When the input to a dynamic system is periodic i.e., a continuously repeating function of time, having period T such as the function shown in Fig. A1.1 it is often useful to describe this function in terms of an infinite series of pure sinusoids known as a Fourier series.

Figure A1.1: A typical periodic function.

One form of such an infinite series is

where is the average, or constant, value of the function, ? ₁=2 ? /T is the radian frequency of the lowest-frequency component, and the amplitudes of the series of component sinusoids at succeeding frequencies k ? ₁ are given by

Alternatively, this function may be expressed as a series of only sine waves or only cosine waves by useof

or

where

The steady response of a dynamic system (i.e., the response remaining after all transients have decayed to zero) to an infinite series of sine waves may also be expressed as an infinite series:

where Y _o is the constant component of output and the coefficients Y _k are the amplitudes of the successive sine waves having frequencies k ? ₁ and phase angles . The values of Y _o, Y _k, and at each frequency are obtained by the methods developed in Chap. 12.

Alternatively, the periodic function x(t) may be expressed in terms of an infinite series of exponentials having the form through the use of Euler s equations:

Substituting these expressions into Eq. (A1.1) yields