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.

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.