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Signal Processing

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    $\begingroup$ Thank you for recalling these important points about wavelet theory, which are necessary indeed to understand how it works. But here the question would be more about building a framework code that would work on audio signal for example. The questions are : how to deal with these infinite sums, how to choose the windows (or rather mother-wavelet), how to do it using pyWavelets in Python (or another equivalent language, I'll translate into Python then), how to choose the parameters (like in my example for audio : sampling rate=44100, fft window = 4096, overlap=4, etc.) $\endgroup$ Commented Nov 19, 2013 at 7:51
  • $\begingroup$ Your notion of overcompleteness is not accurate. A basis being complete means, that the canonical projector onto the basis is the identity operator. But to write is as the sum of the outer product you need orthonormality. For overcompleteness you cannot have orthonormality, so that outer product decomposition does not work. But you can make it work by saying a basis is complete (and possibly overcomplete), if there are coefficients $a_k$ so that $\sum_k \left| k \rangle a_k \langle k \right| = \mathrm{Id}$ $\endgroup$ Commented Nov 19, 2013 at 10:06
  • $\begingroup$ hmm you have introduced a morlet wavelet, but all wavelet doesn't have fb and fc, so they might have constant $K$, also its impossible to make a DWT with morlet, bx its not orthogonal, actually I couldn't get a fine resolution for frequency estimation with DWT compare to the cwt or STFT @apt1002 $\endgroup$ Commented Nov 19, 2013 at 10:24
  • $\begingroup$ In addition, finding these coefficients $a_k$ is non trivial, and also non-unique. The linear dependence of the basis vector implies, that there are infinitely many possible such $a_k$ if the basis is overcomplete. That implies your "filter" is not well defined, in fact, it's not defined at all, because you don't know what you control with your filter function $f$. Each aspect of the signal is realized in many linear dependent wavelet basis vectors. So your theory falls apart. $\endgroup$ Commented Nov 19, 2013 at 11:12
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    $\begingroup$ The best way to see if it works or not would be to provide a minimal code example (with pyWavelet for example it should be possible in a few lines I imagine) (I'll do itas well it once I understand it, I think I need a few more days reading about wavelets!) $\endgroup$ Commented Nov 19, 2013 at 16:04