The Cramer-Rao bound for missing samples scenario in Hermite transform domain

Abstract

Using Cramer-Rao theoretical approach, the minimum variance bound for the Hermite transform, as optimal estimator, is derived. The form of the Gauss-Hermite approximation is analyzed as well. It results as optimal estimator for Hermite-like signals scaled by Hermite function of order $N-1$. In this case, the variance is unevenly distributed in the Hermite domain. The analysis is further extended for the signal with missing samples, showing that the Cramer-Rao minimum variance equation retains the validity under some constraints. Namely, the relation holds only if the number of available samples is greater than a certain value that follows from the equation derived in this paper. The theoretical consideration and results are proven by various numerical and real world examples.