ENSEMBLE AVERAGES of WIGNER DISTRIBUTIONS of NOISE and TELECOMMUNICATION SIGNALS with EMPHASIS on the ROLE of CROSS-TERMS

Abstract: The paper presents a study of properties of ensemble averages of Wigner time-frequency distributions (WDs) of random processes defined by statistically independent samples of stationary and nonstationary Gaussian noise and of radio-frequency telecommunication signals. We used samples of PSK and FSK signals transmitting random telegraph signals. The WDs of single samples of these random processes have the form of random bipolar fields W(t, f), while ensemble averages E{W(t, f)} are well defined deterministic functions. The WDs of real signals have the form of a sum of an even term and a cross-term. In selected cases, theoretical forms of ensemble averages of these terms are derived and compared with computer simulations. In other cases, only computer simulations are applied. It was shown that ensemble averages of even terms are usually (but not always) unipolar and of cross-terms bipolar well-defined deterministic functions. For socalled proper processes the ensemble averages of cross-terms are cancelled. In computer simulations, their level decreases with increasing number of samples. In other cases, cross-terms carry an information about some properties of a random process. Derivations show that the notion of a cross-term coincides with the double value of the real part of the so-called complementary Wigner distribution. In consequence, the paper yields the answer to the question, in which cases the complementary distribution matters and yields a deeper insight into the properties and role of cross-terms.