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I'm looking for analytical justification of linearity or non linearity of the wavelet transform with the real Haar mother wavelet function.

I have googling already. But I can't find and understand the meaning

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    $\begingroup$ do you know the definition of linearity? Can you apply it to the definition of the Wavelet transform? If not, maybe these should be the questions you should ask yourself first (you'll find them answered here, probably, already). If you know that already, where exactly are you stuck applying the definition of linearity to the transform? $\endgroup$ – Marcus Müller Apr 8 at 7:25
  • $\begingroup$ Indeed, there are linear functions ($ax+b$ by abuse), linear transformations, linear systems. $\endgroup$ – Laurent Duval Apr 8 at 8:19
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The wavelet function is preset and continuous signal defined in a time interval from 0 to N-1. The characteristics of this function is to be continuous (defined in any point). Important to remember that the function is applied with the convolution to your signal.

Then, the concept of linearity refers to, by definition, means that the output follows a linear trend with respect to the input (y=m*x+n).

Conclussion, wavelets functions are by definition continuous in time but not linear.

Examples of common wavelet functions. Hope this helps!

G.

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  • $\begingroup$ I would disagree. Wavelets don't need to be continuous, and don't necessary have ($0$ to $N-1$) compact support. $\endgroup$ – Laurent Duval Apr 8 at 9:25
  • $\begingroup$ And not symmetric $\endgroup$ – Laurent Duval Apr 8 at 9:26
  • $\begingroup$ Well, they are continuous by definition from 0 to N-1. $\endgroup$ – Gabriel Galeote-Checa Apr 8 at 9:49
  • $\begingroup$ All your examples have infinite support. And the very admissible Haar wavelet (and others, like piecewise continuous Alpert bases) is not continuous $\endgroup$ – Laurent Duval Apr 8 at 10:11
  • $\begingroup$ Actually I have never worked with those that you mentioned but typically wavelet are continuous in the sense that you must be able to apply Fourier Transform to it. I am referring to mathematical continuity, even though that the signal is discrete, it is defined in kT time samples. Isn't it that true? $\endgroup$ – Gabriel Galeote-Checa Apr 8 at 10:29
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Linearity can play at different levels:

  • The Haar wavelet functions are piece-wise affine (constant), not linear.
  • Any standard wavelet decomposition is linear, as it decomposes a signal into a sum of coefficients times a wavelet
  • Wavelet approximations are often nonlinear, as they consist in choosing the $K$ best (largest) coefficients to represent the signal. And choosing the $K$ largest does not obey the linearity principle.
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