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Variable Delay Lines

Time varying delay lines are fundamental building blocks for delay effects, synthesis algorithms, and computational acoustic models of musical instruments.

Let A denote an array of length $ N$ . Then we can implement an $ M$ -sample variable delay line in the C programming language as shown in Fig.5.1. We require, of course, $ M\leq N$ .

Figure 5.1: The $ M$ -sample variable delay line using separate read- and write-pointers.

   static double A[N];
   static double *rptr = A; // read ptr
   static double *wptr = A; // write ptr

   double setdelay(int M) {
       rptr = wptr - M;
       while (rptr < A) { rptr += N }

   double delayline(double x)
     double y;
     A[wptr++] = x; 
     y = A[rptr++];
     if ((wptr-A) >= N) { wptr -= N }
     if ((rptr-A) >= N) { rptr -= N }
     return y;

The Synthesis Tool Kit, Version 4 [86] contains the C++ class ``Delay'' which implements this type of variable (but non-interpolating) delay line. There are additional subclasses which provide interpolating reads by various methods. In particular, the class DelayL implements continuously variable delay lengths using linear interpolation. The code listing in Fig.5.1 can be modified to use linear interpolation by replacing the line

  y = A[rptr++];
  long rpi = (long)floor(rptr);
  double a = rptr - (double)rpi;
  y = a * A[rpi] + (1-a) * A[rpi+1];
  rptr += 1;

To implement a continuously varying delay, we add a ``delay growth parameter'' g to the delayline function in Fig.5.1, and change the line

  rptr += 1; // pointer update
above to
  rptr += 1 - g; // pointer update
When g is 0, we have a fixed delay line. When $ \texttt{g}>0$ , the delay grows $ \texttt{g}$ samples per sample, which we may also interpret as seconds per second, i.e., $ {\dot D_t}=\texttt{g}$ . In §5.7.2, this will be applied to simulation of the Doppler effect.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University