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 are 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.