Stochastic time-frequency analysis using the analytic signal: why the complementary distribution matters

We challenge the perception that we live in a "proper world", where complex random signals can always be assumed to be proper (also called circularly symmetric). Rather, we stress the fact that the analytic signal constructed from a nonstationary real signal is, in general, improper, which means that its complementary correlation function is nonzero. We explore the consequences of this finding in the context of stochastic time-frequency analysis in Cohen\´s class. There, the analytic signal plays a prominent role because it reduces interference terms. However, the usual time-frequency representation (TFR) based on the analytic signal gives only an incomplete signal description. It must be augmented by a complementary TFR whose properties we develop in detail. We show why it is still advantageous to use the pair of standard and complementary TFRs of the analytic signal rather than the TFR of the corresponding real signal.