Sinusoidal frequency estimation via sparse zero crossings

We consider estimation of the period of a sinusoid in additive Gaussian noise, based on observations of the zero-crossing (ZC) times. The problem is treated in a continuous-time framework. It is assumed that the signal-to-noise ratio is sufficient (approximately ⩾8 dB) such that the noise may be approximated as additive in the phase. An exact mean-square error analysis is provided for this approximation. We apply modified Euclidean algorithms (MEAs) and their least-squares refinements in this framework, to estimate the period of the sinusoid, with low complexity. Unlike linear regression methods based on phase samples, the proposed approach works with very sparse ZC measurements, and is resistant to outliers. The MEA-based approach is motivated by the fact that, in the noise-free case, the greatest common divisor (gcd) of a sparse set of the first differences of the zero crossing times is very highly likely to be the half-period of the sinusoid. The MEA acts to robustly estimate the gcd of the observed noisy data. The MEA period estimate may be refined via a least-squares approach, that asymptotically achieves the appropriate Cramer–Rao bound. Simulation results illustrate the algorithms with as few as 10 zero-crossing times. The algorithm behavior is also studied using Bernoulli and random burst models for the missing ZC times, and good performance is demonstrated with very sparse observations.