Authors

Investigation of the functional dependence between the phases of diagonal coefficients and the phases of eigenvectors of Hermitian matrices

An effective way to ensure the required level of structural secrecy of code sequences used in wireless data transmission systems with multiple access and code division multiplexing is to employ a representative set of code sequence structures that can be changed according to a specific rule. A variant of increasing the structural secrecy of code sequences used in the telecommunication systems under consideration is to employ the necessary arsenal of non-repeating ensembles of multiphase orthogonal code sequences modeled by the eigenvectors of a Hermitian matrix. In known works, the functional relationship between the diagonal coefficients of a tridiagonal Hermitian matrix and the systems of its eigenvectors modeling ensembles of multiphase orthogonal code sequences has not previously been established. This precluded the demonstration of a strict deterministic relationship between the pseudo-random assignment of the phases of the diagonal coefficients of a Hermitian matrix and the properties of its eigenvectors, which determines the relevance of this study. This article proves the existence of a functional relationship between the coefficients of a Hermitian matrix and the system of its eigenvectors. It is shown that the pseudorandom uniform distribution law of the phase values of the coefficients of the second diagonal of Hermitian matrices, with constant values of their moduli, is preserved for the arguments (phases) of the coordinates of the eigenvector systems of the matrices under consideration, based on estimates by the Wald – Wolfowitz criteria, chi-square, and Welch t-criterion. Based on the analysis of the conducted experiment, it was established that the spectrum of the Hermitian matrix does not change with a pseudorandom change in the phases of its diagonal coefficients using the Mersenne Twister pseudorandom number generator with fixed values of the moduli of its diagonal coefficients. Analytical and experimental proof of the relationship between the arguments of the diagonal coefficients of a tridiagonal Hermitian matrix and the arguments of the coordinates of its eigenvector systems, considered as models of ensembles of multiphase orthogonal code sequences, shows that the statistical properties of the latter are determined by the statistical properties of the pseudorandom number generator that specifies the initial data for the stochastic transformation. A conclusion is made about the independence of the spectrum of the Hermitian matrix from the pseudo-random assignment of phase values to its diagonal coefficients with constant values of their moduli. Read more...