Unsupervised Frequency Tracking beyond the Nyquist Limit using Markov Chains

TL;DR

This paper introduces an unsupervised Bayesian method using Markov chains to accurately track frequencies from very short, noisy signals, surpassing the Nyquist limit and reducing variance in estimates.

Contribution

It presents a novel unsupervised approach combining Markov models, Viterbi, and EM algorithms for frequency tracking beyond Nyquist, with automatic hyperparameter tuning.

Findings

Outperforms reference methods in variance reduction

Achieves correct frequency tracking beyond Nyquist frequency

Automatically adjusts hyperparameters using EM-based gradient algorithm

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 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.