Prime Power Delay-Line Lengths

When the delay-line lengths need to be varied in real time, or interactively in a GUI, it is convenient to choose each delay-line length as an integer power of a distinct prime number [457]:

where we call the ``multiplicity'' of the prime . With this choice, the delay-line lengths are always coprime (no factors in common other than ), and yet we can lengthen or shorten each delay line individually (by factors of ) without affecting the mutually prime property.

Suppose we are initially given desired delay-line lengths arranged in ascending order so that

Then good prime-power approximations can be expected using the prime numbers in their natural order:

Since (for any logarithmic base), an optimal (in some sense) choice of prime multiplicity is

where is the desired length in samples. That is, can be simply obtained by rounding to the nearest integer (max 1). The prime-power delay-line length approximation is then of course

and the multiplicative approximation error is bounded by (when ).

This prime-power length scheme is used to keep 16 delay lines both variable and mutually prime in Faust's reverb_designer.dsp programming example (via the function prime_power_delays in effect.lib).

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.