Wikipedia / Continuous wavelet transform
Wikipedia / Continuous wavelet transform
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 wavelets vary continuously.
Definition
is expressed by the following integral
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 first inverse continuous wavelet transform can be exploited.
is the dual function and
, then the admissible constant becomes
Traditionally, this constant is called wavelet admissible constant. A wavelet whose admissible constant satisfies
, the second inverse continuous wavelet transform can be exploited.
This inverse transform suggests that a wavelet should be defined as
is a window. Such defined wavelet can be called as an analyzing wavelet, because it admits to time-frequency analysis. An analyzing wavelet is unnecessary to be admissible.
Scale factor
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.
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 (EEG) data analysis to identify epileptic spikes resulting from epilepsy. 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.
See also
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), The Wavelet Tutorial, viewed 19 January 2008
WaveMetrics (2004), Time Frequency Analysis, viewed 18 January 2008
Valens, Clemens (2004), A Really Friendly Guide to Wavelets, viewed 18 September 2018]
External links