Understanding the Morlet wavelet
Lot of resources can be found online (but different formulas, different names for the same parameter etc..).
Here some slides (work in progress): https://docs.google.com/presentation/d/17iaibu6Z-4mruF550cYVrNbj6w2ZjindhKWd7NA0_nE/edit?usp=sharing
Code
-
invcwt_v1.4:
- Wavelet software was provided by C. Torrence and G. Compo, and is available at URL: http://paos.colorado.edu/research/wavelets/''. Reference: Torrence, C. and G. P. Compo, 1998: A Practical Guide to Wavelet Analysis. Bull. Amer. Meteor. Soc., 79, 61-78. paper: http://paos.colorado.edu/research/wavelets/bams_79_01_0061.pdf
- this code was found here: https://it.mathworks.com/matlabcentral/fileexchange/20821-continuous-wavelet-transform-and-inverse
- related and detailed wavelet tutorial on the same paper: https://github.com/chris-torrence/wavelets
-
paper 2015:
- Postnikov, Eugene B., Elena A. Lebedeva, and Anastasia I. Lavrova. "Computational implementation of the inverse continuous wavelet transform without a requirement of the admissibility condition." Applied Mathematics and Computation 282 (2016): 128-136
- can be found here: https://arxiv.org/pdf/1507.04971.pdf
-
wavelet_matlab just to test how the cwt works (Wavelet Toolbox needed)
-
comparing_morlet_formulas is a comparison between cmorwavf and the one present in:
- Tallon-Baudry, Catherine, et al. "Stimulus specificity of phase-locked and non-phase-locked 40 Hz visual responses in human." Journal of Neuroscience 16.13 (1996): 4240-4249. Link:
- Hipp, Joerg F., et al. "Large-scale cortical correlation structure of spontaneous oscillatory activity." Nature neuroscience 15.6 (2012): 884. Link:
- Both refers to this original aricle for the wavelet definition: Analysis of sound patterns through wavelet transforms https://www.researchgate.net/profile/Richard_Kronland-Martinet/publication/263794237_Analysis_of_sound_patterns_through_wavelet_transforms/links/53ece4d40cf26b9b7dbff7a6/Analysis-of-sound-patterns-through-wavelet-transforms.pdf
Other resources:
- Lilly, J. M. jLab: A data analysis package for Matlab, version 1.6.2. 2016. http://www.jmlilly.net/jmlsoft.html.
- The Uncertainty Principle & Time-Bandwidth Product: https://www.ee.iitb.ac.in/uma/~pawar/Wavelet%20Applications/Chapters_review/ch03_Gr3_Gr2.pdf
- Higher-Order Properties of Analytic Wavelets: referred by cwt Matlab help (explains the different normalization (L1) instead of (L2). May be why we have such a discrepancy between Tannon and Matlab? https://arxiv.org/pdf/0802.2377.pdf
- Mike X Cohen:
- refer to: http://mikexcohen.com/lectures.html for a detailed and clear description of wavelets (matlab code is also available there and here:
- New paper: https://www.biorxiv.org/content/biorxiv/early/2018/08/21/397182.full.pdf
- National:
- Mathworks:
- Web pages:
- https://www.dsprelated.com/freebooks/sasp/Wavelet_Filter_Banks.html
- https://dsp.stackexchange.com/questions/41349/complex-morlet-coefficient?answertab=votes#tab-top
- https://ccrma.stanford.edu/~jos/sasp/Continuous_Wavelet_Transform.html : "The center frequency is typically chosen so that second peak is half of first approx = 5.336. In this case psi(0) almost zero: which is close enough to zero-mean for most practical purposes"
- Books:
- Computational Signal Processing with Wavelets (the one on which Matlab complex Morlet formula is based) https://books.google.it/books?id=SMngBwAAQBAJ&lpg=PA66&vq=morlet&hl=it&pg=PA66#v=onepage&q&f=false
- The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science: https://books.google.it/books?id=VrTZDQAAQBAJ&lpg=PP1&dq=The%20Illustrated%20Wavelet%20Transform%20Handbook%3A%20Introductory%20Theory%20and%20Applications%20in%20Science&hl=it&pg=PP1#v=onepage&q=5.83&f=false. Page 35
- A Tutorial of the Morlet Wavelet Transform http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=924D894AEBCA3A939CC1297FF28B1BA4?doi=10.1.1.695.137&rep=rep1&type=pdf
- Wavelet Transforms in Time Series Analysis: https://www.atmos.umd.edu/~ekalnay/syllabi/AOSC630/Wavelets_2010.pdf Nice and simple explanation of wavelet in general and Morlet in particular.
- Morlet Wavelets in Quantum Mechanics: https://arxiv.org/ftp/arxiv/papers/1001/1001.0250.pdf Complicated but detailed description and calculation of admissibility constant.