A Fast Algorithm for Maximum-Likelihood Estimation of Harmonic Chirp Parameters

The analysis of (approximately) periodic signals is an importantelement in numerous applications. One generalization of standardperiodic signals often occurring in practice are harmonic chirpsignals where the instantaneous frequency increases/decreases linearlyas a function of time. A statistically efficient estimator forextracting the parameters of the harmonic chirp model in additivewhite Gaussian noise is the maximum likelihood (ML) estimator whichrecently has been demonstrated to be robust to noise and accurate ---even when the model order is unknown. The main drawback of the MLestimator is that only very computationally demanding algorithms forcomputing an estimate are known. In this paper, we give an algorithmfor computing an estimate to the ML estimator for a number ofcandidate model orders with a much lower computational complexity thanpreviously reported in the literature. The lower computationalcomplexity is achieved by exploiting recursive matrix structures,including a block Toeplitz-plus-Hankel structure, the fast Fouriertransform, and using a two-step approach composed of a grid andrefinement step to reduce the number of required functionevaluations. The proposed algorithms are assessed via Monte Carlo andtiming studies. The timing studies show that the proposed algorithm isorders of magnitude faster than a recently proposed algorithm forpractical sizes of the number of harmonics and the length of thesignal.