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==History==
{{see also|History of wavelets}}
In the early development of time-frequency analysis, the proposed concepts are mostly not applied to signal processing. Instead, they are developed mainly for physics (quantum or acoustics) and mathematics theories.
Early work in time–frequency analysis can be seen in the [[Haar wavelet]]s (1909) of [[Alfréd Haar]], though these were not significantly applied to signal processing. More substantial work was undertaken by [[Dennis Gabor]], such as [[Gabor atom]]s (1947), an early form of [[wavelet]]s, and the [[Gabor transform]], a modified [[short-time Fourier transform]]. The [[Wigner–Ville distribution]] (Ville 1948, in a signal processing context) was another foundational step.
1909 marks the beginning of the development of the wavelet transform families. In that year, [[Alfréd Haar]] proposed the [[Haar transform]] to give an example of an orthonormal system for the space of [[square-integrable functions]] on the [[unit interval]] [0, 1] (The term ''wavelet'' hasn't been invented yet). Later in 1946, [[Dennis Gabor]] proposed [[Gabor atoms]] which are constructed similarly to wavelets and have similar applications. In 1975, George Zweig discovered the [[continuous wavelet transform]]. In 1988, [[Stephane Mallat]] and [[Yves Meyer]] proposed the multiresolution structure of the wavelet transform (which is the backbone of [[fast wavelet transform]]) while [[Ingrid Daubechies]] proposed the compact support [[orthogonal wavelet]]. Since then, the discrete wavelet transform started to be widely used in image processing. In 1992, Wilson et. al. proposed the generalized wavelet transform. In 1996, [[Ingrid Daubechies]] and Maes proposed the synchrosqueezing transform. In about 2000s, various wavelets were developed: [[chirplet]] by Bultan (1999), [[curvelet]] by Donoho and Candes (2000), [[bandlet]] by Mallet and Peyre (2002), [[contourlet]] by Do and Vetterli (2005) and [[shearlet]] by Kutyniok and Labate (2005). JPEG 2000, one of the image compressing standard proposed by ISO, is developed also in 2000.
Particularly in the 1930s and 1940s, early time–frequency analysis developed in concert with [[quantum mechanics]] (Wigner developed the Wigner–Ville distribution in 1932 in quantum mechanics, and Gabor was influenced by quantum mechanics – see [[Gabor atom]]); this is reflected in the shared mathematics of the position-momentum plane and the time–frequency plane – as in the [[Heisenberg uncertainty principle]] (quantum mechanics) and the [[Gabor limit]] (time–frequency analysis), ultimately both reflecting a [[Symplectic geometry|symplectic]] structure.
Another families of time-frequency analysis – [[bilinear time–frequency distribution]] have their development started in 1932. At that time, [[Eugene Wigner]] proposed the [[Wigner distribution function]] to provide quantum correction for classical statistical mechanics in physics. Skipping to 1989, Hyung-Ill Choi and William J. Williams proposed the [[Choi-Williams distribution]]. Later in 1990, Zhao, Atlas, and Marks proposed the cone-shape distribution. Then in 1994, Boashash and O’Shea developed polynomial Wigner-Ville distributions.
An early practical motivation for time–frequency analysis was the development of radar – see [[ambiguity function]].
The short-time Fourier transform families have their development started in 1946. In that year, [[Dennis Gabor]] proposed the [[Gabor transform]]. The shared mathematics (symplectic structure) of the [[Heisenberg uncertainty principle]] (quantum mechanics) in position-momentum plane and the [[Gabor limit]] (time–frequency analysis) in the time–frequency plane is then explored. After 1965, the development of the [[Cooley-Tukey FFT algorithm]] allows faster computations for STFT. Skipping to 1996, Stockwell, Mansinha, and Lowe proposed the [[S transform]]. This is then generalized by Pinnegar and Mansinha in 2003. In 2007, Zhong and Zeng proposed the multiscale STFT while Pei and Ding proposed the Gabor-Wigner transform.
The [[Hilbert-Huang transform]] is yet another time-frequency analysis technique that does not belong to the three families above. It is proposed by [[Norden E. Huang]] in 1998, and see its application developments in signal processing, climate analysis, geology, economics, and speech in the 2000s.
Some useful techniques for filter designs in time-frequency analysis, like the [[fractional Fourier transform]] and [[linear canonical transform]], are developed and connected to signal processing applications in the 1990s and 1970s respectively.
One of the recent application of time-frequency analysis is signal identification with [[deep learning]] technique (2015 ~). Kang et. al. also proposed the wavelet convolutional neural network in 2017.
== See also ==
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