
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the n ...
s vary continuously.
Definition
The continuous wavelet transform of a function
at a scale
and translational value
is expressed by the following integral
:
where
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
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
. 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
, the first inverse continuous wavelet transform can be exploited.
:
is the
dual function of
and
:
is admissible constant, where hat means Fourier transform operator. Sometimes,
, then the admissible constant becomes
:
Traditionally, this constant is called wavelet admissible constant. A wavelet whose admissible constant satisfies
:
Scale factor

The scale factor
a 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
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
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
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in ...
(FFT) algorithm. Normally, the output
X_w(a,b) 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
\frac\cdot, X_w(a,b), ^2.
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. 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. Wavelet transforms can also be used in
Electroencephalography
Electroencephalography (EEG)
is a method to record an electrogram of the spontaneous electrical activity of the brain. The biosignal, bio signals detected by EEG have been shown to represent the postsynaptic potentials of pyramidal neurons in ...
(EEG) data analysis to identify epileptic spikes resulting from
epilepsy
Epilepsy is a group of Non-communicable disease, non-communicable Neurological disorder, neurological disorders characterized by a tendency for recurrent, unprovoked Seizure, seizures. A seizure is a sudden burst of abnormal electrical activit ...
. Wavelet transform has been also successfully used for the interpretation of time series of landslides and land subsidence, and for calculating the changing periodicities of epidemics.
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.
[Slavic, J and Simonovski, I and M. Boltezar]
Damping identification using a continuous wavelet transform: application to real data
/ref>
See also
* Continuous wavelet
* S transform
* Time-frequency analysis
* Cauchy wavelet
References
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, 723–736.
* Lintao Liu and Houtse Hsu (2012) "Inversion and normalization of time-frequency transform" AMIS 6 No. 1S pp. 67S-74S.
* Stéphane Mallat, "A wavelet tour of signal processing" 2nd Edition, Academic Press, 1999,
*Ding, Jian-Jiun (2008)
Time-Frequency Analysis and Wavelet Transform
viewed 19 January 2008
*Polikar, Robi (2001)
viewed 19 January 2008
*WaveMetrics (2004)
viewed 18 January 2008
*Valens, Clemens (2004)
A Really Friendly Guide to Wavelets
viewed 18 September 2018]
Mathematica Continuous Wavelet Transform
External links
*
{{DEFAULTSORT:Continuous Wavelet Transform
Theory of continuous functions
Integral transforms
fr:Ondelette#Transformée en ondelettes continue