Continuous wavelet transform
Encyclopedia
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. In mathematics, the continuous wavelet transform of a continuous, square-integrable 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 
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
, inverse continuous wavelet transform can be exploited.
is the dual function of 
.
And the dual function should satisfy
Sometimes,
, where
is called the admissibility constant and
is the Fourier transform
of
.
For a successful inverse transform, the admissibility constant has to satisfy the admissibility condition:
.
It is possible to show that the admissibility condition implies that
, so that a wavelet must integrate to zero.
is defined by the following two functions: the wavelet function
, and the scaling function 
. 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 
moments can be expressed by the following equation
If
, we say 
has 
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 
globally decays as 
, where 
is the order of the transform. In other words, a wavelet transform with higher order will result in better signal approximations.
and the scaling function 
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.
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.
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
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 
.
(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.
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
, the continuous wavelet transform possesses the ability to construct a time-frequency representation
Time-frequency representation
A time–frequency representation is a view of a signal represented over both time and frequency. Time–frequency analysis means analysis into the time–frequency domain provided by a TFR...
of a signal that offers very good time and frequency localization. In mathematics, the continuous wavelet transform of a continuous, square-integrable function
at a scale and translational value is expressed by the following integralwhere
is a continuous function in both the time domain and the frequency domain called the mother wavelet and represents operation of complex conjugateComplex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
. 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
, inverse continuous wavelet transform can be exploited. is the dual function of .And the dual function should satisfy
Sometimes,
, whereis called the admissibility constant and
is the Fourier transformFourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of
.For a successful inverse transform, the admissibility constant has to satisfy the admissibility condition:
.It is possible to show that the admissibility condition implies that
, 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 analysisMultiresolution analysis
A multiresolution analysis or multiscale approximation is the design method of most of the practically relevant discrete wavelet transforms and the justification for the algorithm of the fast wavelet transform...
is defined by the following two functions: the wavelet function
, and the scaling function . 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 moments can be expressed by the following equationIf
, we say has 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 globally decays as , where 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 and the scaling function 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 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 convolutionConvolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
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
Fast Fourier transform
A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...
(FFT). Normally, the output
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 .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, ElectrocardiogramElectrocardiogram
Electrocardiography is a transthoracic interpretation of the electrical activity of the heart over a period of time, as detected by electrodes attached to the outer surface of the skin and recorded by a device external to the body...
(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.