Chapter 3
Delays and Effects
Most acoustic systems have some component where waves can propagate, such
as a membrane, a string, or the air in an enclosure. If propagation in these media
is ideal, i.e., free of losses, dispersion, and nonlinearities, it can be simulated by
delay lines.
A delay line is a linear time-invariant, single-input single-output system,
whose output signal is a copy of the input signal delayed by seconds. In
continuous time, the frequency response of such system is
H
D
s
(j) = e
-j
.
(1)
Equation (1) tells us that the magnitude response is unitary, and that the phase
is linear with slope .
3.1
The Circular Buffer
A discrete-time realization of the system (1) is given by a system that imple-
ments the transfer function
H
D
(z) = z
- F
s
= z
-m
,
(2)
where m is the number of samples of delay. When the delay is an integral
multiple of the sampling quantum, m is an integer number and it is straight-
forward to implement the system (2) by means of a memory buffer. In fact, an
m-samples delay line can be implemented by means of a circular buffer, that is
a set of M contiguous memory cells accessed by a write pointer IN and a read
pointer OUT, such that
IN = (OUT + m)%M ,
(3)
where the symbol % is used for the quotient modulo M . At each sampling
instant, the input is written in the location pointed by IN, the output is taken
from the location pointed by OUT, and the two pointers are updated with
IN
= (IN + 1)%M
OUT = (OUT + 1)%M
.
(4)
In words, the pointers are incremented respecting the circularity of the buffer.
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