Random Signal Estimation by Ergodicity associated with Linear Canonical Transform

The linear canonical transform (LCT) provides a general mathematical tool for solving problems in optical and quantum mechanics. For random signals, which are bandlimited in the LCT domain, the linear canonical correlation function and the linear canonical power spectral density can form a LCT pair. The linear canonical translation operator, which is used to define the convolution and correlation functions, also plays a significant role in the analysis of the random signal estimation. Firstly, the eigenfunctions which are invariant under the linear canonical translation and the unitarity property of it are discussed. Secondly, it shows that all of these connect the LCT sampling theorem and the von Neumann ergodic theorem in the sense of distribution, which will develop an estimation method for the power spectral density of a chirp stationary random signal from one sampling signal in the LCT domain. Finally, the potential applications and future work are discussed.