Probabilistic representation of high-dimensional random signals via octonion linear canonical transform

The octonion linear canonical transform (OCLCT) extends the traditional linear canonical transform (LCT) to the octonion algebra, enabling effective processing of higher-dimensional signals. Emerging as a cutting-edge tool for high-dimensional signal analysis, OCLCT offers enhanced capabilities for handling high-dimensional non-stationary signals. This paper explores the properties of OCLCT and introduces probability theory in the OCLCT domain. Firstly, the basic properties of OCLCT, such as boundedness, parity, and shift, are presented, and the convolution theorem of OCLCT is also derived. Secondly, we establish the probabilistic framework for OCLCT, defining the mean, characteristic function in the octonion domain. In addition, the probability theory in the three-dimensional OCLCT domain is also discussed. Finally, numerical simulations validate the proposed theory, including characteristic function computation and distribution visualization for octonion-valued densities.