Digital Signal Processing Reference
In-Depth Information
Table 16.3. Pole locations for lowpass IIR filter specified as item
(1) in the list of filters in Section 16.5.1 before and after coefficient
quantization
Before quantization
After aquantization
0
.
906248860
+
j0
.
374726030
1
.
052267965
+
j0
.
282343949
0
.
906248860
−
j0
.
374726030
1
.
052267965
−
j0
.
282343949
0
.
868476456
+
j0
.
325406471
0
.
884886889
+
j0
.
435649276
0
.
868476456
−
j0
.
325406471
0
.
884886889
−
j0
.
435649276
0
.
816276165
+
j0
.
206545545
0
.
720252455
+
j0
.
304944386
0
.
816276165
−
j0
.
206545545
0
.
720252455
−
j0
.
304944386
0.784371333
0.651185382
(3) Bandpass filter (Example 16.6):
H
(
z
)
=
0.001(8
.
317
z
8
−
6
.
94
z
7
+
4
.
236
z
6
−
5
.
952
z
5
+
13
.
52
z
4
−
5.952
z
3
+
4
.
236
z
2
−
6
.
94
z
+
8
.
317)
z
8
−
1
.
389
z
7
+
3
.
714
z
6
−
3
.
356
z
5
+
4
.
685
z
4
−
2
.
693
z
3
+
2
.
397
z
2
−
0
.
7107
z
+
0
.
4106
.
(4) Bandstop filter (Example 16.7):
H
(
z
)
=
0
.
2887
z
8
−
0
.
4484
z
7
+
1
.
363
z
6
−
1
.
372
z
5
+
2
.
149
z
4
−
1
.
372
z
3
+
1
.
363
z
2
−
0
.
4484
z
+
0
.
2887
z
8
−
1
.
096
z
7
+
1
.
977
z
6
−
1
.
519
z
5
+
1
.
78
z
4
−
0
.
8638
z
3
+
0
.
6172
z
2
−
0
.
1739
z
+
0
.
09751
.
The poles and zeros of the four filters are plotted separately in Figs. 16.11(a)-
(d). Since in all cases the poles lie within the unit circle, the four filters are
absolutely BIBO stable when they are implemented with full precision.
Now, let us consider the effect of quantization on the stability of the lowpass
filter. Although most digital systems use binary arithmetic, we will use decimal
arithmetic for simplicity and assume that the coefficients of the lowpass filter
(item (1) above) are implemented up to an accuracy of three decimal places
leading to the following approximated transfer function:
0.001(4
z
7
−
14
z
6
+
21
z
5
−
11
z
4
−
11
z
3
+
21
z
2
−
14
z
+
4)
z
7
−
5
.
966
z
6
+
15
.
538
z
5
−
22
.
859
z
4
+
20
.
494
z
3
−
11
.
188
z
2
+
3
.
442
z
−
0
.
46
.
H
(
z
)
=
Although the filters
H
(
z
) and
H
(
z
) look similar, they are not identical. The
location of poles can be found by calculating the roots of the characteristic
equations of
H
(
z
) and
H
(
z
), and these are listed in Table 16.3. The pole-zero
locations are shown in Fig. 16.12. It is observed that the two poles in
H
(
z
),
which lie close to (but inside) the unit circle, moved outside the unit circle after
coefficient quantization. Therefore, although
H
(
z
) behaves as a lowpass filter
after quantization, the filter is no longer absolutely BIBO stable.
Different implementations of IIR filters can be compared to determine relative
stability by observing how close the poles lie to the unit circle. The highpass
filter, the with pole-zero plot shown in Fig. 16.11(b), has four poles, which are
well inside the unit circle. The pole-zero plot of the bandpass filter is shown
in Fig. 16.11(c). Four of the eight poles in the bandpass filter are close to the
unit circle, which reduces its relative stability. The bandstop filter has eight
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