A recursive fast algorithm for the linear canonical transform
Bryan M. Hennelly, John T. Sheridan · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 2005
The Linear Canonical Transform (LCT) describes the effect of any Quadratic Phase System (QPS) on an input optical wavefield. Special cases of the LCT include the fractional Fourier transform (FRT), the Fourier transform (FT) and the Fresnel Transform (FST) describing free space propagation. We have recently published theory for the Discrete Linear Canonical Transform (DLCT), which is to the LCT what the Discrete Fourier Transform (DFT) is to the FT and we have derived the Fast Linear Canonical Transform (FLCT), a NlogN, algorithm for its numerical implementation using an approach similar to that used in deriving the FFT from the DFT. The algorithm is significantly different to the FFT and is based purely on the properties of the LCT and can be used for fast FT, FRT and FST calculations and in the most general case to rapidly calculate the effect of any QPS. In this paper we develop theory making the algorithm recursive for ease of implementation. We derive the FLCT butterfly and graph a flowchart for the recursive algorithm.