Fast Maximum Likelihood Joint Estimation of Frequency and Frequency Rate

A fast maximum likelihood algorithm is presented that jointly estimates the frequency and frequency rate of a sinusoid corrupted by additive Gaussian white noise. It consists of a coarse search and a fine search. First the two-dimensional frequency-frequency rate plane is subdivided into parallelograms whose size depends on the region of convergence of Newton's method used in maximizing the log-likelihood function (LLF). The size of the parallelogram is explicitly computed and is optimal for the method used. The coarse search consists of maximizing the LLF over the vertices of the parallelograms. Then starting at the vertex where the LLF attained its maximum, a two-dimensional Newton's method to find the absolute maximum of the LLF is implemented. This last step consists of the fine search. The rate of convergence of Newton's method is cubic, and is extremely fast. Furthermore Newton's method will converge after two iterations when the starting point used in the method lies within 75 percent of the distances defined by the parallelogram of convergence whose center coincides with the true values of frequency and frequency rate. In this case, the root mean square error (RMSEs) for frequency and frequency rate are practically equal to the Cramer-Rao bound at all signal-to-noise ratio (SNR)≥15 dB. The frequency-frequency rate ambiguity function is shown to be even and its periodicities are extracted.