Digital Signal Processing Reference
In-Depth Information
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(a) (b)
Figure 6.2
Examples of pole-zero patterns: (a) real-valued IIR filter (note the sym-
metry around the x-axis); (b) linear phase FIR (each zero appears with its recipro-
cal).
Linear-Phase FIR Filters.
First of all, note that the pole-zero plot for an
FIR filter is actually just a zero plot, since FIR's have no poles.
(2)
Aparticu-
larly important case is that of linear phase FIR filters; as we will see in detail
in Section 7.2.2, linear phase imposes some symmetry constraints on the
CCDE coefficients (which, of course, coincide with the filter taps). These
constraints have a remarkable consequence: if
z
0
is a (complex) zero of the
system, 1
z
0
is a zero as well. Since we consider real-valued FIR filters ex-
clusively, the presence of a complex zero in
z
0
implies the existence of three
other zeros, namely in 1
/
z
0
,
z
∗
0
and 1
z
∗
0
(Fig. 6.2b). See also the discussion
/
/
in Section 7.2.2
6.2.2
Pole-Zero Cancellation
We have seen in Section 5.2.1 that the effect of a cascade of two or more
filters is that of a single filter whose impulse response is the convolution of
all of the filters' impulse responses. By the convolution theorem, this means
that the overall transfer function of a cascade of
K
filters
i
,
i
=
1,...,
K
is
simply the product of the single transfer functions
H
i
(
z
)
:
K
H
(
z
)=
H
i
(
z
)
i
=
1
If all filters are realizable, we can consider the factored form of each
H
i
)
as in (6.15). In the product of transfer functions, it may happen that some of
(
z
(2)
Technically, since we use the notation
z
−
1
to express a delay, causal FIR filters have a
multiple pole in the origin (
z
=
0). This is of no consequence for stability, however, so
we will not consider it further.
