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.