Theory of prediction, interpolation and filtering of α-stationary random signals

The Wiener filter, considered as the optimal filter for stationary signals, have an important place in the filtering theory and allows many applications. However, when it comes to the filtering of non-stationary signals, it is necessary to replace it, for example, by the Kalman filter which achieves better results. This paper develops an alternative version of the Wiener filter, called fractional Wiener filter, based on the fractional convolution and the fractional correlation together with the notion of α-stationarity. The fractional Wiener filter, through a parameter called fractional order, can deal with the non-stationary signals as well as the stationary signals in a natural way, and set the basis for a theory of prediction and interpolation from a generalization of the Wiener-Hopf equation. Computational simulations were carried out to show the capabilities of the fractional Wiener filter, where the filtering was shown to be very sensitive to small variations of the fractional order.