Hermite Polynomials and UWB Pulse Shaping Functions: Analysis Across Derivative Orders

This article provides a comprehensive examination of Hermite polynomials and Ultra-Wideband (UWB) pulse shaping functions, with specific focus on their behavior under different derivative orders. Hermite polynomials constitute a set of orthogonal functions frequently employed in quantum mechanics applications, while UWB pulse shaping functions represent specialized impulse signals optimized for wireless communication systems. We will analyze the mathematical definitions, key properties, and practical applications of these functions, with particular emphasis on how their characteristics evolve across successive derivatives. From a computational perspective, implementing Hermite polynomials typically involves recurrence relations (Hₙ₊₁(x) = 2xHₙ(x) - 2nHₙ₋₁(x)) or symbolic differentiation, whereas UWB pulse shaping often utilizes Gaussian derivative pulses generated through successive differentiation of the fundamental Gaussian function. The article includes discussions on numerical stability considerations when computing higher-order derivatives and demonstrates how these mathematical transforms impact pulse characteristics like bandwidth and temporal localization. Through detailed explanations and implementation insights, readers will develop deeper understanding of these functions' behavior and gain practical knowledge for their effective application in engineering and physics domains.