Unsupervised Frequency Tracking beyond the Nyquist Limit using Markov Chains

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 specificity 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 postariori 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 information is done automatically by ML using an 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.

💡 Research Summary

The paper addresses the challenging problem of estimating a time‑varying frequency sequence from extremely short, noisy observations—a situation typical in Doppler imaging where only four samples per range bin are available and the sampling rate may induce severe aliasing. Classical approaches based on periodograms, mean‑frequency estimates, or two‑step alias‑inversion procedures suffer from high variance, bias near the Nyquist limit, and a reliance on second‑order statistics that are unreliable with such limited data.

To overcome these limitations, the authors formulate a fully Bayesian model. The observed complex vector yₜ at time (or range) index t is modeled as yₜ = aₜ·z(νₜ) + bₜ, where aₜ is a complex amplitude, bₜ is additive white Gaussian noise, and z(νₜ) =