Unsupervised frequency tracking beyond the Nyquist frequency using Markov chains

Abstract—This paper deals with the estimation of a sequence of frequencies from a corresponding sequence of signals. This problem arises in fields such as Doppler imaging, where its speci-ficity is twofold. First, only short noisy data records are available (typically four sample long), and experimental constraints may cause spectral aliasing so that measurements provide unreliable, ambiguous information. Second, the frequency sequence is smooth. Here, this information is accounted for by a Markov model, and application of the Bayes rule yields the a posteriori density. The maximum a posteriori is computed by a combination of Viterbi and descent procedures. One of the major features of the method is that it is entirely unsupervised. Adjusting the hyperparameters that balance data-based and prior-based infor-mation is done automatically by maximum likelihood (ML) using an expectation-maximization (EM)-based gradient algorithm. We compared the proposed estimate to a reference one and found that it performed better: Variance was greatly reduced, and tracking was correct, even beyond the Nyquist frequency. Index Terms—Aliasing inversion, Bayesian statistic, EM al-gorithm, forward-backward procedure, frequency tracking, hyperparameter estimation, maximum a posteriori, maximum likelihood, meteorological Doppler radar, regularization, ultra-sonic Doppler velocimetry, Viterbi algorithm