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.

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.

and is admissible constant, where hat means Fourier transform operator.

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.

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.

is a real valued function except when the mother wavelet is complex.

The power spectrum of the continuous wavelet transform can be represented by

[1][2] 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.

[4] Wavelet transforms can also be used in Electroencephalography (EEG) data analysis to identify epileptic spikes resulting from epilepsy.

[5] Wavelet transform has been also successfully used for the interpretation of time series of landslides[6] and land subsidence,[7] and for calculating the changing periodicities of epidemics.

[8] Continuous Wavelet Transform (CWT) is very efficient in determining the damping ratio of oscillating signals (e.g. identification of damping in dynamic systems).

Continuous wavelet transform of frequency breakdown signal. Used symlet with 5 vanishing moments.
Visualizing the effect of changing a Morlet wavelet's parameter, which interpolates between the original time-series and a Fourier transform . Here, a frequency-modulated tone (plus noise) is analyzed; is adjusted from 1 to 200, in steps of unity.