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.
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Pei, SC., Ding, JJ. (2016). Eigenfunctions of the Linear Canonical Transform. In: Healy, J., Alper Kutay, M., Ozaktas, H., Sheridan, J. (eds) Linear Canonical Transforms. Springer Series in Optical Sciences, vol 198. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3028-9_3
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