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
(j) = e
Equation (1) tells us that the magnitude response is unitary, and that the phase
is linear with slope .
The Circular Buffer
A discrete-time realization of the system (1) is given by a system that imple-
ments the transfer function
(z) = z
- F
= z
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 ,
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 + 1)%M
OUT = (OUT + 1)%M
In words, the pointers are incremented respecting the circularity of the buffer.
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