I have an application that I must use a complex Morlet wavelet function (cmorfb-fc), I can run CWT (continuous wavelet transform) and it's fine, but I want to run DWT(Discrete wavelet transform) with this complex Morlet(cmorfb-fc) function too. then I want to compare the DWT and CWT results.
so Does anyone can help and explain, how can I implement DWT of complex Morlet (with the known Mallat filters)?
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1$\begingroup$ This is not possible. See this very similar question. $\endgroup$– André BergnerOct 18, 2013 at 11:49
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$\begingroup$ @AndréBergner ohhh bad times- Thank you for your answer. $\endgroup$– SAHOct 18, 2013 at 12:19
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$\begingroup$ You got an answer for an ancient question. Is your question solved, or you expect more contributions? $\endgroup$– Laurent DuvalJul 16, 2017 at 14:53
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A continuous wavelet frame can be discretized with perfect reconstruction under some technical conditions. If this is possible, you obtain a discrete wavelet frame, often with some amount of redundancy (depending on the wavelet shape). This is a result formalized by I. Daubechies, for instance.
For a long time, apart from the Haar or the Franklin wavelets, little was known about the possibility of obtaining a critical (non-redundant) discrete wavelet transform. Until the inception of the Meyer wavelet, and the subsequent ones (Daubechies, Symmlets, Coiflets, etc.). Their somewhat weird shapes is the sign that not every wavelet shape is possible. This is even truer as the complex Morlet is complex, hence somewhat already twice redundant for real signals.
So, Morlet cannot be critically discretized with 2-band (dyadic), as you already guessed, as you wanted to compare CWT and DWT. However, multi-band filter banks are an option to build orthogonal counterparts of "Morlet"-like design. Malvar wavelets, Modulated complex lapped transforms or Lapped orthogonal transforms are instances (see a Panorama of wavelets for references), and their can be further decomposed across scales as $M$-band wavelets.
Or you can accept a non perfect reconstruction dyadic decomposition akin to Morlet.
answered Jul 10, 2017 at 17:25
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1$\begingroup$ Which critical DWT wavelet is the most sinusoid-like? $\endgroup$– endolithAug 9, 2017 at 19:49
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1$\begingroup$ What a darn hard question. Inherently, IMHO, none for finite support 2-band wavelets: forget about Haar, and the others lack symmetry. You could try Malvar-Wilson wavelets, described in Chapter 6 of Wavelets: Tools for Science and Technology, by Stéphane Jaffard, Yves Meyer, Robert Dean Ryan $\endgroup$ Aug 9, 2017 at 20:05
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$\begingroup$ What are the technical conditions mentioned in your first sentence? Given that we can discretize some continuous wavelet frame for perfect reconstruction, can we always implement this discretized wavelet frame by using filter banks? $\endgroup$– ArianJAug 9 at 15:02