On Convergence and Bias Correction of a Joint Estimation Algorithm for Multiple Sinusoidal Frequencies

Abstract

Twenty years ago, Kay proposed an iterative filtering algorithm (IFA) for jointly estimating the frequencies of multiple complex sinusoids from noisy observations. IFA is based on the fact that the noiseless signal is an autoregressive (AR) process, so the frequency estimation problem can be reformulated as the problem of estimating the AR coefficients. By iterating the cycle of AR coefficient estimation and AR filtering, IFA provides a computationally simple procedure yet capable of accurate frequency estimation especially at low signal-to-noise ratio (SNR). However, the convergence of IFA has not been established beyond simulation and a very special case of a single frequency and infinite sample size. This article provides a statistical analysis of the algorithm and makes several important contributions. It shows that the poles of the AR filter must be reduced by an extra shrinkage parameter to accommodate poor initial values and avoid being trapped into false solutions. It also shows that the AR estimates in each iteration must be bias-corrected to produce a more accurate frequency estimator; a closed-form expression is provided for bias correction. Finally, it shows that for a sufficiently large sample size, the resulting algorithm, called new IFA (NIFA), converges to the desired fixed point, which constitutes a consistent frequency estimator. Numerical examples, including a real data example in radar applications, are provided to demonstrate the findings. It is shown in particular that the shrinkage parameter not only controls the estimation accuracy, but also determines the requirements of initial values. It is also shown that the proposed bias-correction method considerably improves the estimation accuracy, especially for high SNR.