Periodic, Finite, and Aperiodic signalsWe have seen that periodic signals and finite signals have alot in common. One can be defined in terms of the other. Thus, a Fourier series can be used to describe a finite signal as well as a periodic one. The "period" is simply the extent of the finite signal. Thus, if the domain of the signal is [a, b] ⊂ Reals, then p = b − a. The fundamental frequency, therefore, is just ω 0 = 2π /(b − a).
An aperiodic signal, like an audio signal, can be partitioned into finite segments, and a Fourier series can be constructed from each segment. Consider the train whistle that we have heard before:
In the above applet, segments of 16 milliseconds are displayed at any one time. Notice that within each segment, the sound clearly has somewhat periodic structure. It is not hard to envision how it could be described as sums of sinusoids. If we construct a Fourier series representation for each 16msec segment, the value of the coefficients Ak is shown on the following applet:
Notice that there are three dominant frequency components that persist throughout.