On Optimal Forms of the Hermite Transform

The Hermite transform has been widely studied in the signal processing field due to its desirable properties. It has applicability in various fields starting from digital image processing, biomedical applications to the ultra-wide band communications. This paper explores the Hermite-like signals in the context of the signal estimation and variance reduction. By deriving an optimal estimator and the Cramer-Rao bound, we demonstrate that Hermite functions sampled at the roots of the Hermite polynomial provide an efficient framework for signal representation in the presence of Gaussian noise. The results indicate that the variance of the optimal form is lower than that of the Gauss-Hermite quadrature approximation, highlighting its benefits for signal estimation. Additionally, as expected, the variance increases with both the order of the Hermite function and the number of sampling points. The theory is illustrated with several examples.