Fast algorithm for determination of linear canonical transform and fractional fourier transform

Abstract

The fractional Fourier transform (FRT) is a recently developed tool for linear space-frequency (or time-frequency) signal processing. It is a generalisation of the Fourier transform, FT, has a simple optical implementation and can potentially find use wherever the FT is used. We discuss and compare the most popular fast algorithms currently being used (in the area of optical signal processing, OSP) to calculate the digital FRT. We develop theory for the discrete Linear Canonical Transform (LCT) of which the FRT is a special case. We then derive the Fast Linear Canonical Transform (FLCT), a fast algorithm for its implementation using a similar approach as was done with the derivation of the Fast Fourier Transform (FFT) for implementation of the Discrete Fourier Transform (DFT). The new algorithm is entirely independent of the FFT and is based purely on the properties of the LCT and FRT.