Skip to content Skip to navigation Skip to collection information

Connexions

You are here: Home » Content » Signals and Systems » Types of Bases

Navigation

Table of Contents

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 ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.

Also in these lenses

  • Lens for Engineering

    This module and collection are included inLens: Lens for Engineering
    By: Sidney Burrus

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

  • richb's DSP display tagshide tags

    This collection is included inLens: richb's DSP resources
    By: Richard Baraniuk

    Comments:

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

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

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

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

Types of Bases

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

Summary: This module discusses the different types of basis that leads up to the definition of an orthonormal basis. Examples are given and the useful of the orthonormal basis is discussed.

Normalized Basis

Definition 1: Normalized Basis
a basis b i b i where each b i b i has unit norm
∥ b i ∥=1  ,   i∈Z    i i b i 1
(1)

Note:

The concept of basis applies to all vector spaces. The concept of normalized basis applies only to normed spaces.
You can always normalize a basis: just multiply each basis vector by a constant, such as 1∥ b i ∥ 1 b i

Example 1

We are given the following basis: b 0 b 1 =( 1 1 )( 1 -1 ) b 0 b 1 1 1 1 -1 Normalized with ℓ 2 ℓ 2 norm: b ~ 0 =12⁢( 1 1 ) b ~ 0 1 2 1 1 b ~ 1 =12⁢( 1 -1 ) b ~ 1 1 2 1 -1 Normalized with ℓ 1 ℓ 1 norm: b ~ 0 =12⁢( 1 1 ) b ~ 0 1 2 1 1 b ~ 1 =12⁢( 1 -1 ) b ~ 1 1 2 1 -1

Orthogonal Basis

Definition 2: Orthogonal Basis
a basis b i b i in which the elements are mutually orthogonal 〈 b i , b j 〉=0  ,   i≠j    i i j b i b j 0

Note:

The concept of orthogonal basis applies only to Hilbert Spaces.

Example 2

Standard basis for ℝ 2 ℝ 2 , also referred to as ℓ 2 ⁢ 0 1 ℓ 2 0 1 : b 0 =( 1 0 ) b 0 1 0 b 1 =( 0 1 ) b 1 0 1 〈 b 0 , b 1 〉=∑ i =01 b 0 ⁢i⁢ b 1 ⁢i=1×0+0×1=0 b 0 b 1 i 1 0 b 0 i b 1 i 1 0 0 1 0

Example 3

Now we have the following basis and relationship: ( 1 1 )( 1 -1 )= h 0 h 1 1 1 1 -1 h 0 h 1 〈 h 0 , h 1 〉=1×1+1×-1=0 h 0 h 1 1 1 1 -1 0

Orthonormal Basis

Pulling the previous two sections (definitions) together, we arrive at the most important and useful basis type:

Definition 3: Orthonormal Basis
a basis that is both normalized and orthogonal ∥ b i ∥=1  ,   i∈Z    i i b i 1 〈 b i , b j 〉  ,   i≠j    i i j b i b j

Notation:

We can shorten these two statements into one: 〈 b i , b j 〉= δ i j b i b j δ i j where δ i j ={1  if  i=j0  if  i≠j δ i j 1 i j 0 i j Where δ i j δ i j is referred to as the Kronecker delta function and is also often written as δ⁢i−j δ i j .

Example 4: Orthonormal Basis Example #1

b 0 b 2 =( 1 0 )( 0 1 ) b 0 b 2 1 0 0 1

Example 5: Orthonormal Basis Example #2

b 0 b 2 =( 1 1 )( 1 -1 ) b 0 b 2 1 1 1 -1

Example 6: Orthonormal Basis Example #3

b 0 b 2 =12⁢( 1 1 )12⁢( 1 -1 ) b 0 b 2 1 2 1 1 1 2 1 -1

Beauty of Orthonormal Bases

Orthonormal bases are very easy to deal with! If b i b i is an orthonormal basis, we can write for any xx

x=∑i α i ⁢ b i x i α i b i
(2)
It is easy to find the α i α i :
〈x, b i 〉=〈∑k α k ⁢ b k , b i 〉=∑k α k ⁢〈( b k , b i )〉 x b i k α k b k b i k α k b k b i
(3)
where in the above equation we can use our knowledge of the delta function to reduce this equation: 〈 b k , b i 〉= δ i k ={1  if  i=k0  if  i≠k b k b i δ i k 1 i k 0 i k
〈x, b i 〉= α i x b i α i
(4)
Therefore, we can conclude the following important equation for xx:
x=∑i〈(x, b i )〉⁢ b i x i x b i b i
(5)
The α i α i 's are easy to compute (no interaction between the b i b i 's)

Example 7

Given the following basis: b 0 b 1 =12⁢( 1 1 )12⁢( 1 -1 ) b 0 b 1 1 2 1 1 1 2 1 -1 represent x=( 3 2 ) x 3 2

Example 8: Slightly Modified Fourier Series

We are given the basis 1T⁢ej⁢ ω 0 ⁢n⁢t| n =−∞∞ n 1 T ω 0 n t on L 2 ⁢ 0 T L 2 0 T where T=2⁢π ω 0 T 2 ω 0 . f⁢t=∑ n =−∞∞〈(f,ej⁢ ω 0 ⁢n⁢t)〉⁢ej⁢ ω 0 ⁢n⁢t⁢1T f t n f ω 0 n t ω 0 n t 1 T Where we can calculate the above inner product in L 2 L 2 as 〈f,ej⁢ ω 0 ⁢n⁢t〉=1T⁢∫0Tf⁢t⁢ej⁢ ω 0 ⁢n⁢t*d t =1T⁢∫0Tf⁢t⁢e−(j⁢ ω 0 ⁢n⁢t)d t f ω 0 n t 1 T t T 0 f t ω 0 n t 1 T t T 0 f t ω 0 n t

Orthonormal Basis Expansions in a Hilbert Space

Let b i b i be an orthonormal basis for a Hilbert space HH. Then, for any x∈H x H we can write

x=∑i α i ⁢ b i x i α i b i
(6)
where α i =〈x, b i 〉 α i x b i .
  • "Analysis": decomposing x x in term of the b i b i
    α i =〈x, b i 〉 α i x b i
    (7)
  • "Synthesis": building x x up out of a weighted combination of the b i b i
    x=∑i α i ⁢ b i x i α i b i
    (8)

Collection Navigation

Content actions

Download:

Collection 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 ...

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:

Collection 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

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