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Fourier Series: Eigenfunction Approach

Module by: Justin Romberg. E-mail the author

Summary: This module will introduce the Fourier Series and its Fourier coefficients using the concepts of eigenfunctions and basis. We will show several examples of how to decompose a signal and find the Fourier coefficients.

Introduction

Since complex exponentials are eigenfunctions of linear time-invariant (LTI) systems, calculating the output of an LTI system given est s t as an input amounts to simple multiplcation, where HsC H s is a constant (that depends on s). In the figure below we have a simple exponential input that yields the following output:

yt=Hsest y t H s s t
(1)

Figure 1: Simple LTI system.
Figure 1 (simpleLTIsys.png)

Using this and the fact that is linear, calculating yt 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 HH on the left side and the output, yt y t , on the right:

  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 c 1 s 1 t c 2 s 2 t c 1 H s 1 s 1 t c 2 H s 2 s 2 t
  2. n c n e s n tn c n H s n e s n t n c n s n t n c n H s n s n t

The action of HH on an input such as those in the two equations above is easy to explain: independently scales each exponential component e s n t s n t by a different complex number H s n C H s n . As such, if we can write a function ft f t as a combination of complex exponentials it allows us to:

  • easily calculate the output of given ft f t as an input (provided we know the eigenvalues Hs H s )
  • interpret how manipulates ft f t

Fourier Series

Joseph Fourier demonstrated that an arbitrary T-periodic function ft f t can be written as a linear combination of harmonic complex sinusoids

ft= n = c n ei ω 0 nt f t n c n ω 0 n t
(2)
where ω 0 =2πT ω 0 2 T is the fundamental frequency. For almost all ft f t of practical interest, there exists c n c n to make Equation 2 true. If ft f t is finite energy ( ftL20T f t L 0 T 2 ), then the equality in Equation 2 holds in the sense of energy convergence; if ft f t is continuous, then Equation 2 holds pointwise. Also, if ft f t meets some mild conditions (the Dirichlet conditions), then Equation 2 holds pointwise everywhere except at points of discontinuity.

The c n c n - called the Fourier coefficients - tell us "how much" of the sinusoid ei ω 0 nt ω 0 n t is in ft f t . Equation 2 essentially breaks down ft f t into pieces, each of which is easily processed by an LTI system (since it is an eigenfunction of every LTI system). Mathematically, Equation 2 tells us that the set of harmonic complex exponentials n ,nZ:ei ω 0 nt n n ω 0 n t form a basis for the space of T-periodic continuous time functions. Below are a few examples that are intended to help you think about a given signal or function, ft f t , in terms of its exponential basis functions.

Fourier Coefficients

In general ft f t , the Fourier coefficients can be calculated from Equation 2 by solving for c n c n , which requires a little algebraic manipulation (for the complete derivation see the Fourier coefficients derivation). The end results will yield the following general equation for the fourier coefficients:

c n =1T0Tfte(i ω 0 nt)d t c n 1 T t T 0 f t ω 0 n t
(3)
The sequence of complex numbers n ,nZ: c n n n c n is just an alternate representation of the function ft f t . Knowing the Fourier coefficients c n c n is the same as knowing ft f t explicitly and vice versa. Given a periodic function, we can transform it into it Fourier series representation using Equation 3. Likewise, we can inverse transform a given sequence of complex numbers, c n c n , using Equation 2 to reconstruct the function ft f t .

Along with being a natural representation for signals being manipulated by LTI systems, the Fourier series provides a description of periodic signals that is convenient in many ways. By looking at the Fourier series of a signal ft f t , we can infer mathematical properties of ft f t such as smoothness, existence of certain symmetries, as well as the physically meaningful frequency content.

Summary: Fourier Series Equations

Our first equation is the synthesis equation, which builds our function, ft f t , by combining sinusoids.

Synthesis

ft= n = c n ei ω 0 nt f t n c n ω 0 n t
(4)
And our second equation, termed the analysis equation, reveals how much of each sinusoid is in ft f t .

Analysis

c n =1T0Tfte(i ω 0 nt)d t c n 1 T t T 0 f t ω 0 n t
(5)
where we have stated that ω 0 =2πT ω 0 2 T .

Note:

Understand that our interval of integration does not have to be 0 T 0 T in our Analysis Equation. We could use any interval a a+T a a T of length TT.

Example 1

This demonstration lets you synthesize a signal by combining sinusoids, similar to the synthesis equation for the Fourier series. See here for instructions on how to use the demo.

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