Fast Discrete Linear Canonical Transform Based on CM-CC-CM Decomposition and FFT

TL;DR

This paper introduces a fast, sampling-period-independent discrete LCT method based on CM-CC-CM decomposition and FFT, offering lower complexity, higher accuracy, and perfect reversibility compared to previous approaches.

Contribution

The paper presents a novel DLCT method using CM-CC-CM decomposition and FFT that avoids sampling issues and provides perfect reversibility, improving efficiency and accuracy.

Findings

Lower computational complexity than previous methods.

Improved approximation accuracy over CDDHFs-based DLCT.

Achieves perfect reversibility without additional inverse transform.

Abstract

In this paper, a discrete LCT (DLCT) irrelevant to the sampling periods and without oversampling operation is developed. This DLCT is based on the well-known CM-CC-CM decomposition, that is, implemented by two discrete chirp multiplications (CMs) and one discrete chirp convolution (CC). This decomposition doesn't use any scaling operation which will change the sampling period or cause the interpolation error. Compared with previous works, DLCT calculated by direct summation and DLCT based on center discrete dilated Hermite functions (CDDHFs), the proposed method implemented by FFTs has much lower computational complexity. The relation between the proposed DLCT and the continuous LCT is also derived to approximate the samples of the continuous LCT. Simulation results show that the proposed method somewhat outperforms the CDDHFs-based method in the approximation accuracy. Besides, the proposed method has approximate additivity property with error as small as the CDDHFs-based method. Most importantly, the proposed method has perfect reversibility, which doesn't hold in many existing DLCTs. With this property, it is unnecessary to develop the inverse DLCT additionally because it can be replaced by the forward DLCT.