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Unformatted text preview: Connexions module: m10496 1 Fourier Series: Eigenfunction Approach * Justin Romberg This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License Abstract This module will introduce the Fourier Series and its Fourier coe cients using the concepts of eigen- functions and basis. We will show several examples of how to decompose a signal and nd the Fourier coe cients. 1 Introduction Since complex exponentials 1 are eigenfunctions of linear time-invariant (LTI) systems 2 , calculating the output of an LTI system H given e st as an input amounts to simple multiplcation, where H ( s ) C is a constant (that depends on s). In the gure (Figure 1) below we have a simple exponential input that yields the following output: y ( t ) = H ( s ) e st (1) Figure 1: Simple LTI system. Using this and the fact that H is linear, calculating y ( t ) for combinations of complex exponentials is also straightforward. This linearity property is depicted in the two equations below - showing the input to the linear system H on the left side and the output, y ( t ) , on the right: * Version 2.22: Sep 30, 2009 3:59 pm GMT-5 http://creativecommons.org/licenses/by/1.0 1 "The Complex Exponential" <http://cnx.org/content/m10060/latest/> 2 "Eigenfunctions of LTI Systems" <http://cnx.org/content/m10500/latest/> http://cnx.org/content/m10496/2.22/ Connexions module: m10496 2 1. c 1 e s 1 t + c 2 e s 2 t c 1 H ( s 1 ) e s 1 t + c 2 H ( s 2 ) e s 2 t 2. X n ( c n e s n t ) X n ( c n H ( s n ) e s n t ) The action of H on an input such as those in the two equations above is easy to explain: H indepen- dently scales each exponential component e s n t by a di erent complex number H ( s n ) C . As such, if we can write a function f ( t ) as a combination of complex exponentials it allows us to: easily calculate the output of H given f ( t ) as an input (provided we know the eigenvalues H ( s ) ) interpret how H manipulates f ( t ) 2 Fourier Series Joseph Fourier 3 demonstrated that an arbitrary T-periodic function 4 f ( t ) can be written as a linear combi- nation of harmonic complex sinusoids f ( t ) = X n =- ( c n e i nt ) (2) where = 2 T is the fundamental frequency. For almost allis the fundamental frequency....
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This document was uploaded on 03/26/2010.
- Spring '09