Discrete Estimations of Cross-Correlation Components of Periodically Correlated Random Signals I. N. Yavorskyj 1,2,* , R. Yuzefovych 1,** , I. Y. Matsko 1 , and Z. Zakrzewski 2 1 Karpenko Physico-Mechanical Institute of NASU, Lviv, Ukraine 2 University of Technology and Life Sciences, Bydgoszcz, Poland *e-mail: iavor@ipm.lviv.ua **e-mail: abzac@ipm.lviv.ua Received in final form December 9, 2013 Abstract—Properties of cross-correlation components estimations of two periodically correlated random signals are analyzed. The conditions of absence of the first and the second genus aliasing effects are obtained. The formulas for estimations variance which allow to select a valid sampling step in dependences on realization length and signal properties are derived. The results are concretized for amplitude modulated signals. DOI: 10.3103/S0735272714020034 Sampling is necessary procedure for statistical signals processing by means of computer equipment. One of the most important problem for its realization is a selection of sampling step. Traditionally this step is selected using Shannon–Kotelnikov theorem and its generalizations. Correspondent the least sampling frequency is called Nyquist frequency. In this connection there is a question, whether the sampling step, selected in such way, satisfies the conditions for solution the problems of signal statistical analysis. Especially it is concern of statistics of non-stationery processes, where it is necessary to operate the correlation functions of two variables even in the second order theory, and these functions behavior properties can be different with regard to each variable. In this paper there are represented results of analysis of discrete estimations of Fourier coefficients of cross-correlation function of periodically correlated random processes (PCRP), i.e. mathematical model of the signals, describing their stochasticity, and repeatability of their properties [1–5]. Properties of repeatability and stochasticity are typical for most signals applying in the systems of communication, telemetry, radio and hydrolocation. These properties are achieved by signals after processes of modulation scanning, encoding, antenna rotation, etc. [3–5]. Mathematical expectation of PCRP is m t E t x x () () = and its correlation b tu E t t u x x x (,) ()( ) = + o o , x x x o () () () t t m t = - are time periodical functions m t m t T x x () ( ) = + , b tu b t Tu x x (,) ( ,) = + . Signals models in form of stationery random processes can be considered as stationery approximation of PCRP: their characteristics are determined by time averaging of the last characteristics. Application of periodical non-stationerity gives an opportunity for creation of such facilities for detection, filtering, classification, and also estimation the parameters, exceeding analogous ones, based on stationery approach [3, 4]. Analysis of single- and multi-channel communication systems, identification random processes propagation paths, definition of their destinations and sources require analysis of interrelation of two or several PCRP. In [6] there are represented main properties of cross- correlation function of such signals, and also properties of coherent estimations by continuous realization was carried out. This paper develops these results on discrete case. Mutual correlation function of interrelated PCRP b tu E t t u xh x h (,) ()( ) = + o o , h h h o () () () t t m t = - , m t E t h h () () = is periodical time t function b t Tu b tu xh xh ( ,) (,) + = and it can be represented in form of Fourier series: 78 ISSN 0735-2727, Radioelectronics and Communications Systems, 2014, Vol. 57, No. 2, pp. 78–91. © Allerton Press, Inc., 2014. Original Russian Text © I.N. Yavorskyj, R. Yuzefovych, I.Y. Matsko, Z. Zakrzewski, 2014, published in Izv. Vyssh. Uchebn. Zaved., Radioelektron., 2014, Vol. 57, No. 2, pp. 29–42.