60
D. Rocchesso: Sound Processing
resonances
0
1
2
3
0
1
2
frequency [rad/sample]
magnitude
Figure 4: Magnitude of the frequency response of the comb FIR filter having
coefficient g = 0.9 and delay length m = 11
-1
0
1
-1
-0.5
0
0.5
1
Re
Im
Figure 5: Zeros and poles of an FIR comb filter
and the transfer function is
H(z) =
z
-m
1 - gz
-m
.
(21)
Whenever g < 1, the stability is ensured. In the case that g = 1, the frequency
response of the filter has the following magnitude and group delay:
|H()| =
1
2 sin (m/2)
gr,H
() =
m
2
,
(22)
and it is easy to verify that the frequency band ranging from dc to the Nyquist
rate comprises m vertical asymptotes (resonances), equally spaced by F
s
/mHz.
If g = 1 the filter is at the limit of stability, and this is the only case when the
phase response is piecewise linear
, starting with the value -/2 at dc, with
discontinuities of at the even multiples of F
s
/2m.
4
The reader is invited to calculate and plot the phase response.