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.
Read more about this topic: Almost Periodic Function
Famous quotes containing the words signals, music and/or synthesis:
“The term preschooler signals another change in our expectations of children. While toddler refers to physical development, preschooler refers to a social and intellectual activity: going to school. That shift in emphasis is tremendously important, for it is at this age that we think of children as social creatures who can begin to solve problems.”
—Lawrence Kutner (20th century)
“The train was crammed, the heat stifling. We feel out of sorts, but do not quite know if we are hungry or drowsy. But when we have fed and slept, life will regain its looks, and the American instruments will make music in the merry cafe described by our friend Lange. And then, sometime later, we die.”
—Vladimir Nabokov (1899–1977)
“The spider-mind acquires a faculty of memory, and, with it, a singular skill of analysis and synthesis, taking apart and putting together in different relations the meshes of its trap. Man had in the beginning no power of analysis or synthesis approaching that of the spider, or even of the honey-bee; but he had acute sensibility to the higher forces.”
—Henry Brooks Adams (1838–1918)