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What are the advantages of the higher-order Generalized Morse Wavelets in comparison with the first-order ones?

Are they also more computationally expensive?

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I'm no expert nor experienced with higher-order GMWs, but I did implement them in Python and tested some of their claims.

What I do know is, the different orders are designed specifically to completement each other, which enables weighted scalograms. From the original paper, cropped Figure 4:

enter image description here

Left is zeroth-order ("first") GMW, right is first, each showing scalogram of the sum of their real and imaginary parts. Left we observe the typical high concentration around the time-frequency point of interest, while right actually has a hole (minimum) in that very spot. They hence adhere to different notions of center frequency, but more importantly, in a complementing manner. Though, I don't know where first-order is useful by itself.

Authors show that, averaging scalograms produced by these wavelets have a denoising effect. I have confirmed this in ssqueezepy, though the effect is weak, although I'm unsure whether the implementation is correct (whether it should be same scale or same center freq, and whether averaging is over complex or absolute values):

enter image description here

Are they also more computationally expensive?

  • Yes: in that they have much more complicated mathematical expressions than something like a Morlet, so generation (sampling) takes longer
  • No: it doesn't matter once pre-computed
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  • $\begingroup$ About the computation, when applied to real-time, it can be speed up once pre-computed, isn't it? $\endgroup$ Jun 24, 2022 at 12:15
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    $\begingroup$ @EddyPiedad Once pre-computed there's no difference between this wavelet and Morlet. It's just ifft(x_f*wavelet_f) (convolution theorem). $\endgroup$ Jun 24, 2022 at 18:23

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