Periodic Functions

Abstract

Periodic functions are functions which repeat: f (t + P) = f (t) for all t. For example, if f (t) is the amount of time between sunrise and sunset at a certain lattitude, as a function of time t, and P is the length of the year, then f (t + P) = f (t) for all t, since the Earth and Sun are in the same position after one full revolution of the Earth around the Sun. We state this explicitly as the following defintion: a function f (t) is periodic with period P > 0 if f (t + P) = f (t) for all t. Example. f (t) = sin(2t) is periodic with period P = π. This is true because, for all t, f (t + π) = sin(2(t + π) = sin(2t + 2π) = sin(2t) = f (t). Notice, though, that in the example above f (t) = sin(2t) also has period P = 2π and period P = 3π. In fact, it has period P = nπ for any integer n = 1, 2, 3. .. . Graphically, a function with period P is one whose graph stays the same if it is shifted P to the left or right. Base Period Most periodic functions have a minimal period, which is often called either the period or the base period. For example, sin t has minimal period is 2π. It follows from this that the minimal period for sin(2t) is π. The only exception is the constant function. Every value of P > 0 is a period and so it has no minimal period. (We don't allow P = 0 to be a period because then every function would be periodic with period P = 0.) Windows To fully describe a periodic function you only need to specify the period and the value of the function over one full period. We call an interval containing one full period a window. Typical choices for windows are [−P/2, P/2) and [0, P), but any interval of length P will work. Frequency Terminology Angular frequency, also called circular frequency has units of radians/unit time. Frequency has units of cycles/unit time. Since one cycle is 2π radians the relationship is