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Modern Control System Theory and Design, 2nd Edition by Stanley M. Shinners

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2.3.  REVIEW OF FOURIER SERIES AND FOURIER TRANSFORM

In the analysis of the behaviour of systems evolving in time, it is often convenient to introduce mathematical transformations that take us from the time domain to a new domain called the frequency domain. Such transformations are called transforms. Here we will focus on the Fourier series, which is used to analyze periodic functions of time, and the Fourier integral which is used to examine aperiodic time functions of a restricted class. Section 2.4 introduces the Laplace transform, which is of great value in the analysis of time functions that vanish for negative time.

A.  Fourier Series

Given a single-valued periodic function f(t) whose period is T (and fundamental frequency is ω = 2π/T), then

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Functions that satisfy Eq. (2.23) can be represented by a Fourier series provided the function is bounded and contains only a finite number of discontinuities in a finite interval. The classical trigonometric form of the Fourier series is given by

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where ω represents the fundamental frequency, and represents the Kth harmonic:

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If f(t) = ...

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