Activity of order n in continuous systems

Abstract

In this work we generalize the concept of activity of continuous time signals. We define the activity of order n of a signal and show that it allows us to estimate the number of sections of polynomials up to order n which are needed to represent that signal with a certain accuracy. Then we apply this concept to obtain a lower bound for the number of steps performed by quantization-based integration algorithms in the simulation of ordinary differential equations. We perform an exhaustive analysis over two examples, computing the activity of order n and comparing it with the number of steps performed by different integration methods. This analysis corroborates the theoretical predictions and also allows us to measure the suitability of the different algorithms depending on how close to the theoretical lower bound they perform.