improve CWT docs

[grlee77]

Document the equations for the CWT wavelets.

see #386

mathjax renders the equations like:

[codecov-io]

Codecov Report

Merging #387 into master will not change coverage.
The diff coverage is n/a.

@@           Coverage Diff           @@
##           master     #387   +/-   ##
=======================================
  Coverage   84.26%   84.26%           
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  Files          22       22           
  Lines        3540     3540           
  Branches      600      600           
=======================================
  Hits         2983     2983           
  Misses        489      489           
  Partials       68       68

Impacted Files	Coverage Δ
pywt/_extensions/_pywt.pyx	71.7% <ø> (ø)	⬆️

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[rgommers]

Good idea. Covers all wavelet types and looks correct from a quick read.

[rgommers]

LGTM. I've added a commit with some typos and more doc formatting fixes.

[grlee77]

thanks. the edits you made look good to me

[rgommers]

Okay, in it goes then. Thanks @grlee77!

[ggrrll]

Hi,

I also agree that an improvement is needed, especially since the definition of the morlet function differs from the one in the seminal paper on CWT.

...is this already live? it doesn't look so here

Thanks

[rgommers]

Thanks for asking. There was an issue in this PR that broke the doc build on readthedocs. I just pushed a fix in commit a057128, so it should appear within 15 minutes or so.

[ggrrll]

Hi,

it's live thanks @rgommers

however, why you opted for such different definition ? even 1/pi^(1/2) ... is that correct or it's a typo (1/pi^(1/4)) ? , like in the usual definition

It seems impossible to find a B, such that the two definitions are equivalent (referring of course to this paper)

Thanks

[grlee77]

however, why you opted for such different definition ? even 1/pi^(1/2) ... is that correct or it's a typo

The normalization used here is consistent with the definition in Matlab and their documentation references the following source:
Teolis, A. (1998), Computational signal processing with wavelets, Birkhäuser.

I accessed that text electronically via my university library and they state that this form makes the L1-norm of the wavelet equal to 1 (and that it's Fourier transform will have a peak value of 1). A quick numerical check seems to verify this claim:

import pywt
import numpy as np
wav = pywt.ContinuousWavelet('cmor')
psi, x = wav.wavefun()
dx = x[1]  - x[0]

L1 = np.trapz(np.abs(psi)) * dx  # L1-norm is the integral of the magnitude
print("Integral of magnitude = {}".format(L1))

I am not sure the reason one would use this normalization vs. others. Unfortunately my expertise in the CWT is limited and the author of the CWT routines in PyWavelets hasn't been active here lately. I think in typical applications the absolute normalization shouldn't be too important? Isn't one typically looking for features in a spectrogram where a global scaling of the plots shouldn't matter?

[grlee77]

We could perhaps add a reference to the Teolis text to the docs as the source of the equations stated in that section. I had transcribed the equations based on inspection of the numpy functions in the test suite. I tried to double check things, so hopefully there are no typos.