A Simple Physical Model

• Waves of sound travel with a velocity of approximately $c$ = 345 meters / second in air.

• The propagation of sound over a distance $d$ thus involves a time delay of $d/c$ seconds.

  Figure 1: A digital delay line of length $M$.

  

• A digital delay line, as represented in Fig. 1, is defined by the difference equation
  $$y[n] = x[n - M], \hspace{0.2in} n = 0, 1, 2, \ldots,$$
  where the input signal $x[n] \ensuremath{\stackrel{\Delta}{=}}0$ for $n < 0$, $y[n]$ is the output signal and $M$ is the delay length in samples.

• Thus, a digital delay line can be interpreted as a discrete-time physical model of traveling-wave propagation over a distance $d$, where $M = f_{s} d / c$ is the number of discrete-time samples corresponding to a distance $d$ at a sample rate $f_{s}$.