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The complex Morlet function is defined as:

$$Ψ(t,f_c,f_b)= \frac{1}{\sqrt[]{ \pi f_{b} } }\exp(-t^2/f_b)\exp(\jmath 2πf_ct)$$

where $f_b$ and $f_c$ are two important parameters in modifying the complex Morlet wavelet. It seems that Mathematica doesn't support complex Morlet transform and Its only support real morlet function that I am not interested to use. I'm into complex wavelet function. Mathematica only has Gabor transform for complex wavelets, and Gabor transform only has one parameter to be tuned.
so I need complex morlet function to run continues wavelet transform. Also I want to define $f_b$ and $f_c$ of the complex morlet function myself.
Can I make a complex Morlet wavalet transform by changing Gabor's parameter? How can I change $f_b$ and $f_c$ in it?
can I define a new wavelet exactly like the equation of complex morlet?

P.S: Actually I am a MATLAB user and as such I don't really know anything about the flexibility of Mathematica, but the reason why I came here is that Mathematica has the InverseContinuousWaveletTransform.

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I've edited your question to make it more readable... please check the equation you've entered and make sure it is correct. Secondly, please quantify what you mean by "It seems that Mathematica doesn't support complex Morlet transform." Please include your code so that we can understand what you tried to do. –  rm -rf Oct 6 '13 at 21:33
    
@rm-rf thanks for editing . –  Electricman Oct 7 '13 at 3:39
    
is your question not answered by the relevant documentation of MeyerWavelet‌​? Have you seen the section on basic uses and wavelet transforms? –  gpap Oct 7 '13 at 13:19
    
@gpap I am looking for complex Morlet , not other wavelet basis. look at my edit plz –  Electricman Oct 7 '13 at 22:01
    
Look here for a tutorial about defining your own wavelet. reference.wolfram.com/mathematica/tutorial/… –  bill s Oct 7 '13 at 22:06

1 Answer 1

up vote 9 down vote accepted

EDIT:

First, a note: As the usage of options, parameters and functions listed below is not documented, be advised that they still need proper tuning and/or may not work at all.

CMorletWavelet[]["WaveletQ"] := True
CMorletWavelet[]["OrthogonalQ"] := False
CMorletWavelet[]["BiorthogonalQ"] := False
CMorletWavelet[]["WaveletFunction"] := 1/Sqrt[π] Exp[2 I π 2 #1] Exp[-#1^2] &
CMorletWavelet[]["FourierFactor"] := 4 π/(6 + Sqrt[2 + 6^2])
CMorletWavelet[]["FourierTransform"] := Function[{Wavelets`NonOrthogonalWaveletsDump`wt, 
   Wavelets`NonOrthogonalWaveletsDump`s},
  π^(-1/4)HeavisideTheta[Wavelets`NonOrthogonalWaveletsDump`wt + $MachineEpsilon] 
  Exp[-(1/2) (Wavelets`NonOrthogonalWaveletsDump`wt Wavelets`NonOrthogonalWaveletsDump`s
  - π Sqrt[2/Log[2]])^2]]

Now you can use the built-in wavelet-related functions:

Plot[{Re@WaveletPsi[CMorletWavelet[], x], Im@WaveletPsi[CMorletWavelet[], x]},
     {x, -5, 5}, PlotRange -> All, Frame -> True, GridLines -> Automatic, 
     PlotStyle -> {Blue, {Red, Dashed}}]

wavelet_plot

snd = Play[Sum[Sin[2000 2^t n t], {n,5 }], {t, 2, 3}]

csd = ContinuousWaveletTransform[snd, CMorletWavelet[]]

WaveletScalogram[csd]

Sound scalogram

InverseContinuousWaveletTransform[csd, CMorletWavelet[]]

Play sound

This sound compression works just fine !

(* A simple example *)
cwd = ContinuousWaveletTransform[Range[10], CMorletWavelet[]]
WaveletScalogram[cwd]

Scalogram

 InverseContinuousWaveletTransform[cwd, CMorletWavelet[]]
{1., 2., 3., 4., 5., 6., 7., 8., 9., 10.}

This works just as expected, but using numbers larger than 63 results in ..

 cwd = ContinuousWaveletTransform[Range[64], CMorletWavelet[]]
 WaveletScalogram[cwd]

Scalogram

 InverseContinuousWaveletTransform[cwd, CMorletWavelet[]]
{0.500005, 4.38214, 6.69958, 10.625, 12.6907, 16.5033, 18.2989, 
21.8762, 23.3564, 26.6196, 27.7395, 30.6377, 31.3658, 33.8706, 
34.1929, 36.2965, 36.2168, 37.9296, 37.4675, 38.8152, 38.0038, 
39.0243, 37.9069, 38.647, 37.274, 37.7859, 36.2116, 36.551, 34.8323, 
35.0564, 33.2508, 33.4173, 31.5827, 31.7492, 29.9436, 30.1677, 
28.449, 28.7884, 27.2141, 27.726, 26.353, 27.0931, 25.9757, 26.9962, 
26.1848, 27.5325, 27.0704, 28.7832, 28.7035, 30.8071, 31.1294, 
33.6342, 34.3623, 37.2605, 38.3804, 41.6436, 43.1238, 46.7011, 
48.4967, 52.3093, 54.375, 58.3004, 60.6179, 64.5}

One of the reasons for this lies in the fact that I used the Fourier Transform of the original MorletWavelet which is a built-in predicate and quite different in implementation from the one I used. There are probably other parameters I need to set up properly, but I can't seem to find them, because, like I said, the usage is undocumented.


I know you came here because of the InverseContinuousWaveletTransform, but at that time of the day, or should I say night, I can't really think any more and will continue when I have more time to do so, unfortunately...

Note: As you are a MATLAB user I implemented the Complex Morlet wavelet according to THEIR documentation.

Preliminaries

For simplicity we assume that smallest wavelet scale is equal to 1 and we use a rather short data set.

I also used the following pages from the documentation (A-Z)

Implementation

(* Example data set *)

data = {1, 2, 3, 4};

(* Parameters *)

noct = Floor@Log[2, (data // Length)/2]
1
nvoc = 4;

(* Scaling parameter *)

s[oct_, voc_] := N[2^(oct - 1) 2^(voc/nvoc)]

(* Defining the wavelet function *)

ComplexMorlet[n_, band_, centerFreq_] := 
    1/Sqrt[π band] Exp[2 I π centerFreq n] Exp[-n^2/band]

(* Example expansion *)

ComplexMorlet[x, 1, 2]
E^(4 I π x - x^2)/Sqrt[π]
Plot[{Re@ComplexMorlet[x, 1, 2], Im@ComplexMorlet[x, 1, 2]}, {x, -3, 3},
     PlotStyle -> {Blue, {Red, Dashed}}, PlotRange -> All, 
     Frame -> True, GridLines -> Automatic]

Plot

(* Wavelet transform of a sampled sequence *)

 w[u_, oct_, voc_] := 1/s[oct, voc] Sum[data[[k]]
     Conjugate[ComplexMorlet[(k - u)/s[oct, voc], 1, 2]], {k, 1, data // Length}]

(* Performing the wavelet transform on our example data set *)

Table[w[k, 1, voc], {k, data // Length}, {voc, 4}]
{{0.228074 + 0.361025 I, 0.0610598 - 0.123408 I, 
     0.283659 - 0.583475 I, 1.15175 + 3.47516*10^-16 I},
   {0.486587 + 0.340747 I, 0.0693978 - 0.058132 I, 0.786587 - 0.662852 I, 
     1.85808 + 3.10964*10^-16 I}, 
   {0.821662 + 0.446737 I, -0.0236108 - 0.295969 I, 1.47435 - 0.380752 I, 
     2.26824 + 5.67838*10^-17 I}, 
   {1.57014 - 0.595682 I, 1.02407 + 0.281895 I, 1.47482 + 0.762858 I, 
     2.02475 - 2.84949*10^-16 I}}
(* Wavelet Scalogram using ComplexMorlet[x, 1, 2] *)

WaveletScalogram@ContinuousWaveletData[
 {{1, 1} -> {0.22807383843702972` + 0.36102529036876024` I, 
       0.06105984372279422` - 0.12340783119864777` I, 
       0.28365883675526904` - 0.5834746966816698` I, 
       1.1517469935306757` + 3.4751640646106677`*^-16 I},
  {1, 2} -> {0.4865866432814967` + 0.3407467247569226` I, 
       0.06939782717412021` - 0.05813200432524761` I, 
       0.7865874222126943` - 0.6628516103818837` I, 
       1.8580796599037956` + 3.1096385445125467`*^-16 I},
  {1, 3} -> {0.8216617511105463` + 
       0.44673675942817265` I, -0.02361080340458542` - 
       0.2959689122870983` I, 
       1.4743517412825382` - 0.3807516306374966` I, 
       2.26823511807995` + 5.678382044215492`*^-17 I},
  {1, 4} -> {1.570143054029254` - 0.5956822545417808` I, 
       1.024067417876664` + 0.2818946441776095` I, 
       1.4748223337693926` + 0.7628582023394818` I, 
       2.024752422313301` - 2.849488941725102`*^-16 I}}]

WaveletScalogram

(* Wavelet Scalogram using ComplexMorlet[x, 1, 10] *)

WaveletScalogram@ContinuousWaveletData@
 {{1, 1} -> {0.11634486079523618` - 0.17990847470866217` I, 
       0.9410569485064904` - 0.3524175549056541` I, 
       0.9995892268140318` + 0.3575695443712028` I, 
       1.1517469935306757` + 2.5826325630023094`*^-15 I},
 {1, 2} -> {0.2085276338912312` - 0.15114828701865127` I, 
      1.8062819251440743` - 0.3772206439472593` I, 
      1.813592761954768` + 0.36136020250254647` I, 
      1.8580796599037956` + 1.5548192722562736`*^-15 I},
 {1, 3} -> {0.2547509048762912` - 0.27877696228455096` I, 
      2.5401537117071564` - 0.16692666476822` I, 
      2.402824979378204` + 0.10553538050034861` I, 
      2.26823511807995` + 2.8391910221077465`*^-16 I},
 {1, 4} -> {1.3309683457126755` + 0.3296339838999044` I, 
      2.319228847343012` + 0.4019097092762081` I, 
      2.1426745757435186` - 0.3492240227193354` I, 
      2.024752422313301` - 1.6360071035367952`*^-15 I}}

WaveletScalogram 2

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I will improve this, if I have time tomorrow, so we can benefit from the "WaveletQ" and "OrthogonalQ" properties and make the whole process more refined for the end user. –  Sektor Oct 10 '13 at 23:37
    
Thanks loads for your neat and specified answer. would you help me for inverse continues_wavelet transform please? why the scalogram plot looks like that? I used to work with more clear and continues plots. I guess its for division between scale, I used to use this relationship for scales: $$ s_{j}= s_{0} 2^{j \delta _{j} } , j=0,1,2,... J$$ $$J= \delta j^{-1} log_{2} ( \frac{N \delta t}{ s_{0} } )$$ and i was using $\delta _{j}$ = 0.01 Thanks for answering –  Electricman Oct 13 '13 at 10:51
    
Yes, I will, I am sorry for the delay, but was on a short trip. I have already written everything, so expect it in a few hours. The scalogram looks like this because of the data set I used. –  Sektor Oct 13 '13 at 11:11
    
Thanks a lot my mate. :) –  Electricman Oct 13 '13 at 11:21
1  
@NikolaDimitrov It appears that way because it is defined in a package. For example, if you have a package BeginPackage["Foo`"] and then a context Begin["`Private`"], then if you define Function[{x}, ...] inside, it will appear (in a standard notebook) as Function[{Foo`Private`x}, ...] –  rm -rf Oct 13 '13 at 18:39

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