Eigenfunctions of the linear canonical transform

Abstract

In this chapter, the eigenfunctions and the eigenvalues of the linear canonical transform are discussed. The style of the eigenfunctions of the LCT is closely related to the parameters {a, b, c, d} of the LCT. When |a + d| < 2, the LCT eigenfunctions are the scaling and chirp multiplication of Hermite-Gaussian functions. When |a + d| = 2 and b = 0, the eigenfunctions are the impulse trains. When |a + d| = 2 and b≠0, the eigenfunctions are the chirp multiplications of periodic functions. When |a + d| > 2, the eigenfunctions are the chirp convolution and chirp multiplication of scaling-invariant functions, i.e., fractals. Moreover, the linear combinations of the LCT eigenfunctions with the same eigenvalue are also the eigenfunctions of the LCT. Furthermore, the two-dimensional case is also discussed. The eigenfunctions of the LCT are helpful for analyzing the resonance phenomena in the radar system and the self-imaging phenomena in optics.