Extended formulas of sampling theorem on band‐limited waves and their application to the design of digital filters for data transmission

Abstract

AbstractThe paper presents an extension of the sampling theorem with nonuniform sampling points to band‐limited waves where the spectrum of the obtained interpolation functions is differentiable to any desired degree. The adopted series of sampling points, in general, is a periodic repetition of K nonuni‐formly spaced sampling points. It is then shown that, neglecting shift along the time axis, there is no need of K different interpolation functions and that these functions are discrete orthogonal under certain conditions. Next, general formulas are derived to evaluate interpolation error due to moderate band limitation and simplified forms of the formulas are derived at K = 2 for each frequency interval. These formulas are also used as examples for an intuitive explanation. As a typical example of data transmission filters, a digital Nyquist filter is considered and a design method for it is introduced by applying Bellanger's method to some of the low‐rate subsystems while the remaining ones are supplied by interpolation. As an example, two out of five low‐rate subsystems are designed by the proposed method and are shown to require about 40% less hardware than the filters designed by Bellanger's method.