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Symmetric and general forms of the two-dimensional and cyclic autocorrelation functions of non-stationary random processes

DOI 10.18127/j00338486-201903-05

Keywords:

T.Ya. Shevgunov – Ph.D.(Eng.), Associate Professor, Department «Theoretical Radio Engineering», Moscow Aviation Institute (National Research University)
E-mail: shevgunov@gmail.com


The paper deals with the choice of the analytic forms of two-dimensional and cyclic autocorrelation functions used for describing wide-sense cyclostationary random processes. It is shown that all the forms of autocorrelation functions with an arbitrary chosen moment for the current time can be transformed to the symmetric form. The technique for the synthesis of symmetric autocorrelation in discrete time avoiding the interpolation and fractional time indexing is proposed. The analysis of time and time-frequency second-order characteristics used for describing probabilistic features of wide-sense cyclostationary random processes is carried out in the paper.
The general parametrized representation of the two-dimensional autocorrelation function (2D-ACF) is obtained for the arbitrary choice of the current time; the most important particular cases are derived for the symmetric form of the 2D-ACF and two non-symmetric forms related to the initial and final time instants. The correlation diagrams of three above-mentioned forms of the 2D-ACF are presented for the example of a pulse train with random uncorrelated amplitudes showing the clear cyclostationary behavior. The equivalent representations of the symmetric and general forms of the 2D-ACF are obtained via Fourier series expansion which coefficients make up the cyclic autocorrelation function (CACF).
It is shown that 2D-ACF can be transformed into the symmetric form by the translation of the current time regardless the initial choice of the current time moment. The link between the components of the CACF is established for the transformation from one form of the 2D-ACF to another. The absolute values of the CACF components are proved to be invariable about any time point chosen as the current time. A technique for the discrete-time synthesis of the symmetric two-dimensional autocorrelation function is proposed; it is based on the Fourier series representation avoiding the interpolation or fractional sample indexing.

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