Generalised discretisation of continuous-time distributions

Abstract

In this study, the definition of discretisation that was proposed recently for continuous-time distributions is made applicable not only to ordinary functions but to a variety of distributions including weak derivatives such that they could be viewed from a unified perspective under useful theorems. While it is not absolutely necessary to introduce distributions for discrete-time signals having finite values, it turns out that it is insightful to introduce discrete-time equivalents in appreciating their richness, which culminates into continuous-time distributions as the sampling-interval approaches zero. For instance, a discretisation of a derivative of a distribution can be found as a discrete derivative of a discretisation of a distribution. This is much easier than the traditional approach, where an ordinary function must first be found to approximate the derivative of a distribution. Simulations show that, by changing a single parameter of the proposed model, different types of signals that are similar to traditional ones developed separately by approximating distributions by ordinary functions, such as Dirichlet’ kernel, Gaussian distribution and sinc approximation, can be obtained.

Keywords

• approximation theory
• signal sampling
• discrete time systems
• gaussian distribution
• continuous time systems
• ordinary function
• discrete-time signals
• discrete-time equivalents
• continuous-time distributions
• discrete derivative
• gaussian distribution
• generalised discretisation