Bayesian Frequency Estimation

In the literature, the problem of frequency estimation is usually cast in a nonlinear parametric fashion. We show in this paper that it can also be formulated in a Bayesian nonparametric framework by assigning a uniform a priori probability distribution to the unknown frequency. We find that the covariance matrix of signal model is the discrete-time analogue of the integral operator whose eigenfunctions are the famous prolate spheroidal wave functions, introduced by Slepian and coworkers in the 1960's. Two methods are proposed to estimate the hyperparameters of the prior distribution. One uses techniques which are essentially linear from subspace identification. The other is based on Prediction Error Minimization. An explicit formula for the predictor is derived exploiting the exceptional decay property of the eigenvalues of the covariance matrix. In addition, a covariance estimation scheme is suggested assuming panel data in response to the nonergodicity of the process. The approach seems quite promising.