On the moments of the Wigner distribution and the fractional Fourier transform

A Fourier transformation maps a one-dimensional time signal into a one-dimensional frequency function, the signal spectrum. Although the Fourier transform provides the signal's spectral content, it fails to indicate the time location of the spectral components, which is important, for example, when we consider non-stationary or time-varying signals. In order to describe such signals, time-frequency representations (TFRs) are used [1], which map one-dimensional time signal into a two-dimensional functions of time and frequency. In this paper we consider the fractional Fourier transform [2], which belongs to the class of linear TFRs, and establish its connection to the Wigner distribution, which is one of the most widely used quadratic TFRs in electrical engineering. In particular we will use the Radon-Wigner transform, which relates projections of TFRs to the squared modulus of the fractional Fourier transform. The application of the different TFRs often depends on how informative their moments are and how easily these moments can be measured or calculated. The established connection between the Wigner distribution and the Radon-Wigner transform permits to find an optimal way for the calculation of the known Wigner distribution moments and to introduce fractional Fourier transform moments which can be useful for signal analysis. We conclude that all frequently used moments of the Wigner distribution can be obtained from the Radon-Wigner transform. [1] F. Boudreaux-Bartels, "Mixed time-frequency signal transformations," in The Transforms and Applications Handbook, A.D. Poularikas, Ed., CRC Press, Alabama (1996) 887-962. [2] L.B. Almeida, "The fractional Fourier transform and time-frequency representations," IEEE Trans. Signal Process. 42 (1994) 3084-3091.