Skip to content Skip to navigation

Connexions

You are here: Home » Content » Fourier Series in a Nutshell

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

    • External bookmarks
  • E-mail the authors

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

This content is ...

In these lenses

  • richb's DSP

    This module is included inLens: richb's DSP resources
    By: Richard BaraniukAs a part of collection:"Signals and Systems"

    Comments:

    "My introduction to signal processing course at Rice University."

    Click the "richb's DSP" link to see all content selected in this lens.

Recently Viewed

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Fourier Series in a Nutshell

Module by: Michael Haag, Justin Romberg

Summary: This module will give a brief over of the key concepts involving the Fourier series and the tools used to decompose and approximate a given signal.

Introduction

The convolution integral is the fundamental expression relating the input and output of an LTI system. However, it has three shortcomings:

  1. It can be tedious to calculate.
  2. It offers only limited physical interpretation of what the system is actually doing.
  3. It gives little insight on how to design systems to accomplish certain tasks.
The Fourier Series, along with the Fourier Transform and Laplace Transofrm, provides a way to address these three points. Central to all of these methods is the concept of an eigenfunction (or eigenvector). We will look at how we can rewrite any given signal, ft f t , in terms of complex exponentials.

In fact, by making our notions of signals and linear systems more mathematically abstract, we will be able to draw enlightening parallels between signals and systems and linear algebra.

Eigenfunctions and LTI Systems

The action of a LTI system ℋ… ℋ … on one of its eigenfunctions ⅇst s t is

  1. extremely easy (and fast) to calculate
    ℋst=Hsⅇst ℋ s t H s s t (1)
  2. easy to interpret: ℋ… ℋ … just scales ⅇst s t , keeping its frequency constant.
If only every function were an eigenfunction of ℋ… ℋ … ...

LTI System

... of course, not every function can be, but for LTI systems, their eigenfunctions span the space of periodic functions, meaning that for (almost) any periodic function ft f t we can find c n c n where n∈ℤ n and c i ∈ℂ c i such that:

ft=∑n=-∞∞ c n ⅇⅈ ω 0 nt f t n c n ω 0 n t (2)
Given Equation 2, we can rewrite ℋt=yt ℋ t y t as the following system

Figure 1: Transfer Functions modeled as LTI System.
Figure 1 (Transferfunc.png)

where we have: ft=∑n c n ⅇⅈ ω 0 nt f t n c n ω 0 n t yt=∑n c n Hⅈ ω 0 nⅇⅈ ω 0 nt y t n c n H ω 0 n ω 0 n t This transformation from ft f t to yt y t can also be illustrated through the process below. Note that each arrow indicates an operation on our signal or coefficients.

ft→ c n → c n Hⅈ ω 0 n →yt f t c n c n H ω 0 n y t (3)
where the three steps (arrows) in the above illustration represent the following three operations:
  1. Transform with analysis ( Fourier Coefficient equation): c n =1T∫0Tftⅇ-ⅈ ω 0 ntdt c n 1 T t T 0 f t ω 0 n t
  2. Action of ℋℋ on the Fourier series - equals a multiplication by Hⅈ ω 0 n H ω 0 n
  3. Translate back to old basis - inverse transform using our synthesis equation from the Fourier series: yt=∑n=-∞∞ c n ⅇⅈ ω 0 nt y t n c n ω 0 n t

Physical Interpretation of Fourier Series

The Fourier series c n c n of a signal ft f t , defined in Equation 2, also has a very important physical interpretation. Coefficient c n c n tells us "how much" of frequency ω 0 n ω 0 n is in the signal.

Signals that vary slowly over time - smooth signals - have large c n c n for small nn.

Figure 2: We begin with our smooth signal ft f t on the left, and then use the Fourier series to find our Fourier coefficients - shown in the figure on the right.
Subfigure 2.1Subfigure 2.2
Subfigure 2.1 (fsnut_1.png)Subfigure 2.2 (fsnut_2.png)

Signals that vary quickly with time - edgy or noisy signals - will have large c n c n for large nn.

Figure 3: We begin with our noisy signal ft f t on the left, and then use the Fourier series to find our Fourier coefficients - shown in the figure on the right.
Subfigure 3.1Subfigure 3.2
Subfigure 3.1 (fsnut_3.png)Subfigure 3.2 (fsnut_4.png)

Example 1: Periodic Pulse

We have the following pulse function, ft f t , over the interval -T2T2 T 2 T 2 :

Figure 4: Periodic Signal ft f t
Figure 4 (fsnut_e1.png)

Using our formula for the Fourier coefficients,

c n =1T∫0Tftⅇ-ⅈ ω 0 ntdt c n 1 T t T 0 f t ω 0 n t (4)
we can easily calculate our c n c n . We will leave the calculation as an exercise for you! After solving the the equation for our ft f t , you will get the following results:
c n =2 T 1 Tifn=02sin ω 0 n T 1 nπifn≠0 c n 2 T 1 T n 0 2 ω 0 n T 1 n n 0 (5)
For T 1 =T8 T 1 T 8 , see the figure below for our results:

Figure 5: Our Fourier coefficients when T 1 =T8 T 1 T 8
Figure 5 (fsnut_e2.png)

Our signal ft f t is flat except for two edges (discontinuities). Because of this, c n c n around n=0 n 0 are large and c n c n gets smaller as nn approaches infinity.

question:

Why does c n =0 c n 0 for n=…-44816… n … -4 4 8 16 … ? (What part of ⅇ-ⅈ ω 0 nt ω 0 n t lies over the pulse for these values of nn?)

Comments, questions, feedback, criticisms?

Send feedback