In mathematics, a quasiperiodic function is a function that has a certain similarity to a periodic function.[1] A function
f
\omega
f(z+\omega)=g(z,f(z))
g
f
A simple case (sometimes called arithmetic quasiperiodic) is if the function obeys the equation:
f(z+\omega)=f(z)+C
Another case (sometimes called geometric quasiperiodic) is if the function obeys the equation:
f(z+\omega)=Cf(z)
An example of this is the Jacobi theta function, where
\vartheta(z+\tau;\tau)=e-2\pi\vartheta(z;\tau),
shows that for fixed
\tau
\tau
Functions with an additive functional equation
f(z+\omega)=f(z)+az+b
\zeta(z+\omega,Λ)=\zeta(z,Λ)+η(\omega,Λ)
for a z-independent η when ω is a period of the corresponding Weierstrass ℘ function.
In the special case where
f(z+\omega)=f(z)
Λ
Quasiperiodic signals in the sense of audio processing are not quasiperiodic functions in the sense defined here; instead they have the nature of almost periodic functions and that article should be consulted. The more vague and general notion of quasiperiodicity has even less to do with quasiperiodic functions in the mathematical sense.
A useful example is the function:
f(z)=\sin(Az)+\sin(Bz)
If the ratio A/B is rational, this will have a true period, but if A/B is irrational there is no true period, but a succession of increasingly accurate "almost" periods.