Almost Periodic Function - Quasiperiodic Signals in Audio and Music Synthesis

Quasiperiodic Signals in Audio and Music Synthesis

In speech processing, audio signal processing, and music synthesis, a quasiperiodic signal, sometimes called a quasiharmonic signal, is a waveform that is virtually periodic microscopically, but not necessarily periodic macroscopically. This does not give a quasiperiodic function in the sense of the Wikipedia article of that name, but something more akin to an almost periodic function, being a nearly periodic function where any one period is virtually identical to its adjacent periods but not necessarily similar to periods much farther away in time. This is the case for musical tones (after the initial attack transient) where all partials or overtones are harmonic (that is all overtones are at frequencies that are an integer multiple of a fundamental frequency of the tone).

When a signal is fully periodic with period, then the signal exactly satisfies

or

The Fourier series representation would be

or

where is the fundamental frequency and the Fourier coefficients are

where can be any time: .

The fundamental frequency, and Fourier coefficients, or, are constant, not functions of time. The harmonic frequencies are exact integer multiples of the fundamental frequency.

When is quasiperiodic then

or

where

Now the Fourier series representation would be

or

or

where is the possibly time-varying fundamental frequency and the Fourier coefficients are

and the instantaneous frequency for each partial is

Whereas in this quasiperiodic case, the fundamental frequency, the harmonic frequencies, and the Fourier coefficients, or are not necessarily constant, and are functions of time albeit slowly varying functions of time. Stated differently these functions of time are bandlimited to much less than the fundamental frequency for to be considered to be quasiperiodic.

The partial frequencies are very nearly harmonic but not necessarily exactly so. The time-derivative of, that is, has the effect of detuning the partials from their exact integer harmonic value . A rapidly changing means that the instantaneous frequency for that partial is severely detuned from the integer harmonic value which would mean that is not quasiperiodic.