Maximum likelihood estimation of the parameters of multiple sinusoids from noisy measurements

Abstract

The problem of estimating the frequencies, phases, and amplitudes of sinusoidal signals is considered. A simplified maximumlikelihood Gauss-Newton algorithm which provides asymptotically efficient estimates of these parameters is proposed. Initial estimates for this algorithm are obtained by a variation of the overdetermined Yule-Walker method and a periodogram-based procedure. Use of the maximum-likelihood Gauss-Newton algorithm is not, however, limited to this particular initialization method. Some other possibilities to get suitable initial estimates are briefly discussed. An analytical and numerical study of the shape of the likelihood function associated with the sinusoids-in-noise process reveals its multimodal structure and clearly sets the importance of the initialization procedure. Some numerical examples are presented to illustrate the performance of the proposed estimation procedure. Comparison to the performance corresponding to the Cramer-Rao lower bound is also presented, using a simple expression for the asymptotic Cramer-Rao bound covariance matrix derived in the paper.

Efficient Maximum Likelihood Estimation for Multiple and Coupled Harmonics

“…More recently, the results have been presented by Stoica et al [3] and Porat [4] and extended to include multiplicative noise by Zhou and Giannakis [5]. The estimation algorithms presented by Stoica et al [3] do not exploit a key observation made originally by Rife and Boorstyn [1], and more recently by Porat [4], on the effect of defining the phase of the signal in the middle of the time interval rather than at the beginning, which is more commonly done. Perhaps some reluctance in using the less common definition is due to the misperception (as in Porat [4]) that odd and even numbers of samples need to be treated separately.…”

“…Perhaps some reluctance in using the less common definition is due to the misperception (as in Porat [4]) that odd and even numbers of samples need to be treated separately. With the phase in the middle of the time interval, the information matrix is approximately diagonal as compared to the results in Stoica et al [3] and elsewhere, where the phase and frequency estimates have significant correlation, and the formulation and analysis of the maximum likelihood estimates (MLEs) and the CRBs are simplified. More significantly, this simple modification decouples the estimations of phase and frequency and leads to more efficient MLE gradient descent algorithms.…”

“…The preference, debatable at best, is certainly not a big issue. However, with more complicated analysis such as that presented by Stoica et al [3] and Zhou and Giannakis [5], the small increment of simplicity in this approach is probably welcome. Unless noted otherwise, the results presented here reflect the phase defined in the middle of the time window.…”

Computational Aspects of Efficient Estimation of Harmonically Related Sine-Waves

“…Since, under assumption of Gaussian noise, maximum likelihood (ML) principle [2] leads to nonlinear least squares (LS) problem, most of algorithms for computing ML estimates are iterative search routines in their nature [3][4][5]. As such, they are usually two-stage procedures, the first stage being a coarse initial estimation that provides seed values for the second stage, which iteratively approaches maximum likelihood solution.…”

Structured total least squares approach for efficient frequency estimation

“…Well known frequency estimation algorithms include maximum likelihood (ML) estimator [6], subspacebased techniques [7], Yule-Walker method [8] and linear prediction (LP) [9] approach. The ML method is statistically efficient in the sense that its estimation performance can attain the Cram´r-Rao lower bound (CRLB) asymptotically under additive white Gaussian noise (AWGN).…”