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.
Let the input signal be denoted , and let the delay-line length be samples. Then the output signal is specified by the relation
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 samples is trivially implemented, and non-integer delays can be implemented using interpolation techniques, as discussed later in §4.1.