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I am calculating approximations using wavelets but outside the multi resolution analysis framework; particularly, I am not using the built-in wavelet transforms.

Still, I need to calculate coefficients and therefore I am wondering how to calculate the support of a given wave in order to know where to start and stop calculating coefficients for WaveletPhi[wave] and WaveletPsi[wave] for a given resolution level. Therefore, I need to know the support for a given wavelet.

I suspect that I could use WaveletFilterCoefficients[wave,"PrimalHighpass"] and, similarly for "PrimalLowpass"; but it is just a hunch, not sure how.

Basically, I am after something along the lines of: WaveletSupport[wave] -> {a,b} support of given wavelet (For the moment just orthogonal wavelets, not biorthogonal).

Thanks.

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What families are you using ? –  Sektor Jun 10 at 0:16
    
I want to use DaubechiesWavelet. –  caya Jun 10 at 0:37

1 Answer 1

up vote 4 down vote accepted

Both the scaling and wavelet functions of the DaubechiesWavelet are compactly supported on the interval $\left [ 0, N-1 \right ]$, where N is the number of "taps", the impulse response length or just the length of the filter. Choose the one that suits you ^^ Bear in mind that the notation in Mathematica is slightly different - The wavelets are classified by the number of vanishing moments, rather than filter lengths, so the DaubechiesWavelet[2] has 2 vanishing moments and the filter length is 4

There is another definition specifying the bounds of the support:


If the scaling function is supported on $\left [ LB, UB \right ]$ then the wavelet function is supported on $\left [ \frac{LB-UB+1}{2}, \frac{UB-LB+1}{2} \right ]$


Just a simple experiment to confirm the statement (undocumented function):

ListLinePlot[Wavelets`ScalingAndWaveletFunction[DaubechiesWavelet[8], 
      "PrimalScalingFunction", 8, WorkingPrecision -> MachinePrecision], 
      PlotRange -> All]

Mathematica graphics

ListLinePlot[Wavelets`ScalingAndWaveletFunction[DaubechiesWavelet[8], 
      "PrimalWaveletFunction", 8, WorkingPrecision -> MachinePrecision], 
      PlotRange -> All]

Mathematica graphics

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Thanks a lot @Sektor, but the support of second graph (wavelet) seems to be around [-6,4], isn't it? This is what confuses me, because the values for $x$ where $\psi _{j,k}(x)$ is guaranteed to be zero don't seem be just outside of [0,$2^{-j}$]. –  caya Jun 10 at 1:47
    
No need to ! :) Look again :) –  Sektor Jun 10 at 8:31
    
Thanks a lot @Sektor. Your comment about the "impulse response" got me thinking and I was able to arrive to something similar to yours using the "PrimalHighpass" coefficients (it is most likely the same, but I will double check with yours; fantastic.) –  caya Jun 10 at 11:53
1  
@caya Thank you for the question and I am glad to help :) BTW Do not hesitate to share something about wavelets or x-lets, in general - something you are working on, etc, if you can of course :) –  Sektor Jun 10 at 12:02

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