Continuous wavelet transform

Continuous wavelet transform of frequency breakdown signal. Used symlet with 5 vanishing moments.

In mathematics, a continuous wavelet transform (CWT) is used to divide a continuous-time function into wavelets. Unlike Fourier transform, the continuous wavelet transform possesses the ability to construct a time-frequency representation of a signal that offers very good time and frequency localization. The continuous wavelet transform of a continuous, square-integrable function $x(t)$ at a scale $a>0$ and translational value $b\in \mathbb {R}$ is expressed by the following integral

$X_{w}(a,b)={\frac {1}{\sqrt {|a|}}}\int _{-\infty }^{\infty }x(t)\psi ^{\ast }\left({\frac {t-b}{a}}\right)\,dt$

where $\psi (t)$ is a continuous function in both the time domain and the frequency domain called the mother wavelet and $^{\ast }$ 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 $x(t)$, inverse continuous wavelet transform can be exploited.

$x(t)=\int _{0}^{\infty }\int _{-\infty }^{\infty }{\frac {1}{a^{2}}}X_{w}(a,b){\frac {1}{\sqrt {|a|}}}{\tilde {\psi }}\left({\frac {t-b}{a}}\right)\,db\ da$

${\tilde {\psi }}(t)$ is the dual function of $\psi (t)$. And the dual function should satisfy

$\int _{0}^{\infty }\int _{-\infty }^{\infty }{\frac {1}{|a^{3}|}}\psi \left({\frac {t_{1}-b}{a}}\right){\tilde {\psi }}\left({\frac {t-b}{a}}\right)\,db\ da=\delta (t-t_{1}).$

Sometimes, ${\tilde {\psi }}(t)=C_{\psi }^{-1}\psi (t)$, where

$C_{\psi }={\frac {1}{2}}\int _{-\infty }^{+\infty }{\frac {\left|{\hat {\psi }}(\zeta )\right|^{2}}{\left|\zeta \right|}}d\zeta$

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.

Mother wavelet

In general, it is preferable to choose a mother wavelet that is continuously differentiable with compactly supported scaling function and high vanishing moments. A wavelet associated with a multiresolution analysis is defined by the following two functions: the wavelet function $\psi (t)$, and the scaling function $\varphi (t)$. The scaling function is compactly supported if and only if the scaling filter h has a finite support, and their supports are the same. For instance, if the support of the scaling function is [N1,N2], then the wavelet is [(N1-N2+1)/2,(N2-N1+1)/2]. On the other hand, the $k^{th}$ moments can be expressed by the following equation conditions of mother wavelet 1) admisibility 2) regularity 3) no of vanishing moments

$m_{k}=\int t^{k}\psi (t)\,dt.$

If $m_{0}=m_{1}=m_{2}=.....=m_{p-1}=0$, we say $\psi (t)$ has $p$ vanishing moments. The number of vanishing moments of a wavelet analysis represents the order of a wavelet transform. According to the Strang-Fix conditions, the error for an orthogonal wavelet approximation at scale $a=2^{-i}$ globally decays as $a^{L}$, where $L$ is the order of the transform. In other words, a wavelet transform with higher order will result in better signal approximations.

Scaling function

The wavelet function $\psi (t)$ and the scaling function $\varphi (t)$ define a wavelet. The scaling function is primarily responsible for improving the coverage of the wavelet spectrum. This could be difficult since time is inversely proportional to frequency. In other words, if we want to double the spectrum coverage of the wavelet in the time domain, we would have to sacrifice half of the bandwidth in the frequency domain. Instead of covering all the spectrum with an infinite number of levels, we use a finite combination of the scaling function to cover the spectrum. As a result, the number of wavelets required to cover the entire spectrum has been greatly reduced.

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 of the input data sequence with a set of functions generated by the mother wavelet. The convolution can be computed by using the Fast Fourier Transform (FFT). 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 $|X_{w}(a,b)|^{2}$ .

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. 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.

Continuous Wavelet Transform (CWT) is very efficient in determining the damping ratio of oscillating signals (e.g. identification of damping in dynamical systems). CWT is also very resistant to the noise in the signal.

See also

• Continuous_wavelet
• S transform
• Time-frequency analysis

References

• Stéphane Mallat, "A wavelet tour of signal processing" 2nd Edition, Academic Press, 1999, ISBN 0-12-466606-X
• 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 January 2008]
• Mathematica Continuous Wavelet Transform
• Lewalle, Jacques: Continuous wavelet transform, viewed 6 February 2010