22
D. Rocchesso: Sound Processing
zeros of the filter
poles of the filter
points where the transfer function vanishes, and the points where it diverges to
infinity. Let us rewrite the transfer function as the ratio of two polynomials in
z
H(z) =
1
2
z - z
0
z
,
(10)
where z
0
= -1 is the root of the numerator. The roots of the numerator of a
transfer function are called zeros of the filter, and the roots of the denominator
are called poles of the filter. Usually, for reasons that will emerge in the fol-
lowing, only the nonzero roots are counted as poles or zeros. Therefore, in the
example (10) we have only one zero and no pole.
In order to evaluate the frequency response of the filter it is sufficient to
replace the variable z with e
j
and to consider e
j
as a geometric vector whose
head moves along the unit circle. The difference between this vector and the
vector z
0
gives the cord drawn in fig. 2. The cord length doubles
the magnitude
response of the filter. Such a chord, interpreted as a vector with the head in e
j
,
has an angle that can be subtracted from the vector angle of the pole at the
origin, thus giving the phase response of the filter at the frequency .
-1.5
-1
-0.5
0
0.5
1
1.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Figure 2: Single zero () and pole in the origin (×)
The following general rules can be given, for any number of poles and zeros:
· Considered a point e
j
on the unit circle, the magnitude of the frequency
response (regardless of constant factors) at the frequency is obtained by
multiplication of the magnitudes of the vectors linking the zeros with the
point e
j
, divided by the magnitudes of the vectors linking the poles with
the point e
j
.
· The phase response is obtained by addition of the phases of the vectors
linking the zeros with the point e
j
, and by subtraction of the phases of
the vectors linking the poles with the point e
j
.
It is readily seen that poles or zeros in the origin do only contribute to the phase
of the frequency response, and this is the reason for their exclusion from the
total count of poles and zeros.
The graphic method, based on pole and zero placement on the complex plane
is very useful to have a rough idea of the frequency response. For instance, the
reader is invited to reconstruct fig. 1 qualitatively using the graphic method..
3
Do not forget the scaling factor
1
2