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

The analysis of (approximately) periodic signals is an important element in numerous applications. One generalization of standard periodic signals often occurring in practice is harmonic chirp signals where the instantaneous frequency increases/decreases linearly as a function of time. A statistically efficient estimator for extracting the parameters of the harmonic chirp model in additive white Gaussian noise is the maximum-likelihood (ML) estimator, which recently has been demonstrated to be robust to noise and accurate-even when the model order is unknown. The main drawback of the ML estimator is that only very computationally demanding algorithms for computing an estimate are known. In this paper, we give an algorithm for computing an estimate to the ML estimator for a number of candidate model orders with a much lower computational complexity than previously reported in the literature. The lower computational complexity is achieved by exploiting recursive matrix structures, including a block Toeplitz-plus-Hankel structure, the fast Fourier transform, and using a two-step approach composed of a grid and refinement step to reduce the number of required function evaluations. The proposed algorithms are assessed via Monte Carlo and timing studies. The timing studies show that the proposed algorithm is orders of magnitude faster than a recently proposed algorithm for practical sizes of the number of harmonics and the length of the signal.