Editing Time–frequency analysis (section)

==Motivation==

In [[signal processing]], time–frequency analysis<ref>P. Flandrin, "Time–frequency/Time–Scale Analysis," ''Wavelet Analysis and its Applications'', Vol. 10 ''Academic Press'', San Diego, 1999.</ref> is a body of techniques and methods used for characterizing and manipulating signals whose statistics vary in time, such as [[Transient (acoustics)|transient]] signals.

It is a generalization and refinement of [[Fourier analysis]], for the case when the signal frequency characteristics are varying with time. Since many signals of interest – such as speech, music, images, and medical signals – have changing frequency characteristics, time–frequency analysis has broad scope of applications.

Whereas the technique of the [[Fourier transform]] can be extended to obtain the frequency spectrum of any slowly growing [[locally integrable]] signal, this approach requires a complete description of the signal's behavior over all time.  Indeed, one can think of points in the (spectral) frequency domain as smearing together information from across the entire time domain.  While mathematically elegant, such a technique is not appropriate for analyzing a signal with indeterminate future behavior.  For instance, one must presuppose some degree of indeterminate future behavior in any telecommunications systems to achieve non-zero entropy (if one already knows what the other person will say one cannot learn anything).

To harness the power of a frequency representation without the need of a complete characterization in the time domain, one first obtains a time–frequency distribution of the signal, which represents the signal in both the time and frequency domains  simultaneously.  In such a representation the frequency domain will only reflect the behavior of a temporally localized version of the signal.  This enables one to talk sensibly about signals whose component frequencies vary in time.

For instance rather than using [[tempered distributions]] to globally transform the following function into the frequency domain one could instead use these methods to describe it as a signal with a time varying frequency.

: <math>x(t)=\begin{cases}
\cos(  \pi t);  & t  <10 \\
\cos(3 \pi t);  & 10 \le t < 20 \\
\cos(2 \pi t);  & t  > 20
\end{cases}</math>
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Once such a representation has been generated other techniques in time–frequency analysis may then be applied to the signal in order to extract information from the signal, to separate the signal from noise or interfering signals, etc.