Bayesian Frequency Estimation on Narrow Bands

Abstract In some recent works, the authors have proposed and developed an Empirical Bayes framework for frequency estimation. The unknown frequencies in a noisy oscillatory signal are modeled as uniform random variables supported on narrow frequency bands. The bandwidth and the relative band centers are known as hyperparameters which can be efficiently estimated using techniques from subspace identification. In the current paper, we examine carefully how the estimated frequency prior can be used to produce a Bayesian estimate of the unknown frequencies based on the same data (for hyperparameter estimation). To this end, we formulate the Bayesian Maximum A Posteriori (MAP) optimization problem and propose an iterative algorithm to compute its solution. Then, we do extensive simulations under various parameter configurations, showing that the MAP estimate of the frequencies are asymptotically close to the band centers of the frequency priors. These results provide an attractive link between the conventional Bayesian method and the Empirical Bayes method for frequency estimation, and in retrospect justify the use of the latter.