So far we have worked almost entirely with signal spaces of the form ℓ2(ℤ+) or L2(0, ∞) and their full-axis analogues; in physical terms, these are spaces of finite-energy signals, which die away in some sense at infinity. In this chapter we shall work with what may loosely be described as finite-power signals or, still more loosely, as persistent signals.

Persistent signals include classes of signals with some regularity properties, such as periodic and almost-periodic signals, as well as much more general spaces of signals in which the notion of “power” is less clearly defined. In particular, we are are able to discuss concepts such as the idea of a white noise signal in a rigorous and largely non-stochastic framework. Persistent signals in general can be taken as the inputs and outputs of linear systems (the term filter is commonly used here), as we shall see.

Almost-periodic functions

Almost-periodic functions defined on the real line form a class of functions that has been much studied since the 1920s. Our aim in this section is to derive their fundamental properties and to bring out their similarities with the theory of periodic functions.