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What could be the arguments of using fractional Fourier transform instead of multiscale wavelet for data analysis ?

  • Optimization of the good time-frequency domain parameter? good in the sens of best time-frequency domains that minimize spectral entropy of the data. But often, best entropy is achieved in a full 100% frequency domain.

  • Due to the physic world behavior, particle-path is locally described by fractional Fourier in N-slit problem context. Usefull for describing binary brownian data or differential binary brownian data ? ( sign(cumsum(randn(n,1))) or abs(diff(sign(cumsum(randn(n,1))))) )

  • Wavelet could be thinking as just a fast and efficient dyadic scheme of a particular sparse fractional transform ?

  • Continuous transformation from time to frequency: What would be the purpose of this transformation? Maybe to bring up patterns?

I really appreciate any strong argument about advantage of fractional Fourier Transform in comparison to wavelet.

(Our universe choosed fractional transform coding scheme for particles, there must be a reason of efficiency somewhere...)

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  • $\begingroup$ Can you define what "fractional Fourier transform" is? $\endgroup$ – Wolfgang Bangerth Feb 15 '20 at 0:08
  • $\begingroup$ FrFT is the eigendecomposition of Fourier Matrix via Hermite Functions Basis (solution of Schrodinger equation for a harmonic oscillator in quantum mechanics). FrFT allow put an exponent $\alpha$ on the eigenvalue matrix. $$ F = H E H', F^2 x(t) = x(-t), F^4 = Id, F^5 = F$$, $$ F^\alpha = H E^\alpha H' $$ That give you a continuous transformation from Time to Frequency as a 90° degrees rotation. There is a trick, when $\alpha$ is low ($< 45°$), you can fast approximate $F^\alpha$ via Two Traditional FFT. $$F^\alpha = D_1 F D_2 F' D_1 $$ $\endgroup$ – sharl Feb 15 '20 at 18:15
  • $\begingroup$ Thanks to this, you can fast recursively approximate the hermite polynomials transform (derivation of $e^{-x^2}$). Physically speaking, it's what happen near a two-slit experiment (and all diffractions variations), $\alpha$ is the distance from the slit. $\endgroup$ – sharl Feb 15 '20 at 18:15

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