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I am working on a time sensitive computer science and fluid dynamics project that requires me to find applications of wavelet analysis. I know that at its core, a wavelet transform simply takes a signal and decomposes it into wavelets. What I am confused by is 1) how this can be represented (i.e. as a series of functions? etc) and 2) how said representation can be analyzed (what kind of information can I extract from it, and how?). I've seen some papers online, but I still can't wrap my head around this stuff.

I personally would like to gain a deeper understanding of wavelets, but again, this is time sensitive, so unfortunately I don't have this luxury at the moment.

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  • $\begingroup$ Look at the stuff on wavelets in the textbook First Course in Wavelets with Fourier Analysis. It is very stripped down and is very good at building intuition with the Haar wavelet. $\endgroup$ – Cameron Williams Jul 24 '13 at 22:05
  • $\begingroup$ Unfortunately I don't really have enough time to even order that book ... really I'm just looking into pushing stuff into Matlab lab functions. I just want to know what the meaning of the output is. The project is exploratory, which explains its open ended nature. $\endgroup$ – William Jul 24 '13 at 22:35
  • $\begingroup$ It is available as a pdf online.. $\endgroup$ – Cameron Williams Jul 24 '13 at 23:10
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Wow this was an old question, but you never know, maybe it could help someone.

Wavelets aren't easy to get an in-depth understanding of very fast. Wavelets are functions which are self-similar at different scales. There is a father and a mother wavelet. The father wavelet captures in some sense low Fourier frequencies (especially mean values - "DC" level) and the mother wavelet captures in some sense high Fourier frequencies. For the discrete wavelet transform there is a recursive relation between the filters and the wavelet functions which make fast transformation possible - instead of having to use larger filters for the lower frequency bands we can use redundancies deliberately introduced in the construction of the wavelet to speed up calculations a lot. You could in some sense compare this to fast Fourier transform (which actually builds on the high-school relation between sines and cosines) allowing fast transformation using the Fourier transform.

For each new scale level of a wavelet transform you get

  1. a bunch of details (captured by the mother wavelet - the wavelet filter) and
  2. a coarser version of the signal (captured by the father wavelet - the scaling filter).

Then you iterate on 2 -- the coarse signal, finding details for the next scale level ending up with an even coarser signal and so on.

If you want to learn this well you will need to spend time on it. It is enough of a challenge implementing the algorithm on a short time frame, but there really is lots of theory to really understand it and not only be able to use it.

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