Editing Time–frequency analysis (section)

==Time–frequency distribution functions==
{{main|Time–frequency distribution}}

===Formulations===

There are several different ways to formulate a valid time–frequency distribution function, resulting in several well-known time–frequency distributions, such as:

*[[Short-time Fourier transform]] (including the [[Gabor transform]]),
*[[Wavelet transform]],
*[[Bilinear time–frequency distribution]] function ([[Wigner distribution function]], or WDF),
*[[Modified Wigner distribution function]], Gabor–Wigner distribution function, and so on (see [[Gabor–Wigner transform]]).
*[[Hilbert–Huang transform]]

More information about the history and the motivation of development of time–frequency distribution can be found in the entry [[Time–frequency representation]].

===Ideal TF distribution function===

A time–frequency distribution function ideally has the following properties:{{Citation needed|date=October 2010}}

#'''High resolution''' in both time and frequency, to make it easier to be analyzed and interpreted.
#'''No cross-term''' to avoid confusing real components from artifacts or noise.
#'''A list of desirable mathematical properties''' to ensure such methods benefit real-life application.
#'''Lower computational complexity''' to ensure the time needed to represent and process a signal on a time–frequency plane allows real-time implementations.

Below is a brief comparison of some selected time–frequency distribution functions.<ref>{{Cite journal|last1=Shafi|first1=Imran|last2=Ahmad|first2=Jamil|last3=Shah|first3=Syed Ismail|last4=Kashif|first4=F. M.|date=2009-06-09|title=Techniques to Obtain Good Resolution and Concentrated Time-Frequency Distributions: A Review|journal=EURASIP Journal on Advances in Signal Processing|language=en|volume=2009|issue=1|pages=673539|doi=10.1155/2009/673539|bibcode=2009EJASP2009..109S |issn=1687-6180|doi-access=free|hdl=1721.1/50243|hdl-access=free}}</ref>

{| class="wikitable"
|-
|
| '''Clarity'''
| '''Cross-term'''
| '''Good mathematical properties'''{{Clarify|date=January 2011}}
| '''Computational complexity'''
|-
| '''Gabor transform'''
| Worst
| No
| Worst
| Low
|-
| '''Wigner distribution function'''
| Best
| Yes
| Best
| High
|-
| '''Gabor–Wigner distribution function'''
| Good
| Almost eliminated
| Good
| High
|-
| '''[[Cone-shape distribution function]]'''
| Good
| No (eliminated, in time)
| Good
| Medium (if recursively defined)
|}
To analyze the signals well, choosing an appropriate time–frequency distribution function is important. Which time–frequency distribution function should be used depends on the application being considered, as shown by reviewing a list of applications.<ref>A. Papandreou-Suppappola, Applications in Time–Frequency Signal Processing (CRC Press, Boca Raton, Fla., 2002)</ref> The high clarity of the Wigner distribution function (WDF) obtained for some signals is due to the auto-correlation function inherent in its formulation; however, the latter also causes the cross-term problem. Therefore, if we want to analyze a single-term signal, using the WDF may be the best approach; if the signal is composed of multiple components, some other methods like the Gabor transform, Gabor-Wigner distribution or Modified B-Distribution functions may be better choices.

As an illustration, magnitudes from non-localized Fourier analysis cannot distinguish the signals:

: <math>x_1 (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>

: <math>x_2 (t)=\begin{cases}
\cos(  \pi t);  & t  <10 \\
\cos(2 \pi t);  & 10 \le t < 20 \\
\cos(3 \pi t);  & t  > 20
\end{cases}</math>
[[File:X1-x2.jpg|thumb]]


But time–frequency analysis can.