Detecting finite bandwidth periodic signals in stationary noise using the signal coherence spectrum

All signals that appear to be periodic have some sort of variability from period to period regardless of how stable they appear to be in a data plot. A true sinusoidal time series is a deterministic function of time that never changes and thus has zero bandwidth around the sinusoid's frequency. A zero bandwidth is impossible in nature since all signals have some intrinsic variability over time. Deterministic sinusoids are used to model cycles as a mathematical convenience. Hinich [IEEE J. Oceanic Eng. 25 (2) (2000) 256-261] introduced a parametric statistical model, called the randomly modulated periodicity (RMP) that allows one to capture the intrinsic variability of a cycle. As with a deterministic periodic signal the RMP can have a number of harmonics. The likelihood ratio test for this model when the amplitudes and phases are known is given in [M.J. Hinich, Signal Processing 83 (2003) 1349-1352]. A method for detecting a RMP whose amplitudes and phases are unknown random process plus a stationary noise process is addressed in this paper. The only assumption on the additive noise is that it has finite dependence and finite moments. Using simulations based on a simple RMP model we show a case where the new method can detect the signal when the signal is not detectable in a standard waterfall spectrogram display.