# Wigner distribution weighted in the fractional Fourier domain

## LJubisa Stankovic, Tatiana Alieva, and Martin J. Bastiaans

Different types of joint time-frequency representations are used in signal processing. The advantages and disadvantages of most representations are well known. Thus, for example, the short-time Fourier transform (STFT) of multicomponent signals, although not stressing the auto-terms very well, is free from cross-terms, if the components do not overlap. On the other hand, the Wigner distribution (WD) of such signals is highly concentrated, but suffers from cross-terms, which may hide some of the auto-terms. In particular the S-method weighted WD (SWWD) of a multicomponent signal leads to a representation with significantly reduced cross-terms, while the auto-terms are close to or exactly the same as the ones in the WD. The SWWD is based on weighting the STFT, and two different forms of it have been proposed, based on either an averaging in the time direction or an averaging in the frequency direction.

In this paper we introduce the SWWD in the fractional Fourier transform (FT) domain, where we have to calculate the STFT of the fractional FT of the signal, followed by an averaging with an appropriate weighting window. As the STFT of a fractional FT corresponds to the STFT of the signal itself, but with the window function being the fractional FT of the original one and with an additional rotation of the coordinate system, the STFT in a fractional domain can be calculated with roughly the same computational costs as for the normal STFT.

In order to find the most appropriate fractional domain in which to apply the weighted averaging, the properties of fractional FT moments are used. In particular, we suppose that an optimal SWWD calculation direction corresponds to minimal signal width, i.e., minimal fractional second-order moment. Determination of this moment can be done analytically, based on three known moments for three different directions.

Numerical simulations show a qualitative advantage in the time-frequency representation, when the calculation is done in this optimal fractional domain.

To: Papers by Martin J. Bastiaans