Delay Lines and Effects
59
antiresonances
non-recursive comb filter
FIR comb
resonator
constant not exceeding 1.
The difference equation is expressed as
y(n) = x(n) + g x(n - m) ,
(14)
and, therefore, the transfer function is
H(z) = 1 + gz
-m
.
(15)
In the case that g = 1, it is easy to see by using the De Moivre formula (see
section A.6) that the frequency response of the comb filter has the following
magnitude and group delay:
|H()| =
2(1 + cos (m))
gr,H
() =
m
2
,
(16)
and it is straightforward to verify that the frequency band ranging from dc to the
Nyquist rate comprises m zeros (antiresonances), equally spaced by F
s
/mHz.
The phase response
3
is piecewise linear with discontinuities of at the odd
multiples of F s/2m.
If g < 1, it is easy to see that the amplitude of the resonances is
P = 1 + g ,
(17)
while the amplitude of the points of minimum (halfway between contiguous
resonances) is
V = 1 - g .
(18)
An important parameter of this filtering structure, called non-recursive comb
filter (or FIR comb), is the peak-to-valley ratio
P
V
=
1 + g
1 - g
.
(19)
Fig. 4 shows the response of a non-recursive comb filter having length m =
11samples and a reflection attenuation g = 0.9. The shape of the frequency
response justifies the name comb given to the filter.
The zeros of the comb filter are evenly distributed along the unit circle at
the m-th roots of -g, as shown in figure 5.
3.4
The Recursive Comb Filter
A simple model of one-dimensional resonator can be constructed using the basic
blocks presented in this and in the preceding chapters. It is composed by an
m-samples delay line, with the incidental fractional part of m obtained by FIR
interpolation or allpass filtering, in feedback loop with an attenuation coefficient
g, possibly replaced by a filter in order to give different decay times at different
frequencies. Let us analyze the whole filtering structure in the case that m is
integer and g is a positive constant not exceeding 1.
The difference equation is expressed as
y(n) = x(n - m) + g y(n - m) ,
(20)
3
The reader is invited to calculate and plot the phase response.
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