Eigenfunctions of the linear canonical transform

The linear canonical transform (LCT) is a powerful tool for signal processing applications and is also the generalized form of the well-known transforms such as Fourier, fractional Fourier and Fresnel transforms, and some operators such as scaling, chirp multiplication and chirp convolution. The LCT is characterized by a 2 × 2 unit-determinant matrix of parameters {a, b, c, d}. When studying linear operators, one of the crucial steps in order to understand what the operators do, is to find their eigenfunctions. Eigenfunctions of the LCT may be useful for applications such as self-imaging and resonance problems. In this paper, we derive eigenfunctions of the LCT for different cases. In order to expose eigenfunctions of the LCT, we split our investigation into three parts: (a + d) >; 2, (a + d) <; -2, and |a+d| <; 2. The complete set of eigenfunctions is known to be chirp functions modulated by scaled harmonic oscillator (also known as Hermite-Gaussian) functions for the case |a+d| <; 2. However, we propose that when |a + d| >; 2, the eigenfunction set of the LCT are chirp functions modulated with imaginary-width Hermite polynomials.