A fractional-Fourier-domain realization of the weighted Wigner distribution producing auto-terms close to the ones in the Wigner distribution itself, but with reduced cross-terms, is presented. Improvement over the standard time-frequency representations is achieved when the principal axes of a signal (defined as mutually orthogonal directions in the time-frequency plane for which the width of the signal's fractional power spectrum is minimum or maximum) do not correspond to time and frequency. The computational cost of this fractional-domain realization is the same as the computational cost of the realizations in the time or the frequency domain, since the windowed Fourier transform of the fractional Fourier transform of a signal corresponds to the windowed Fourier transform of the signal itself, with the window being the fractional Fourier transform of the initial one. The appropriate fractional domain is found from the knowledge of three second-order fractional Fourier transform moments. Numerical simulations confirm a qualitative advantage in the time-frequency representation, when the calculation is done in the optimal fractional domain. The approach is generalized to the time-frequency distributions from the Cohen class. Rotation of the kernel is proposed in order to align the kernel's preferred axes to the signal's principal axes. It is shown that the resulting time-frequency representations show a better reduction of cross-terms without too severely degrading the auto-terms than the corresponding, original time-frequency representations.
Stankovic, L., Alieva, T., & Bastiaans, M. J. (2003). Time-frequency signal analysis based on the windowed fractional Fourier transform. Signal Processing, 83(11), 2459-2468. https://doi.org/10.1016/S0165-1684(03)00197-X