Phase-Continuous Frequency Line Track-Before-Detect of a Tone with Slow Frequency Variation

We consider optimal Bayesian detection of a slowly varying tone of unknown amplitude in situations characterized by very low signal-To-noise ratio (SNR) and a large number of measurements, as found in certain gravitational wave and passive sonar problems. We use a hidden Markov model (HMM) framework but, unlike typical HMM-based frequency line tracking methods, we develop a true track-before-detect algorithm, which does not threshold the blocked Fourier data and only considers frequency trails that have phase continuity across all HMM steps. We model the frequency and phase evolution as a phase-wrapped Ornstein-Uhlenbeck process. The resulting optimal detector is computationally efficient. The detectability improvement arising from phase continuity is characterized via comparative simulation for a mock, simplified gravitational wave search problem.