This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

ContinuousWaveletTransform

 ContinuousWaveletTransform gives the continuous wavelet transform of a list of values . ContinuousWaveletTransformgives the continuous wavelet transform using the wavelet wave. ContinuousWaveletTransformgives the continuous wavelet transform using noct octaves with nvoc voices per octave. ContinuousWaveletTransformgives the continuous wavelet transform of sampled sound.
• Properties of the ContinuousWaveletData cwd can be found using cwd["prop"]. A list of available properties can found using cwd["Properties"].
• The resulting wavelet coefficients are arrays of the same dimensions as the input data.
• The possible wavelets wave include:
 MorletWavelet[...] Morlet cosine times Gaussian GaborWavelet[...] complex Morlet wavelet DGaussianWavelet[...] derivative of Gaussian MexicanHatWavelet[...] second derivative of Gaussian PaulWavelet[...] Paul wavelet
• The default value for noct is given by , where is the length of the input.  »
• The default value for nvoc is 4.
• The continuous wavelet transform of a function is given by .
• The continuous wavelet transform of a uniformly sampled sequence is given by .
• The scaling parameter is given by equal-tempered scale where is the octave number, the voice number, and the smallest wavelet scale.
• The following options can be given:
 Padding None how to extend data beyond boundaries SampleRate Automatic samples per unit WaveletScale Automatic smallest resolvable scale WorkingPrecision MachinePrecision precision to use in internal computations
• Padding pads the input data to the next higher power of 2 to reduce boundary effects. The settings for Padding are the same as for the padding argument used in ArrayPad.
Compute a continuous wavelet transform using MexicanHatWavelet:
Plot the coefficients:
Perform an inverse continuous wavelet transform:
Transform a sampled Sound object:
Plot a scalogram:
Compute a continuous wavelet transform using MexicanHatWavelet:
 Out[1]=
Plot the coefficients:
 Out[2]=
Perform an inverse continuous wavelet transform:
 Out[3]=

Transform a sampled Sound object:
 Out[1]=
 Out[2]=
Plot a scalogram:
 Out[3]=
 Scope   (18)
Compute a continuous wavelet transform:
Show all the voices for the 8octave:
Use Normal to get all wavelet coefficients explicitly:
Also use All as an argument to get all coefficients:
Use to find out what wavelet coefficients are available:
Extract specific coefficient arrays:
Extract several wavelet coefficients corresponding to the list of wavelet index specifications:
Extract all coefficients whose wavelet indexes match a pattern:
WaveletScalogram gives a time scale representation of wavelet coefficients:
More voices per octave increases the scale resolution:
Higher number of octaves gives a wider spectrum of scale range:
A single frequency shows up as a horizontal band at the equivalent scale:
Multiple frequencies show up as multiple bands at the equivalent scales:
Sinusoid with linearly increasing frequency:
Wavelet transform gives a good time localization of features:
Higher frequencies are resolved at lower octaves and lower frequencies at higher octaves:
Resolve time and frequency features of a signal:
Use GaborWavelet to perform a continuous wavelet transform:
There is an inverse relationship between scale values and frequency values:
Find pairs of that resolve frequencies 20 Hz and 70 Hz:
Verify using a WaveletScalogram:
Compute the wavelet transform using different wavelet families:
A narrow wavelet function will have good time and scale resolution:
A broad wavelet function will have poor time and scale resolution:
Use different families of wavelets to capture different features:
Speech analysis using ContinuousWaveletTransform:
The orange patches correspond to the words "You will return safely to Earth":
Extract octaves 5 and 6:
 Options   (9)
The settings for Padding are the same as the methods for ArrayPad, including :
:
:
:
:
:
:
Padding has no effect on the length of wavelet coefficients:
Padding pads the input data to the next higher power of 2 to reduce boundary effects:
Boundary effects at the start:
Boundary effects at the end:
WaveletScale indicates the smallest resolvable scale used for the transform:
The scales used are given as with wavelet scale, octave, and voice:
For lists, the Automatic value of SampleRate is set to 1:
Explicitly set the sample rate:
For Sound data, the Automatic value of SampleRate is extracted from the Sound data object:
SampleRate is used for normalizing wavelet transform coefficients:
By default, WorkingPrecision is used:
Use higher-precision computation:
 Applications   (4)
Real wavelet functions can be used to isolate peaks or discontinuities:
Complex wavelets can be used to capture oscillatory behavior:
Amplitude of wavelet coefficients:
Phase of wavelet coefficients:
ContinuousWaveletTransform can be used to filter frequencies:
Filter the cosine with frequency :
Perform InverseContinuousWaveletTransform on a thresholded data object:
The final filtered signal:
Identify musical notes using a scalogram:
Generate a sequence of pitches corresponding to an equal-tempered scale at 300 Hz:
Compute frequencies resolved corresponding to octaves and voices:
Find pairs of that resolve frequencies 300 Hz:
The default value for octave is given by :
Default value of voices is 4:
Low-frequency data is resolved at higher octaves:
Based on the length of input data, the Automatic setting for octaves resolved 8 octaves:
Increase the number of octaves to resolve the low-frequency component:
Scalogram of a Zeta function:
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