Editing Short-time Fourier transform (section)

=== Continuous-time STFT ===
Given the width and definition of the window function ''w''(''t''), we initially require the area of the window function to be scaled so that

:<math> \int_{-\infty}^{\infty} w(\tau) \, d\tau  = 1.</math>

It easily follows that

:<math> \int_{-\infty}^{\infty} w(t-\tau) \, d\tau  = 1 \quad \forall \ t </math>

and

:<math> x(t) = x(t) \int_{-\infty}^{\infty} w(t-\tau) \, d\tau  = \int_{-\infty}^{\infty} x(t) w(t-\tau) \, d\tau. </math>

The continuous Fourier transform is

:<math> X(\omega) = \int_{-\infty}^{\infty} x(t) e^{-i \omega t} \, dt. </math>

Substituting ''x''(''t'') from above:

:<math> X(\omega) = \int_{-\infty}^{\infty} \left[ \int_{-\infty}^{\infty} x(t) w(t-\tau) \, d\tau \right] \, e^{-i \omega t} \, dt </math>

:::<math> = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} x(t) w(t-\tau) \, e^{-i \omega t} \, d\tau \, dt. </math>

Swapping order of integration:

:<math> X(\omega) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} x(t) w(t-\tau) \, e^{-i \omega t} \, dt \, d\tau </math>

:::<math> = \int_{-\infty}^{\infty} \left[ \int_{-\infty}^{\infty} x(t) w(t-\tau) \, e^{-i \omega t} \, dt \right] \, d\tau </math>

:::<math> = \int_{-\infty}^{\infty} X(\tau, \omega) \, d\tau. </math>

So the Fourier transform can be seen as a sort of phase coherent sum of all of the STFTs of ''x''(''t'').  Since the inverse Fourier transform is

:<math> x(t)  = \frac{1}{2 \pi} \int_{-\infty}^{\infty} X(\omega) e^{+i \omega t} \, d\omega, </math>

then ''x''(''t'') can be recovered from ''X''(τ,ω) as

:<math> x(t)  = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} X(\tau, \omega) e^{+i \omega t} \, d\tau \, d\omega. </math>

or

:<math> x(t)  = \int_{-\infty}^{\infty} \left[ \frac{1}{2 \pi} \int_{-\infty}^{\infty} X(\tau, \omega) e^{+i \omega t} \, d\omega \right] \, d\tau. </math>

It can be seen, comparing to above that windowed "grain" or "wavelet" of ''x''(''t'') is

:<math> x(t) w(t-\tau)  = \frac{1}{2 \pi} \int_{-\infty}^{\infty} X(\tau, \omega) e^{+i \omega t} \, d\omega. </math>

the inverse Fourier transform of ''X''(τ,ω) for τ fixed.

An alternative definition that is valid only in the vicinity of τ, the inverse transform is:
:<math>x(t) = \frac{1}{w(t-\tau)}\frac{1}{2 \pi} \int_{-\infty}^{\infty} X(\tau, \omega) e^{+i \omega t} \, d\omega.</math>

In general, the window function <math>w(t)</math> has the following properties:

:(a) even symmetry: <math>w(t) = w(-t)</math>;<br />
:(b) non-increasing (for positive time): <math>w(t) \geq w(s)</math> if <math>|t| \leq |s|</math>;<br />
:(c) compact support: <math>w(t)</math> is equal to zero when |t| is large.