Editing Almost periodic function (section)

== Quasiperiodic signals in audio and music synthesis ==

In [[speech processing]], [[audio signal processing]], and [[synthesizer|music synthesis]], a '''quasiperiodic''' signal, sometimes called a '''quasiharmonic''' signal, is a [[waveform]] that is virtually [[Frequency|periodic]] microscopically, but not necessarily periodic macroscopically. This does not give a [[quasiperiodic function]], 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 [[Harmonic series (music)#Partial|partial]]s or [[overtone]]s are [[harmonic]] (that is all overtones are at frequencies that are an integer multiple of a [[fundamental frequency]] of the tone).

When a signal <math> x(t) \ </math> is '''fully periodic''' with period <math> P \ </math>, then the signal exactly satisfies

: <math> x(t) = x(t + P) \qquad \forall t \in \mathbb{R} </math>

or

: <math> \Big| x(t) - x(t + P) \Big| = 0 \qquad \forall t \in \mathbb{R}. \ </math>

The [[Fourier series]] representation would be

: <math>x(t) = a_0 + \sum_{n=1}^\infty \big[a_n\cos(2 \pi n f_0 t) - b_n\sin(2 \pi n f_0 t)\big]</math>

or

: <math>x(t) = a_0 + \sum_{n=1}^\infty r_n\cos(2 \pi n f_0 t + \varphi_n)</math>

where <math> f_0 = \frac{1}{P} </math> is the fundamental frequency and the Fourier coefficients are

:<math>a_0 = \frac{1}{P} \int_{t_0}^{t_0+P} x(t) \, dt  \ </math>

:<math>a_n = r_n \cos \left( \varphi_n \right) = \frac{2}{P} \int_{t_0}^{t_0+P} x(t) \cos(2 \pi n f_0 t) \, dt  \qquad n \ge 1 </math>

:<math>b_n = r_n \sin \left( \varphi_n \right)  = - \frac{2}{P} \int_{t_0}^{t_0+P} x(t) \sin(2 \pi n f_0 t) \, dt \ </math>

:where  <math> t_0 \ </math> can be any time:  <math> -\infty < t_0 < +\infty \ </math>.

The [[fundamental frequency]] <math> f_0 \ </math>, and Fourier [[coefficient]]s <math> a_n \ </math>,  <math> b_n \ </math>, <math> r_n \ </math>, or <math> \varphi_n \ </math>,  are constants, i.e. they are not functions of time. The harmonic frequencies are exact integer multiples of the fundamental frequency.

When <math> x(t) \ </math> is '''quasiperiodic''' then

: <math> x(t) \approx x \big( t + P(t) \big) \ </math>

or

: <math> \Big| x(t) - x \big( t + P(t) \big) \Big| < \varepsilon \ </math>

where

: <math> 0 < \epsilon \ll \big \Vert x \big \Vert = \sqrt{\overline{x^2}} = \sqrt{ \lim_{\tau \to \infty} \frac{1}{\tau} \int_{-\tau/2}^{\tau/2} x^2(t)\, dt }. \ </math>

Now the Fourier series representation would be

: <math>x(t) = a_0(t) \ + \ \sum_{n=1}^\infty \left[a_n(t)\cos \left(2 \pi n \int_{0}^{t} f_0(\tau)\, d\tau \right) - b_n(t)\sin \left( 2 \pi n \int_0^t  f_0(\tau)\, d\tau \right) \right]</math>

or

: <math>x(t) = a_0(t) \ + \ \sum_{n=1}^\infty r_n(t)\cos \left( 2 \pi n \int_0^t f_0(\tau)\, d\tau + \varphi_n(t) \right) </math>

or

: <math>x(t) = a_0(t) + \sum_{n=1}^\infty r_n(t)\cos \left( 2 \pi \int_0^t f_n(\tau)\, d\tau + \varphi_n(0) \right)</math>

where <math> f_0(t) = \frac{1}{P(t)} </math> is the possibly ''time-varying'' fundamental frequency and the  ''time-varying'' Fourier coefficients are

:<math>a_0(t) = \frac{1}{P(t)} \int_{t-P(t)/2}^{t+P(t)/2} x(\tau) \, d\tau  \  </math>

:<math>a_n(t) = r_n(t) \cos\big(\varphi_n(t)\big) = \frac{2}{P(t)} \int_{t-P(t)/2}^{t+ P(t)/2} x(\tau) \cos\big( 2 \pi n f_0(t) \tau \big) \, d\tau  \qquad n \ge 1  </math>

:<math>b_n(t) = r_n(t) \sin\big(\varphi_n(t)\big) = -\frac{2}{P(t)} \int_{t-P(t)/2}^{t+P(t)/2} x(\tau) \sin\big( 2 \pi n f_0(t) \tau \big) \, d\tau  \  </math>

and the [[instantaneous phase#Instantaneous frequency|instantaneous frequency]] for each [[Harmonic series (music)#Partial|partial]] is

:<math> f_n(t) = n f_0(t) + \frac{1}{2 \pi} \varphi_n^\prime(t). \, </math>

Whereas in this quasiperiodic case, the fundamental frequency <math> f_0(t) \ </math>, the harmonic frequencies <math> f_n(t) \ </math>, and the Fourier coefficients <math> a_n(t) \ </math>,  <math> b_n(t) \ </math>,  <math> r_n(t) \ </math>, or <math> \varphi_n(t) \ </math> 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 <math> x(t) \ </math> to be considered to be quasiperiodic.

The partial frequencies <math> f_n(t) \ </math> are very nearly harmonic but not necessarily exactly so.  The time-derivative of <math> \varphi_n(t) \ </math>, that is <math> \varphi_n^\prime(t) \ </math>, has the effect of detuning the partials from their exact integer harmonic value <math> n f_0(t) \ </math>.  A rapidly changing <math> \varphi_n(t) \ </math> means that the instantaneous frequency for that partial is severely detuned from the integer harmonic value which would mean that <math> x(t) \ </math> is not quasiperiodic.