Editing Short-time Fourier transform (section)

== Resolution issues ==
{{Further|Gabor limit|Küpfmüller's uncertainty principle}}

One of the pitfalls of the STFT is that it has a fixed resolution.  The width of the windowing function relates to how the signal is represented—it determines whether there is good frequency resolution (frequency components close together can be separated) or good time resolution (the time at which frequencies change).  A wide window gives better frequency resolution but poor time resolution.  A narrower window gives good time resolution but poor frequency resolution.  These are called narrowband and wideband transforms, respectively.

[[Image:STFT - windows-en.svg|thumb|400px|none|Comparison of STFT resolution. Left has better time resolution, and right has better frequency resolution.]]

This is one of the reasons for the creation of the [[wavelet transform]] and [[multiresolution analysis]], which can give good time resolution for high-frequency events and good frequency resolution for low-frequency events, the combination best suited for many real signals.

This property is related to the [[Werner Heisenberg|Heisenberg]] [[uncertainty principle]], but not directly – see [[Gabor limit]] for discussion.  The product of the standard deviation in time and frequency is limited.  The boundary of the uncertainty principle (best simultaneous resolution of both) is reached with a Gaussian window function (or mask function), as the Gaussian minimizes the [[Fourier uncertainty principle]].  This is called the [[Gabor transform]] (and with modifications for multiresolution becomes the [[Morlet wavelet]] transform).

One can consider the STFT for varying window size as a two-dimensional domain (time and frequency), as illustrated in the example below, which can be calculated by varying the window size. However, this is no longer a strictly time-frequency representation – the kernel is not constant over the entire signal.

=== 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".

=== Explanation ===
It can also be explained with reference to the sampling and [[Nyquist frequency]].

Take a window of ''N''  samples from an arbitrary real-valued signal at sampling rate ''f''<sub>s</sub> .  Taking the Fourier transform produces ''N'' complex coefficients.  Of these coefficients only half are useful (the last ''N/2'' being the [[complex conjugate]] of the first ''N/2'' in reverse order, as this is a real valued signal).

These ''N/2'' coefficients represent the frequencies 0 to ''f''<sub>s</sub>/2 (Nyquist) and two consecutive coefficients are spaced apart by
''f''<sub>s</sub>/''N'' Hz.

To increase the frequency resolution of the window the frequency spacing of the coefficients needs to be reduced.  There are only two variables, but decreasing ''f''<sub>s</sub> (and keeping ''N'' constant) will cause the window size to increase — since there are now fewer samples per unit time.  The other alternative is to increase ''N'', but this again causes the window size to increase.  So any attempt to increase the frequency resolution causes a larger window size and therefore a reduction in time resolution—and vice versa.