Digital computation of linear canonical transform for local spectra with flexible resolution ability

Dear editor, The linear canonical transform (LCT) is a threeparameter linear integral transform [1]. It has many applications in optical systems, filter design, image watermarking, and other fields [2, 3]. Moreover, the discretization and fastness of the LCT is one of the most important issues in practical applications. Since the continuous LCT was introduced, there has been considerable work on the definition and fast implementation of the LCT [3–6]. These existing discrete LCT (DLCT) algorithms have advantages of high computation speed and accuracy, but they can only calculate the N point input samples to obtain N -point output. In many applications, such as the detection and estimation of peak values, it is of more interest to study the details in a small portion of the linear canonical spectra or to calculate a single or a few output spectra with an arbitrary sampling interval. It is clear that the previous algorithms with fixed resolution cannot meet these requirements. To solve the aforementioned problems, we present two flexible algorithms: a novel type of DLCT with zooming-in ability (ZDLCT) and a singlepoint LCT (Sp-LCT) based on the Goertzel algorithm. Compared to existing digital computation methods, both of our proposed algorithms are more flexible in terms of resolution and observation intervals. Also, the Sp-LCT can be suitable for used to calculate the LCT of a non-uniform sampling signal. Before deriving the new methods for the computation of LCT, some basic preliminaries are introduced in Appendix A.