The derivation Cramer-Rao lower bound for parameters of signals multiplied on window functions

Abstract

The article focuses on the refinement of the Cramer-Rao lower boundary for the single-tone signal amplitude for estimation methods using window functions. The Cramer-Rao lower boundary allows finding the minimum variance of the signal parameter estimate. There are formulas for variances estimations of the amplitude, frequency, and phase of harmonics of harmonic signals, but the known methods for their finding, including algorithms based on the maximum likelihood method, show results above this boundary. The increased variance of the estimate occurs due to the application of window functions on the original signal. The estimation of the accuracy of the parameters harmonics in case using the window functions usually done with a numerical simulation. In the article, the authors derive a formula for the minimum variance for amplitudes of harmonics in the case of the using the window functions. This derivation allows us to understand the mathematical meaning of the broadening the Cramer-Rao lower boundary when using window functions, and the resulting formula brings a faster and more accurate estimation of an amplitude accuration, in comparison with numerical simulation. According to the results of the experiment the calculations by the proposed formula and the numerical experiment data are the same.