namely, more hardware complexity, as the multipliers are doubled in
number when compared to the direct-form implementation.
Fig. 4.5 Direct form—i
realization of an IIR filter
x
(
n
)
y
(
n
)
+
-
z
1
+
+
a
-
1
-
z
1
a
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