IIR filter design - Continuous and Discrete Time Signals and Systems - page 737

Fig. 16.10. Magnitude response
of the DT bandstop filter
designed in Example 16.7.

0

−20

−40

−60

W

− p −0.75 p −0.5 p −0.25 p 0 0.25 p 0.5 p 0.75 pp

>> [numz,denumz]=bilinear(nums,denums,0.5); % DT Filter

The resulting DT filter is given by

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

.

Figure 16.10 shows the magnitude response of the designed bandstop filter. We
observe that both the pass-band and stop-band specifications are satisfied by
the bandstop filter.

16.5 IIR and FIR filters

A classical problem in the design of digital filters is the selection between FIR
and IIR filters since both types of filters can be used to satisfy a given set of
specifications. In this section, we compare IIR and FIR filters with respect to
three criteria: stability, implementation complexity, and delay.

16.5.1 Stability

Stability is a major concern in the design of filters. When designing digital
filters, care must be taken to ensure that the designed filters are absolutely
BIBO stable to prevent infinite outputs. Recall that an LTID system is stable if
its poles lie inside the unit circle in the z-plane. Since the only poles in FIR filters
lie at the origin ( z = 0), FIR filters are always BIBO stable. On the other hand,
IIR filters have non-trivial poles because of the feedback loops and therefore
may run into stability issues.
Use of finite-precision DSP boards places a severe limitation on the type of
IIR filters that can be used. Even if the designed IIR filter is stable, quantization
of the filter coefficients can adversely affect its stability. To illustrate the effect
of quantization on the stability of the filter, consider the following four filters.
(1) Lowpass filter (arbitrary):

H ( z ) = 0.001(3 . 5747 z 7 − 13 . 649 z 6 + 20 . 9446 z 5 − 10 . 7188 z 4 − 10 . 7188 z 3 + 20 . 9446 z 2 − 13 . 649 z + 3 . 5747)
z 7 − 5 . 9664 z 6 + 15 . 5383 z 5 − 22 . 8594 z 4 + 20 . 49 z 3 − 11 . 1881 z 2 + 3 . 4416 z − 0 . 46

.

(2) Highpass filter (Example 16.5):
H ( z ) = 0 . 1725 z 4 − 0 . 6539 z 3 + 0 . 9638 z 2 − 0 . 6539 z + 0 . 1725
z 4 − 0 . 6829 z 3 + 0 . 7518 z 2 − 0 . 138 z + 0 . 0468

.