Editing Short-time Fourier transform (section)

=== Examples ===
When the original function is:
[[File:Window B.png|thumb|]]
:<math>X(t,f) = \int^\infty_{-\infty}w(t-\tau) x(\tau) e^{-j 2 \pi f \tau} d\tau</math>

We can have a simple example:

w(t) = 1     for |t| smaller than or equal B

w(t) = 0     otherwise

B = window

Now the original function of the Short-time Fourier transform can be changed as

:<math>X(t,f) = \int^{t+B}_{t-B}x(\tau) e^{-j 2 \pi f \tau} d\tau</math>

Another example:

Using the following sample signal <math>x(t)</math> that is composed of a set of four sinusoidal waveforms joined together in sequence.  Each waveform is only composed of one of four frequencies (10, 25, 50, 100 [[hertz|Hz]]).  The definition of <math>x(t)</math> is:

:<math>x(t)=\begin{cases}
\cos (2 \pi 10 t)  & 0\,\mathrm{s}  \le t < 5  \,\mathrm{s} \\
\cos (2 \pi 25 t)  & 5\,\mathrm{s}  \le t < 10\,\mathrm{s} \\
\cos (2 \pi 50 t)  & 10\,\mathrm{s} \le t < 15\,\mathrm{s} \\
\cos (2 \pi 100 t) & 15\,\mathrm{s} \le t < 20\,\mathrm{s} \\
\end{cases}</math>

Then it is sampled at 400&nbsp;Hz. The following spectrograms were produced:

{|
|-
|[[Image:STFT colored spectrogram 25ms.png|thumb|300px|25 ms window]]
|[[Image:STFT colored spectrogram 125ms.png|thumb|300px|125 ms window]]
|-
|[[Image:STFT colored spectrogram 375ms.png|thumb|300px|375 ms window]]
|[[Image:STFT colored spectrogram 1000ms.png|thumb|300px|1000 ms window]]
|-
|}

{{clear}}

The 25&nbsp;ms window allows us to identify a precise time at which the signals change but the precise frequencies are difficult to identify.  At the other end of the scale, the 1000 ms window allows the frequencies to be precisely seen but the time between frequency changes is blurred.

Other examples:
[[File:Gausian B.png|thumb|]]
:<math>w(t) = exp(\sigma-t^{2})</math>

Normally we call <math>exp(\sigma-t^{2})</math> a [[Gaussian function]] or Gabor function. When we use it, the short-time Fourier transform is called the "Gabor transform".