Existence and estimation of impropriety in real rhythmic signals

Impropriety in complex signal processing has been studied and used primarily in a communications context, but also in some cases where complex signals are generated by adding real signals in quadrature. We discuss the meaning of impropriety, and the associated use of complementary statistics, when a real-valued random process is improper in the frequency domain. Through the use of modulation signal models, spectral impropriety can be connected explicitly to the frequency and phase of components belonging to a periodic, or more generally rhythmic, modulator waveform. We give theoretical signal models and provide an example of complementary analysis on underwater propeller noise from a merchant ship.