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I am trying to use the discrete wavelet transform for signal processing (time series data from a plant). I would like to use a mother wavelet that is not in Matlab. Mallat and Zhong 1992 described a nonorthogonal quadratic spline wavelet I would like to use. They name some coefficients and impulse responses, but I'm not sure how to use these (reference below). To detect and treat abnormal sudden changes, the wavelet must satisfy a very particular property. We call a smoothing function any function θ(x) whose integral is equal to 1 and that converges to 0 at infinity. This function must be twice differentiable and define ψ_1 the first order derivative and ψ_2 the second order derivative.

ψ_1=(dθ(x))/dx and ψ_2=(d^2 θ(x))/(dx^2 )

The wavelet transform is computed by convolving the signal function f with a wavelet function (dilated at a scale s).

W_s^1 f(x)=fψ_(1,s) (x) and W_s^2 f(x)=fψ_(2,s) (x)

The extrema points of the first wavelet transform (WT) gives us the zero crossings points of the second WT. The extrema of W_s^1 f(x) and the zero-crossings of W_s^2 f(x) are the inflection points or the sharp variations points of f(x) smoothed by θ(x).Those edges in the signal can only be found if the wavelet is the second derivative of a smooth function. Furthermore, the first- and second-order WT are used in the detection of steady-state periods.

So I would like to code a quadratic spline wavelet function and a cubic spline scaling function like those in Mallat and Zhong (1992) article. Could anyone help me ?

enter image description here

Mallat, S., & Zhong, S. (1992). Characterization of Signals From Multiscale Edges. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(7), 710-732. doi:10.1109/34.142909

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I would think it is available in the Matlab codes from the Wavelab toolbox. It hosts codes for papers and books. Especially, there is a directory \Wavelab\Books\WaveTour\ for Stéphane Mallat's book: A Wavelet Tour of Signal Processing. You can check images chapter by chapter from A Wavelet Tour/Images. Those images can be regenerared. For instance, from Chapter 5, Wavelab codes in \Matlab_Wavelab\Books\WaveTour\WTCh05 say from instance in wt05fig04.m:

disp('The dyadic wavelet is a quadratic spline with one vanishing moment.') disp('These graphs display the modulus square of its Fourier transform') disp('dilated by 2^j for 0

which may suit your needs.

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  • $\begingroup$ Thank you very much for your answer. Unfortunately, there is a code to plot the image but there's no code that uses the quadratic spline wavelet to process a given signal. The file you are refering to gives image 5.1 in his book. Thank you anyway, I appreciate the quick answer. $\endgroup$
    – Émilie Thibault
    Commented May 13, 2020 at 21:10
  • $\begingroup$ Indeed it does process a signal, you get the Fourier transform. I have not used Wavelab for a long time, and I believe that there should be codes to process a signal in the Fourier domain, as one often do with biorthogonal splines. $\endgroup$
    – Laurent Duval
    Commented May 14, 2020 at 7:42

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