Systems

This chapter is concerned with setting up the foundations that allow the connection between causality and analyticity to be established. The interplay between these two topics and the Hilbert transform is also treated. The material of this chapter lays the basis for many of the applications discussed in the following chapters.

Consider the arrangement in Figure 17.1, where the input to the system is denoted by i, and r designates the output response. The input and output could be of the same nature, for example a voltage, or very different variables, for example a voltage input, with the output being a physical displacement of a mass. As an example of Figure 17.1, consider the input to be the driving force acting on a simple oscillator arrangement with a mass hanging from a fixed point by a spring. The output is the displacement of the mass. The input has some dependence on time, and likewise the output response. Henceforth, the temporal dependence is made explicit. Often the input and output response are continuous functions of the time variable, but this is not a requirement. In later sections, the focus will include consideration of step function and impulse inputs.

The term “system” refers to a device capable of converting an input into some output response. The output response should be totally characterized by the system input and the characteristics of the system.