The zeros of the transfer function
of a linear-phase filter lie in specific configurations.
We can write the symmetry condition
in the
domain. Taking the
-transform of both sides gives
Recall that we are assuming that
is real-valued. If
is a zero of
,
then
(Because the roots of a polynomial with real coefficients
exist in complex-conjugate pairs.)
Using the symmetry condition,
, it follows that
and
or
If
is a zero of a (real-valued) linear-phase filter, then so
are
,
, and
.
Zeros locations
It follows that
generic zeros of a linear-phase filter exist in sets of 4.
zeros on the unit circle (
) exist in sets of 2. (
)
zeros on the real line (
) exist in sets of 2. (
)
zeros at 1 and -1 do not imply the existence of zeros at
other specific points.
Zero locations: automatic zeros
The frequency response
of a Type II FIR filter always has a zero at
:
always for Type II filters.
Similarly, we can derive the following rules for Type III and
Type IV FIR filters.
always for Type III filters.
always for Type IV filters.
The automatic zeros can also be derived using the
characteristics of the amplitude response
seen earlier.
Type
automatic zeros
I
II
III
IV
Zero locations: examples
The Matlab command
zplane can be used to
plot the zero locations of FIR filters.
Note that the zero locations satisfy the properties noted
previously.