Skip to content Skip to navigation

Connexions

You are here: Home » Content » Fourier Series Wrap-Up

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to 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 member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

In these lenses

  • Lens for Engineering

    This module is included inLens: Lens for Engineering
    By: Sidney Burrus

    Click the "Lens for Engineering" link to see all content selected in this lens.

Recently Viewed

This feature requires Javascript to be enabled.
 

Fourier Series Wrap-Up

Module by: Michael Haag, Justin Romberg. E-mail the authors

Summary: (Blank Abstract)

Below we will highlight some of the most important concepts about the Fourier Series and our understanding of it through eigenfunctions and eigenvalues. Hopefully you are familiar with all of this material, so this document will simply serve as a refresher, but if not, then refer to the many links below for more information on the various ideas and topics.

  1. We can represent a periodic function (or a function on an interval) f⁢t f t as a combination of complex exponentials:
    f⁢t=∑ n =−∞∞ c n ⁢ei⁢ ω 0 ⁢n⁢t f t n c n ω 0 n t
    (1)
    c n =1T⁢∫0Tf⁢t⁢e−(i⁢ ω 0 ⁢n⁢t)d t c n 1 T t T 0 f t ω 0 n t
    (2)
    Where the fourier coefficients, c n c n , approximately equal how much of frequency ω 0 ⁢n ω 0 n is in the signal.
  2. Since ei⁢ ω 0 ⁢n⁢t ω 0 n t are eigenfunctions of LTI systems, we can interpret the action of a system on a signal in terms of its eigenvalues:
    H⁢i⁢ ω 0 ⁢n=∫−∞∞h⁢t⁢e−(i⁢ ω 0 ⁢n⁢t)d t H ω 0 n t h t ω 0 n t
    (3)
    • |H⁢i⁢ ω 0 ⁢n| H ω 0 n is large ⇒ system accentuates frequency ω 0 ⁢n ω 0 n
    • |H⁢i⁢ ω 0 ⁢n| H ω 0 n is small ⇒ system attenuates frequency ω 0 ⁢n ω 0 n
  3. In addition, the c n c n of a periodic function f⁢t f t can tell us about:
    • symmetries in f⁢t f t
    • smoothness of f⁢t f t , where smoothness can be interpreted as the decay rate of | c n | c n .
  4. We can approximate a function by re-synthesizing using only some of the Fourier coefficients (truncating the F.S.)
    f N ′t=∑ n n≤|N| c n ⁢ei⁢ ω 0 ⁢n⁢t f N t n n N c n ω 0 n t
    (4)
    This approximation works well where f⁢t f t is continuous, but not so well where f⁢t f t is discontinuous. This idea is explained by Gibb's Phenomena.

Download LabVIEW Source

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to 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 member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks