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I'm looking for wavelets that are both:

  • causal (a wavelet coefficient $W(j,k)$, where $j$ is scale and $k$ is translation, does not use data from $t > k$ to be estimated).
  • complex (such as the Morlet wavelet, to analyze phase differences between two time-series).

Do any exist?

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You can always just build your own, for instance by taking the Morlet wavelet shift it to the right until, say, 99% of the area below the function is on the positive axis and then set everything on the negative axis to zero. But bear in mind that it's impossible to build a wavelet (or filter which is the same) that is causal and analytic, i.e., a filter that erases all negative frequencies, which you actually need to get that property you need (separation of phase and amplitude). However a pretty good approximation is always possible.

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How about the dual-tree complex wavelet transform. You essentially have a complex wavelet, the real and imaginary parts of which comprise a (approximate) Hilbert pair, so you get something quite akin to the Fourier transform, in which displacement in the time domain manifests as a phase change in the DTCWT domain, with a slowly varying amplitude.

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