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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 iω nt ) (2) where ω = 2 π T is the fundamental frequency. For almost allis the fundamental frequency....
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- Spring '09
- Complex number, Leonhard Euler, Euler's formula, Joseph Fourier, fourier coe 1Ecients