Matrix-based Fourier analysis of matrix signals and systems for polarization optics

Matrix functions are, of course, indispensable and of primary concern in polarization optics when the vector nature of light has been considered. This paper is devoted to investigating matrix-based Fourier analysis of two-dimensional matrix signals and systems. With the aid of the linearity and the superposition integral of matrix functions, the theory of linear invariant matrix systems has been constructed by virtue of six matrix-based integral transformations [i.e., matrix (direct) convolution, matrix (direct) correlation, and matrix element-wise convolution/correlation]. Properties of the matrix-based Fourier transforms have been introduced with some applications including the identity impulse matrix, matrix sampling theorem, width, bandwidth and their uncertainty relation for the matrix signal, and Haagerup's inequality for matrix normalization. The coherence time and the effective spectral width of the stochastic electromagnetic wave have been discussed as an application example to demonstrate how to apply the proposed mathematical tools in analyzing polarization-dependent Fourier optics.