Formulation

In mathematics and signal processing, the continuous wavelet transform (CWT) of a function $f$ is a wavelet transform defined by

\gamma (\tau ,s)=\int _{-\infty }^{+\infty }f(t){\frac {1}{\sqrt {|s|}}}{\overline {\psi \left({\frac {t-\tau }{s}}\right)}}dt

where $\tau$ represents translation, $s$ represents scale and $\psi$ is the mother wavelet. ${\overline {z}}$ is the complex conjugate of $z$ .

The original function $f$ can be reconstructed with the inverse transform

f(t)={\frac {1}{C_{\psi }}}\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }\gamma (\tau ,s)\psi \left({\frac {t-\tau }{s}}\right)d\tau {\frac {ds}{|s|^{2}}}

where

C_{\psi }=\int _{-\infty }^{+\infty }{\frac {\left|{\hat {\psi }}(\zeta )\right|^{2}}{\left|\zeta \right|}}d\zeta

is called the admissibility constant and ${\hat {\psi }}$ is the Fourier transform of $\psi$ . For a successful inverse transform, the admissibility constant has to satisfy the admissibility condition:

0<C_{\psi }<+\infty

.

It is possible to show that the admissibility condition implies that ${\hat {\psi }}(0)=0$ , so that a wavelet must integrate to zero.

The function $\psi$ is the prototype of the pattern the signal is convolved with. It is thus called the mother wavelet. In contrast to that, the scaled and shifted variants of that function are called daughter wavelets. They are obtained as follows:

\psi _{s,\tau }(t)={\frac {1}{\sqrt {|s|}}}\psi \left({\frac {t-\tau }{s}}\right)

.

Computation

The continuous wavelet transform of a discretised signal is typically computed over the temporal domain (translation) of the signal and a range of scales equivalent to the Nyquist range. Computation can either be performed using direct inner products (possibly taking advantage of the sparseness of the wavelet) or via the FFT. In the latter case, the continuous wavelet transform is noted to be a convolution at each scale, which can be performed efficiently via a discrete Fourier transform using the FFT.

Applications

Determination of the fractal dimension

Looks at extrema of the CWT with respect to translation in order to quantify the fractal dimension of a function.

Time-frequency analysis

Relates extrema of the CWT with respect to scale to conventional Fourier components in order to decompose a signal in terms of both time and frequency simultaneously. Continuous wavelets used for time-frequency analysis are designed to mimic the complex sinusoidal basis functions of the Fourier transform.

CWT-based time-frequency analysis has many benefits over other time-frequency methods (such as the short-time or windowed Fourier transform, Wigner-Ville and Choi-Williams distributions).

Time-frequency analysis has applications in many subjects including physics (quantum mechanics, seismic geophysics, turbulence), chemistry (diffraction), biology (EEG, ECG, protein- and DNA-sequence analysis), engineering (electrical transient response, impulse-shock response for non-destructive testing, fatigue analysis), finance, climatology and speech recognition.

References

Robi Polikar, The Engineer'S Ultimate Guide to Wavelet Analysis, The Wavelet Tutorial (1999)
Jon Harrop, Wavelet-based Time-Frequency Analysis (2005)
Ingrid Daubechies, Ten Lectures on Wavelets (CBMS-NSF Regional Conference Series in Applied Mathematics), (1992) Soc for Industrial & Applied Math.

Continuous wavelet transform