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

Abstract

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.

An Optimal Channel Estimation Scheme for Intelligent Reflecting Surfaces Based on a Minimum Variance Unbiased Estimator

“…3.12]). From (15), and without other modifications, we identify that we can attain the bound (22) if Φ H Φ = diag(d) 1 . Furthermore, say that we will try to achieve the same variance for all unknowns θ.…”

“…It is interesting to note that it is not uncommon that design choices influences the CRLB, see e.g [14,15]. where the time index is important for chirp and harmonic chirp models.…”

Estimation of parameters of a harmonic chirp model

“…Finally, in Figures 11,12,13,14,15,16 and 17,18,19,20,21, we evaluate the performance of the estimators with respect to varying α and β on the x-axis, respectively. These plots show that the estimation accuracy of the LSEs for different values of α and β is quite balanced and at par with the corresponding asymptotic results.…”

“…Popular noise models such as AR, MA or ARMA models can be represented as a stationary linear process under certain conditions and hence this assumption covers a large class of random processes. This model has been suggested recently by Christensen and Jensen [1] and thereafter some authors, for instance, Doweck, Amar and Cohen [14,15], Nørholm, Jensen and Christensen [16], Sward, Brynolfsson, Jakobsson and Hansson-Sandsten [17], Jensen, Nielsen, Jensen, Christensen and Jensen [18] and Nielsen, Jensen, Jensen, Christensen and Jensen [19] have considered its parameter estimation. In real life applications, this model captures the deterministic component of the signals more accurately as it takes into account the modulations in the frequency of the observed signal.…”

“…Sward, Brynolfsson, Jakobsson and Hansson-Sandsten [17] consider the same model and provide an iterative sparse reconstruction framework, which is an extension of the work done by Adalbjörnsson, Jakobsson and Christensen [27] on the harmonic model. Jensen, Nielsen, Jensen, Christensen and Jensen [18] consider a real-valued harmonic chirp model and provide an algorithm for computing an estimate to the ML estimator with much lower computational complexity using recursive matrix structures, the fast Fourier transform, and using a two-step approach that reduces the number of required function evaluations. Nielsen, Jensen, Jensen, Christensen and Jensen [19] proposed an accurate and robust fast method for harmonic chirp summation.…”