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{{Short description|Integral transform}}
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[[File:Continuous wavelet transform.svg|thumb|320px|right|Continuous [[wavelet]] transform of frequency breakdown signal. Used [[symlet]] with 5 vanishing moments.]]
In [[mathematics]], the '''continuous wavelet transform''' ('''CWT''') is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the [[wavelet]]s vary continuously.
==Definition==
The continuous wavelet transform of a function <math>x(t)</math> at a scale
:<math display="block">X_w(a,b)=\frac{1}{|a|^{1/2}} \int_{-\infty}^\infty x(t)\overline\psi\left(\frac{t-b}{a}\right)\,
where <math>\psi(t)</math> is a continuous function in both the time ___domain and the frequency ___domain called the mother wavelet and the overline represents operation of [[complex conjugate]]. The main purpose of the mother wavelet is to provide a source function to generate the daughter wavelets which are simply the translated and scaled versions of the mother wavelet. To recover the original signal <math>x(t)</math>, the first inverse continuous wavelet transform can be exploited.
:<math>x(t)=C_\psi^{-1}\int_{0}^{\infty}\int_{-\infty}^{\infty} X_w(a,b)\frac{1}{|a|^{1/2}}\tilde\psi\left(\frac{t-b}{a}\right)\,
<math>\tilde\psi(t)</math> is the [[Dual wavelet|dual function]] of <math>\psi(t)</math> and
:<math>C_\psi=\int_{-\infty}^{\infty}\frac{\overline\hat{\psi}(\omega)\hat{\tilde\psi}(\omega)}{|\omega|}\, \mathrm{d}\omega</math>
is admissible constant, where hat means Fourier transform operator. Sometimes, <math>\tilde\psi(t)=\psi(t)</math>, then the admissible constant becomes
:<math>C_\psi = \int_{-\infty}^{+\infty}
\frac{\left| \hat{\psi}(\omega) \right|^2}{\left| \omega \right|} \, \mathrm{d}\omega
</math>
Traditionally, this constant is called wavelet admissible constant. A wavelet whose admissible constant satisfies
:<math>0<C_\psi <\infty</math>
is called an admissible wavelet
:<math>x(t)=\frac{1}{2\pi\overline\hat{\psi}(1)}\int_{0}^{\infty}\int_{-\infty}^{\infty} \frac{1}{a^2}X_w(a,b)\exp\left(i\frac{t-b}{a}\right)\,
This inverse transform suggests that a wavelet should be defined as
:<math>\psi(t)=w(t)\exp(it) </math>
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==Scale factor==
[[File:Continuous wavelet transform.gif|thumb|300px|right]]
The scale factor <math>a</math> either dilates or compresses a signal. When the scale factor is relatively low, the signal is more contracted which in turn results in a more detailed resulting graph. However, the drawback is that low scale factor does not last for the entire duration of the signal. On the other hand, when the scale factor is high, the signal is stretched out which means that the resulting graph will be presented in less detail. Nevertheless, it usually lasts the entire duration of the signal.
==Continuous wavelet transform properties==
In definition, the continuous wavelet transform is a [[convolution]] of the input data sequence with a set of functions generated by the mother wavelet. The convolution can be computed by using a [[fast Fourier transform]] (FFT) algorithm. Normally, the output <math>X_w(a,b)</math> is a real valued function except when the mother wavelet is complex. A complex mother wavelet will convert the continuous wavelet transform to a complex valued function. The power spectrum of the continuous wavelet transform can be represented by <math>\frac{1}{a}\cdot|X_w(a,b)|^2</math>.<ref>{{cite journal |last1=Torrence |first1=Christopher |last2=Compo |first2=Gilbert |title=A Practical Guide to Wavelet Analysis |journal=Bulletin of the American Meteorological Society |date=1998 |volume=79 |issue=1 |pages=
[[File:Wavelet scale sweep for FM signal.gif|thumb|300px|Visualizing the effect of changing a [[Morlet wavelet|Morlet wavelet's]] <math>\sigma</math> parameter, which interpolates between the original time-series and a [[Fourier transform]]. Here, a [[Frequency modulation|frequency-modulated]] tone (plus noise) is analyzed; <math>1/\sigma</math> is adjusted from 1 to 200, in steps of unity.]]
==Applications of the wavelet transform==
One of the most popular applications of wavelet transform is image compression. The advantage of using wavelet-based coding in image compression is that it provides significant improvements in picture quality at higher compression ratios over conventional techniques. Since wavelet transform has the ability to decompose complex information and patterns into elementary forms, it is commonly used in acoustics processing and pattern recognition, but it has been also proposed as an instantaneous frequency estimator.<ref>{{Cite journal|last1=Sejdic|first1=E.|last2=Djurovic|first2=I.|last3=Stankovic|first3=L.|date=August 2008|title=Quantitative Performance Analysis of Scalogram as Instantaneous Frequency Estimator|journal=IEEE Transactions on Signal Processing|volume=56|issue=8|pages=3837–3845|doi=10.1109/TSP.2008.924856|bibcode=2008ITSP...56.3837S|s2cid=16396084|issn=1053-587X}}</ref> Moreover, wavelet transforms can be applied to the following scientific research areas: edge and corner detection, partial differential equation solving, transient detection, filter design, [[electrocardiogram]] (ECG) analysis, texture analysis, business information analysis and gait analysis.<ref>[https://www.youtube.com/watch?v=DTpEVQSEBBk "Novel method for stride length estimation with body area network accelerometers"], ''IEEE BioWireless 2011'', pp.
Continuous Wavelet Transform (CWT) is very efficient in determining the damping ratio of oscillating signals (e.g. identification of damping in dynamic systems). CWT is also very resistant to the noise in the signal.<ref>Slavic, J and Simonovski, I and M. Boltezar, [http://lab.fs.uni-lj.si/ladisk/?what=abstract&ID=11 Damping identification using a continuous wavelet transform: application to real data]</ref>
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* [[S transform]]
* [[Time-frequency analysis]]
* [[Cauchy wavelet]]
==References==
{{Reflist}}▼
=== Further reading===
*A. Grossmann & J. Morlet, 1984, Decomposition of Hardy functions into square integrable wavelets of constant shape, Soc. Int. Am. Math. (SIAM), J. Math. Analys., 15,
* Lintao Liu and Houtse Hsu (2012) "Inversion and normalization of time-frequency transform" AMIS 6 No. 1S pp.
* [[Stéphane Mallat]], "A wavelet tour of signal processing" 2nd Edition, Academic Press, 1999, {{ISBN|0-12-466606-X}}
*Ding, Jian-Jiun (2008), [http://djj.ee.ntu.edu.tw/TFW.htm Time-Frequency Analysis and Wavelet Transform], viewed 19 January 2008
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*Valens, Clemens (2004), [http://www.polyvalens.com/blog/wavelets/ A Really Friendly Guide to Wavelets], viewed 18 September 2018]
*[http://reference.wolfram.com/mathematica/ref/ContinuousWaveletTransform.html Mathematica Continuous Wavelet Transform]
▲{{Reflist}}
== External links ==
* {{YouTube|jnxqHcObNK4|Wavelets: a mathematical microscope}}
{{DEFAULTSORT:Continuous Wavelet Transform}}
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