Frequency Estimation by Interpolation of Two Fourier Coefficients: Cramér-Rao Bound and Maximum Likelihood Solution

Abstract

Sinusoidal frequency estimation in the presence of white Gaussian noise plays a major role in many engineering fields. Significant research in this area has been devoted to the fine tuning stage, where the discrete Fourier transform (DFT) coefficients of the observation data are interpolated to acquire the residual frequency error ε. Iterative interpolation schemes have recently been designed by employing two q-shifted spectral lines symmetrically placed around the DFT peak, and the impact of q on the estimation accuracy has been theoretically assessed. Such analysis, however, is available only for some specific algorithms and is mostly conducted under the assumption of a vanishingly small frequency error, which makes it inappropriate for the first stage of any iterative process. In this work, further investigation on DFT interpolation is carried out to examine some issues that are still open. We start by evaluating the Cramér-Rao bound (CRB) for frequency recovery by interpolation of two q-shifted spectral lines and assess its dependence on ε and q. Such a bound is of primary importance to check whether existing schemes can provide efficient estimates at any iteration or not. After determining the optimum value of q for a given ε, we eventually derive the maximum likelihood (ML) DFT interpolator. Since the latter exhibits the best performance at any step of the iteration process, it might attain the desired accuracy just at the end of the first iteration, which is especially advantageous in terms of computational load and processing time.

Information-theoretic analysis of Hierarchical Temporal Memory-Spatial Pooler algorithm with a new upper bound for the standard information bottleneck method

“…Using this lower bound, selecting the most appropriate estimator based on possible minimal variance is possible ( Tune, 2012 ; Huang et al, 2020 ; Khorasani et al, 2020 ). The Cramer-Rao Lower Bound (CRLB) is the most popular lower bounds in the literature due to its attractiveness and ease of evaluation ( Rao et al, 1973 ; D’Amico et al, 2022 ). As stated in the Cramer-Rao inequality ( Dogandzic and Nehorai, 2001 ), the diagonal terms of the inverse of the Fisher information Matrix (FIM) (assuming it exists) represent asymptotic lower bounds on any unbiased estimator’s variance.…”