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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula> . Iterative interpolation schemes have recently been designed by employing two <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -shifted spectral lines symmetrically placed around the DFT peak, and the impact of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -shifted spectral lines and assess its dependence on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> . 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> for a given <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula> , 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.

Topics & Concepts