Maximum likelihood estimation of multiple frequencies with constraints to guarantee unit circle roots

An approximate maximum-likelihood estimator (MLE) of multiple exponentials converts the frequency estimation problem into a problem of estimating the coefficients of a z-polynomial with roots at the desired frequencies. Theoretically, the roots of the estimated polynomial should fall on the unit circle, but MLE, as originally proposed, does not guarantee unit circle roots. This drawback sometimes causes merged frequency estimates, especially at low SNR. If all the sufficient conditions for the z-polynomial to have unit circle roots are incorporated, the optimization problem becomes too nonlinear and it loses the desirable weighted-quadratic structure of MLE. In the present paper, the exact constraints are imposed on each of the first-order factors corresponding to individual frequencies for ensuring unit circle roots. The constraints are applied during optimization alternately for each frequency. In the absence of any merged frequency estimates, the RMS values more closely approach the theoretical Cramer-Rao (CR) bound at low SNR levels.<>