Fractional convolution, correlation theorem and its application in filter design

In this paper, fractional convolution and correlation structures are proposed. The corresponding theorems for fractional Fourier transform (FRFT) are derived, which state that fractional convolution in the time domain is equivalent to a simple multiplication operation for FRFT and FT domain; this feature is more instrumental for the multiplicative filter model in FRFT domain. Moreover, the fractional convolution operation proposed in this paper can be expressed as ordinary convolution form in FT domain; such expression is particularly useful and easy to implement in filter design in time domain. Classical convolution and correlation theorems for Fourier transform (FT) are shown to be special case of our achieved result. The potential application of the derived result in filter design is also discussed, and the proposed method has lower computational complexity and can be easily implemented in FRFT domain.