Discrete Linear Canonical Transform Based on Hyperdifferential Operators

TL;DR

This paper introduces a new discrete linear canonical transform (DLCT) based on hyperdifferential operators, providing a structured, operator-theoretic approach that aligns with the discrete Fourier transform and enables straightforward computation.

Contribution

The paper develops a novel DLCT framework using hyperdifferential operators, establishing a consistent and compatible discrete analogue of the continuous LCT.

Findings

DLCT matrix is compatible with DFT and circulant structures

The DLCT can be computed by simple matrix multiplication

Provides a structurally analogous discrete transform to the continuous LCT

Abstract

Linear canonical transforms (LCTs) are of importance in many areas of science and engineering with many applications. Therefore a satisfactory discrete implementation is of considerable interest. Although there are methods that link the samples of the input signal to the samples of the linear canonical transformed output signal, no widely-accepted definition of the discrete LCT has been established. We introduce a new approach to defining the discrete linear canonical transform (DLCT) by employing operator theory. Operators are abstract entities that can have both continuous and discrete concrete manifestations. Generating the continuous and discrete manifestations of LCTs from the same abstract operator framework allows us to define the continuous and discrete transforms in a structurally analogous manner. By utilizing hyperdifferential operators, we obtain a DLCT matrix which is totally compatible with the theory of the discrete Fourier transform (DFT) and its dual and circulant structure, which makes further analytical manipulations and progress possible. The proposed DLCT is to the continuous LCT, what the DFT is to the continuous Fourier transform (FT). The DLCT of the signal is obtained simply by multiplying the vector holding the samples of the input signal by the DLCT matrix.