Editing Time–frequency representation (section)

== Wavelet transforms ==
Wavelet transforms, in particular the [[continuous wavelet transform]], expand the signal in terms of wavelet functions which are localised in both time and frequency. Thus the wavelet transform of a signal may be represented in terms of both time and frequency. Continuous wavelet transform analysis is very useful for identifying non-stationary signals in [[time series]],<ref>{{Cite journal |last1=Torrence |first1=Christopher |last2=Compo |first2=Gilbert P. |date=January 1998 |title=A Practical Guide to Wavelet Analysis |url=http://journals.ametsoc.org/doi/10.1175/1520-0477(1998)0792.0.CO;2 |journal=Bulletin of the American Meteorological Society |language=en |volume=79 |issue=1 |pages=61–78 |doi=10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2 |issn=0003-0007}}</ref> such as those related to climate<ref>{{Cite journal |last1=Grinsted |first1=A. |last2=Moore |first2=J. C. |last3=Jevrejeva |first3=S. |date=2004-11-18 |title=Application of the cross wavelet transform and wavelet coherence to geophysical time series |url=https://npg.copernicus.org/articles/11/561/2004/ |journal=Nonlinear Processes in Geophysics |language=English |volume=11 |issue=5/6 |pages=561–566 |doi=10.5194/npg-11-561-2004 |doi-access=free |issn=1023-5809}}</ref> or landslides.<ref>{{Cite journal |last1=Tomás |first1=R. |last2=Li |first2=Z. |last3=Lopez-Sanchez |first3=J. M. |last4=Liu |first4=P. |last5=Singleton |first5=A. |date=2016-06-01 |title=Using wavelet tools to analyse seasonal variations from InSAR time-series data: a case study of the Huangtupo landslide |url=https://doi.org/10.1007/s10346-015-0589-y |journal=Landslides |language=en |volume=13 |issue=3 |pages=437–450 |doi=10.1007/s10346-015-0589-y |issn=1612-5118|hdl=10045/62160 |hdl-access=free }}</ref>

The notions of time, frequency, and amplitude used to generate a TFR from a wavelet transform were originally developed intuitively. In 1992, a quantitative derivation of these relationships was published, based upon a [[stationary phase approximation]].<ref>
{{cite journal
 | journal = IEEE Transactions on Information Theory
 | title = Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies
 | author = Delprat, N., Escudii, B., Guillemain, P., Kronland-Martinet, R., Tchamitchian, P., and Torrksani, B.
 | volume = 38
 | issue = 2
 | pages = 644–664
 | year = 1992
 | doi = 10.1109/18.119728
 | url = https://hal.archives-ouvertes.fr/hal-01222729/document
 }}</ref>