An Explicit Form of Ramp Function

Abstract

In this paper, an analytical exact form of the ramp function is presented. This seminal function constitutes a fundamental concept of the digital signal processing theory and is also involved in many other areas of applied sciences and engineering. In particular, the ramp function is performed in a simple manner as the pointwise limit of a sequence of real and continuous functions with pointwise convergence. This limit is zero for strictly negative values of the real variable x, whereas it coincides with the independent variable x for strictly positive values of the variable x. Here, one may elucidate beforehand that the pointwise limit of a sequence of continuous functions can constitute a discontinuous function, on the condition that the convergence is not uniform. The novelty of this work, when compared to other research studies concerning analytical expressions of the ramp function, is that the proposed formula is not exhibited in terms of miscellaneous special functions, e.g., gamma function, biexponential function, or any other special functions, such as error function, hyperbolic function, orthogonal polynomials, etc. Hence, this formula may be much more practical, flexible, and useful in the computational procedures, which are inserted into digital signal processing techniques and other engineering practices.