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Delay Lines

The delay line is an elementary functional unit which models acoustic propagation delay. It is a fundamental building block of both delay-effects processors and digital-waveguide synthesis models. The function of a delay line is to introduce a time delay between its input and output, as shown in Fig.2.1.

Figure 2.1: The $ M$ -sample delay line.
\includegraphics{eps/delay}

Let the input signal be denoted $ x(n),\, n=0,1,2,\ldots$ , and let the delay-line length be $ M$ samples. Then the output signal $ y(n)$ is specified by the relation

$\displaystyle y(n) = x(n-M),\quad n=0,1,2,\ldots \protect$ (3.1)

where $ x(n)\isdef 0$ for $ n<0$ .

Before the digital era, delay lines were expensive and imprecise in ``analog'' form. For example, ``spring reverberators'' (common in guitar amplifiers) use metal springs as analog delay lines; while adequate for that purpose, they are highly dispersive and prone to noise pick-up. Large delays require prohibitively long springs or coils in analog implementations. In the digital domain, on the other hand, delay by $ N$ samples is trivially implemented, and non-integer delays can be implemented using interpolation techniques, as discussed later in §4.1.



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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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