Delay Lines

The delay line is an elementary functional unit which models acoustic propagation delay. It is a fundamental building block of both digital waveguide models and delay effects processors. The function of a delay line is to introduce a time delay between its input and output, as shown in Fig. 1.1.

Figure 1.1: The $M$-sample delay line.

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

$$y(n) = x(n-M),\quad n=0,1,2,\ldots \protect$$ (2.1)

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 §3.2.