Class Wiki - User contributions [en]

Orthogonal functions

2006-10-13T05:34:30Z

Nathan: /* Orthogonality for functions */

*[[Signals and systems|Signals and Systems]]
==Introduction==
In this article we will examine another viewpoint for functions than that traditionally taken. Normally we think of a function <math> f(t) </math> as a complicated entity <math> f() </math> in a simple environment (one dimension, or along the t axis). Now we want to think of a function as a vector or point (a simple thing) in a very complicated environment (possibly an infinite dimensional space).

==Vectors==
Recall that vectors consist of an ordered set of numbers. Often the numbers are Real numbers, but we shall allow them to be Complex for our purposes. The numbers represent the amount of the vector in the direction denoted by the position of the number in the list. Each position in the list is associated with a direction. For example, the vector
<math> \vec \bold v = <1, 4, 3> </math> means that the vector <math> \vec \bold v </math> is one unit in the first direction (often the x direction), four units in the second direction (often the y direction), and three units in the last direction (often the z direction). We say the component of <math> \vec \bold v </math> in the second direction is 4. This is often written as <math> v_y = 4 </math>.
====Vector notation====
We don't have to use x, y, and z as the direction names; we can use numbers, like 1, 2, and 3 instead. The advantage of this is that it leads to more compact notation, and extends to more than three dimensions much better. For example we could say <math> v_2 = 4 </math> instead of <math> v_y = 4 </math>. Instead of writing <math> \vec \bold v = <1, 4, 3> </math> we can write <math> \vec \bold v = \sum_{k=1}^3 v_k \hat \bold a_k </math> where the <math>\hat \bold a_k </math> denotes a basis vector in the kth direction, <math>v_1 = 1,</math> <math> v_2 = 4, </math> and <math> v_3 = 3</math>. The idea of basis vectors was implicit in the notation <math> \vec \bold v = <1, 4, 3> </math>.

===Inner products for vectors===
When vectors are real, inner products (sometimes called dot products) give the component of one vector in another vector's direction, scaled by the magnitude (length) of the second vector. Inner products are useful to find components of vectors. We commonly use a dot as the symbol for inner product. For example, the inner product of <math> \vec \bold v </math> and <math> \vec \bold a_n </math> is written:

<math> \vec \bold v \bullet \vec \bold a_n </math>

====Orthogonality for vectors====
It is quite handy to pick the directions used so that they are perpendicular (or orthogonal). With this arrangement the basis vectors have no components in each other's directions, which means that

<math>\vec \bold a_k \bullet \vec \bold a_n = w_k \delta_{k,n} </math>

where the <math> w_k </math> is the square of the length of <math> \vec \bold a_k </math> and the symbol <math> \delta_{k,n} </math>, known as the [http://en.wikipedia.org/wiki/Kronecker_delta Kronecker delta], is one when k = n and zero otherwise.
=====Normalization=====
When the <math> w_k = 1</math> we have an orthonormal basis set, meaning that it is both orthogonal and that the basis vectors are normalized to unity (or have length one). Orthonormal vector systems are very popular. In fact they are the most common vector systems you will find. The reason they are so handy is each direction is uncoupled from the others.

For example, to find <math> v_n </math>, we take the inner product of the vector <math> \vec \bold v </math> with a unit vector in the nth direction, <math> \vec \bold a_n </math>. We write this operation like this:

<math> \vec \bold v \bullet \vec \bold a_n = \sum_{k=1}^3 v_k \vec \bold a_k \bullet \vec \bold a_n = \sum_{k=1}^3 v_k \delta_{k,n} = v_n </math>

Suppose we have two vectors from an orthonormal system, <math> \vec \bold u </math> and <math> \vec \bold v </math>. Taking the inner product of these vectors, we get

<math> \vec \bold u \bullet \vec \bold v = \sum_{k=1}^3 u_k \vec \bold a_k \bullet \sum_{m=1}^3 v_m \vec \bold a_m = \sum_{k=1}^3 u_k \sum_{m=1}^3 v_m \vec \bold a_k \bullet \vec \bold a_m = \sum_{k=1}^3 u_k \sum_{m=1}^3 v_m \delta_{k,m} = \sum_{k=1}^3 v_k u_k </math>

This shows that when we have an orthonormal vector space, inner products boil down to summing the products of like components. Also note that if we take the inner product of <math> \vec \bold v </math> with itself, we get

<math> \vec \bold v \bullet \vec \bold v = \sum_{k=1}^3 v_k \vec \bold a_k \bullet \sum_{m=1}^3 v_m \vec \bold a_m = \sum_{k=1}^3 v_k \sum_{m=1}^3 v_m \vec \bold a_k \bullet \vec \bold a_m = \sum_{k=1}^3 v_k \sum_{m=1}^3 v_m \delta_{k,m} = \sum_{k=1}^3 v_k^2</math>

which is the magnitude of the vector <math> \vec \bold v </math> squared (<math> | \vec \bold v |^2 </math>) from the Pythagorean Theorem.

====Changing vector basis sets====
Sometimes in our studies we find it useful to change basis sets. For example, when solving a physics problem with cylindrical symmetry, it is often easier to use cylindrical coordinates, and the basis vectors that go with that system, rather than the more usual Cartesian coordinates and basis vectors.
=====So, how do I change the basis set?=====
If the new basis set is orthonormal, it is really pretty simple. You need to project the vector you want changed onto each of the new basis vectors. This means that the new components are just the inner product of the vector and the appropriate basis function. If the new basis set is not orthonormal, and if there are n dimensions in each basis set, you will have n linear coupled equations in n unknowns to solve.

===More vector questions===

[[Complex vector inner products|What if the vectors have complex components?]]

[[Vector weighting functions|What if not all components of the vectors have the same units?]]

[[Multiple dimensional vectors|What if there are more than three dimensions?]]

==Functions and vectors, an analogy==
We may think of the number of the direction, <math> k </math>, as the independent variable of a vector and the component in that direction, <math> v_k </math> as the dependent variable of the vector <math> \vec \bold v </math> in a similar way to the way we think of t as the independent variable of a function f(), where f(t) is the dependent variable of f. Probably the biggest difference here is that t often takes on real values from <math> - \infty </math> to <math> \infty </math>, and <math> k \in {1, 2, 3} </math>. Using this analogy, we may think of a function as a vector having an uncountably infinite number of dimensions.

====Can we write functions in an analogous way compared to the way we write vectors?====

Remember we wrote <math> \vec \bold v = \sum_{k=1}^3 v_k \hat \bold a_k </math>. Can we write something similar for a function, f(t) defined for a t element of the reals? Well maybe.... If the sum over the dummy index k becomes an integral over the dummy variable, x, and the unit vectors <math> \vec \bold a_k </math> are replaced with something like <math> \delta(x-t) </math>, the [http://en.wikipedia.org/wiki/Delta_function Dirac delta function]. The result would look something like this:

<math> f(t) = \int_{- \infty}^\infty f(x) \delta (x-t) dx </math>.

This works! The Dirac delta functions, playing the roll of the basis vectors, are called basis functions. The function f(x) plays the roll of the vector coefficients <math>v_k</math>. This gives us another way to think of the function f().

===Inner products for functions===
[[Orthogonal functions#Inner products for vectors|Above]] we found that a vector inner product between <math>\vec \bold u </math> and <math>\vec \bold v </math> could be written as <math> \vec \bold u \bullet \vec \bold v = \sum_{k=1}^3 u_k v_k </math>. If we follow our above analogy, we should be able to replace the sum over k with an integral over x. There is one little notational problem, and that is we don't want to confuse the functional inner product with a simple muliply, so we need some new notation to denote this new inner product. In [http://en.wikipedia.org/wiki/Quantum_mechanics quantum mechanics], physicists use the [http://en.wikipedia.org/wiki/Bra-ket_notation bra-ket] notation. Let's borrow that.

<math> <u|v> = \int_{-\infty}^\infty u^*(x) v(x) dx </math>

Note the complex conjugate on the function u(x). That is in case u(x) is a complex valued function. For the analogous case with vectors see [[Complex vector inner products]].
====Orthogonality for functions====
Two functions, <math>u(t)\!</math> and <math>v(t)\!</math> are said to be orthogonal on the interval <math> (a,b) \!</math> with respect to the weighting function <math> w(t) \!</math> if and only if
<math>\int_a^b w(x) u^*(x) v(x) dx = 0 \!</math>.
The weighting function is often unity, but it is included so that different values of <math>t\!</math> can be weighted appropriately in analogy to the way the <math>w_k\!</math> weight was used when the vector basis set was orthogonal, but not orthonormal (that is, different basis vectors had different numerical lengths), as we discussed [[Vector weighting functions|here]]. Unless otherwise noted we will use <math> w(t) = 1 \!</math>, so that the defining relation for orthogonality of functions <math> u \!</math> and <math> v \!</math> becomes
<math>\int_a^b u^*(x) v(x) dx = 0 </math>.

====Changing basis sets with functions====
====Examples====
*[[Fourier series]]
*[[Reconstructing bandlimited signals from sample points]]

==Other resources on orthogonality==
[http://en.wikipedia.org/wiki/Inner_product Wikipedia inner product]

[http://en.wikipedia.org/wiki/Orthogonal Wikipedia Orthogonality]

Principle author of this page: [[User:Frohro|Rob Frohne]]

Linear Time Invarient System

2006-10-13T05:31:13Z

Nathan: /* LTI system properties */

*[[Signals and systems|Signals and Systems]]
==LTI systems==
LTI System theory is a powerful and widely used concept in electrical engineering. It has applictions in circuit anlysis, control theory , and our main topic of interest signal processing.

===LTI system properties===
A system is considered to be a Linear Time Invariant when it satisfies the two basic criteria implied in its name, one it must be linear and two it must be time invariant. A Linear system is characterized by two properties superposition (additivity) and scaling (homogeneity). The superposition principal says that for any linear system a linear combination of solutions to the system is also a solution to the same linear system. The principal of scaling implies that if you adjust your scale an input by N amount, your output will also be adjusted by N amount. An example of a linear system then would be,

::<math>x_1(t)\!</math>
::<math>x_2(t)\!</math>
::<math>y_1(t) = H(x_1(t))\!</math>
::<math>y_2(t) = H(x_2(t))\!</math>
::<math>Ay_1(t) + By_2(t) = H(Ax_2(t) + Bx_1(t))\!</math>
for any scalar values of A and B.

Time invariance of a system means that for any input <math>x(t)</math> by some amount of time T the out put will also be adjusted by that amount of time. This implies that for,

::<math>x(t - T)\!</math>
::<math>y(t - T) = H(x(t - T))\!</math>

<br>
<br>
Related Links
*[[The Game]]

User:Nathan

2006-10-13T04:43:58Z

Nathan: /* Nathan Ferch */

==Nathan Ferch==

Welcome to my humble page.

[[Image:FerchFamily.jpg|thumb|My Family]]
[[Image:noahferch.jpg|thumb|Noah Ferch]]

User:Nathan

2006-10-13T04:43:52Z

Nathan: /* Nathan Ferch */

==Nathan Ferch==

Welcome to my humble page.

[[Image:FerchFamily.jpg|thumb|My Family]][[Image:noahferch.jpg|thumb|Noah Ferch]]

User:Nathan

2006-10-13T04:43:44Z

Nathan: /* Nathan Ferch */

==Nathan Ferch==

Welcome to my humble page.

[[Image:FerchFamily.jpg|thumb|My Family]]
[[Image:noahferch.jpg|thumb|Noah Ferch]]

File:Noahferch.jpg

2006-10-13T04:43:14Z

Nathan:

User:Nathan

2006-10-13T04:36:14Z

Nathan: /* Nathan Ferch */

==Nathan Ferch==

Welcome to my humble page.

[[Image:FerchFamily.jpg|thumb|My Family]]

User:Nathan

2006-10-13T04:35:50Z

Nathan: /* Nathan Ferch */

==Nathan Ferch==

Welcome to my humble page.

[[Image:FerchFamily.jpg|thumb|My Family]]

--[[User:Nathan|Nathan]] 21:35, 12 Oct 2006 (PDT)

User:Nathan

2006-10-13T04:34:23Z

Nathan: /* Nathan Ferch */

==Nathan Ferch==

Welcome to my humble page.

[[Image:FerchFamily.jpg|thumb|My Family]]

User:Nathan

2006-10-13T04:34:14Z

Nathan:

==Nathan Ferch==
[[Image:FerchFamily.jpg|thumb|My Family]]

User:Nathan

2006-10-13T04:34:02Z

Nathan: /* My Family */

Welcome to my humble page.

==Nathan Ferch==
[[Image:FerchFamily.jpg|thumb|My Family]]

User:Nathan

2006-10-13T04:33:50Z

Nathan: /* My Family */

Welcome to my humble page.

==My Family==
[[Image:FerchFamily.jpg|thumb|My Family]]

Signals and Systems

2006-10-13T04:30:27Z

Nathan:

== Topics ==
[[Fourier series - by Ray Betz|Overview of Signals and Systems]]

===Individual Subjects===
*[[Linear Time Invarient System|Linear Time Invarient Systems]]
**[[The Game|"The Game"]]
*[[Orthogonal functions|Orthogonal Functions]]
*[[Energy in a signal|Finding the Energy in a Signal]]
**[[Rayleigh's Theorem]]
*[[Fourier series|Fourier Series]]
*[[Fourier transform|Fourier Transforms]]
**[[Discrete Fourier transform]]
*[[Sampling]]
*[[FIR Filter Example]]

===Course Pages===
[[2005-2006 Assignments]]

[[2006-2007 Assignments]]

[http://www.wwc.edu/~frohro/ClassNotes/engr455index.htm Class notes for Signals & Systems]

[[User:Frohro|Instructor: Rob Frohne]]

==2004-2005 contributors==

[[User:Barnsa|Sam Barnes]]

[[User:Santsh|Shawn Santana]]

[[User:Goeari|Aric Goe]]

[[User:Caswto|Todd Caswell]]

[[User:Andeda|David Anderson]]

[[User:Guenan|Anthony Guenterberg]]

==2005-2006 contributors==

[[User:GabrielaV|Gabriela Valdivia]]

[[User:SDiver|Raymond Betz]]

[[User:chrijen|Jenni Christensen]]

[[User:wonoje|Jeffrey Wonoprabowo]]

[[User:wilspa|Paul Wilson]]

==2006-2007 contributors==

[[User:Smitry|Ryan J Smith]]

[[User:Nathan|Nathan Ferch]]

[[User:Andrew|Andrew Lopez]]

[[User:Sherna|Nathan Sherman]]

[[User:Adkich|Chris Adkins]]

Signals and Systems

2006-10-13T04:26:45Z

Nathan: /* Course Pages */

[http://www.wwc.edu/~frohro/ClassNotes/engr455index.htm Class notes for Signals & Systems]

== Topics ==
[[Fourier series - by Ray Betz|Overview of Signals and Systems]]

===Individual Subjects===
*[[Linear Time Invarient System|Linear Time Invarient Systems]]
**[[The Game|"The Game"]]
*[[Orthogonal functions|Orthogonal Functions]]
*[[Energy in a signal|Finding the Energy in a Signal]]
**[[Rayleigh's Theorem]]
*[[Fourier series|Fourier Series]]
*[[Fourier transform|Fourier Transforms]]
**[[Discrete Fourier transform]]
*[[Sampling]]
*[[FIR Filter Example]]

===Course Pages===
[[2005-2006 Assignments]]

[[2006-2007 Assignments]]

[[User:Frohro|Instructor: Rob Frohne]]

==2004-2005 contributors==

[[User:Barnsa|Sam Barnes]]

[[User:Santsh|Shawn Santana]]

[[User:Goeari|Aric Goe]]

[[User:Caswto|Todd Caswell]]

[[User:Andeda|David Anderson]]

[[User:Guenan|Anthony Guenterberg]]

==2005-2006 contributors==

[[User:GabrielaV|Gabriela Valdivia]]

[[User:SDiver|Raymond Betz]]

[[User:chrijen|Jenni Christensen]]

[[User:wonoje|Jeffrey Wonoprabowo]]

[[User:wilspa|Paul Wilson]]

==2006-2007 contributors==

[[User:Smitry|Ryan J Smith]]

[[User:Nathan|Nathan Ferch]]

[[User:Andrew|Andrew Lopez]]

[[User:Sherna|Nathan Sherman]]

[[User:Adkich|Chris Adkins]]

Signals and Systems

2006-10-13T04:26:10Z

Nathan: /* Topics */

[http://www.wwc.edu/~frohro/ClassNotes/engr455index.htm Class notes for Signals & Systems]

== Topics ==
[[Fourier series - by Ray Betz|Overview of Signals and Systems]]

===Individual Subjects===
*[[Linear Time Invarient System|Linear Time Invarient Systems]]
**[[The Game|"The Game"]]
*[[Orthogonal functions|Orthogonal Functions]]
*[[Energy in a signal|Finding the Energy in a Signal]]
**[[Rayleigh's Theorem]]
*[[Fourier series|Fourier Series]]
*[[Fourier transform|Fourier Transforms]]
**[[Discrete Fourier transform]]
*[[Sampling]]
*[[FIR Filter Example]]

===Course Pages===
[[2005-2006 Assignments]]

[[2006-2007 Assignments]]

I couldn't figure out how to get to others Users pages easily so I decided to start posting them here, please add yours:

[[User:Frohro|Rob Frohne]]

==2004-2005 contributors==

[[User:Barnsa|Sam Barnes]]

[[User:Santsh|Shawn Santana]]

[[User:Goeari|Aric Goe]]

[[User:Caswto|Todd Caswell]]

[[User:Andeda|David Anderson]]

[[User:Guenan|Anthony Guenterberg]]

==2005-2006 contributors==

[[User:GabrielaV|Gabriela Valdivia]]

[[User:SDiver|Raymond Betz]]

[[User:chrijen|Jenni Christensen]]

[[User:wonoje|Jeffrey Wonoprabowo]]

[[User:wilspa|Paul Wilson]]

==2006-2007 contributors==

[[User:Smitry|Ryan J Smith]]

[[User:Nathan|Nathan Ferch]]

[[User:Andrew|Andrew Lopez]]

[[User:Sherna|Nathan Sherman]]

[[User:Adkich|Chris Adkins]]

Talk:Signals and Systems

2006-10-13T03:57:15Z

Nathan:

===Signals and Systems Discussion===

Energy in a signal

2006-10-11T04:22:53Z

Nathan: /* Energy of a Signal */

*[[Signals and systems|Signals and Systems]]
===Definition of Energy===
Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,
: <math> W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}</math>

Power represents a change in energy.
: <math> P(t) = \frac{dW}{dt} </math>

This means we can also write energy as
: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>

===Energy of a Signal===
From circuit analysis we know that the power generated by a voltage source is,
: <math>P(t) = {v^2(t) \over R}</math>
Assuming that R is 1 then the total energy is just,
: <math>W = \int_{-\infty}^\infty |v|^2(t) \, dt</math>
This can be written using [http://en.wikipedia.org/wiki/Bra-ket_notation bra-ket] notation as
: <math> <v(t) | v(t)> \!</math> or <math> <v|v> \!</math>
By [[Rayliegh's Theroem]],
: <math> <v|v> = \int_{-\infty}^{\infty} |V(f)|^2\,df </math>
This implies that the energy of a signal can be found by the fourier transform of the signal,
: <math> W = \int_{-\infty}^{\infty} |V(f)|^2\,df </math>

Energy in a signal

2006-10-11T04:20:50Z

Nathan: /* Energy of a Signal */

Energy in a signal

2006-10-11T04:20:28Z

Nathan: /* Energy of a Signal */

*[[Signals and systems|Signals and Systems]]
===Definition of Energy===
Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,
: <math> W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}</math>

Power represents a change in energy.
: <math> P(t) = \frac{dW}{dt} </math>

This means we can also write energy as
: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>

===Energy of a Signal===
From circuit analysis we know that the power generated by a voltage source is,
: <math>P(t) = {v^2(t) \over R}</math>
Assuming that R is 1 then the total energy is just,
: <math>W = \int_{-\infty}^\infty |v|^2(t) \, dt</math>
This can be written using [http://en.wikipedia.org/wiki/Bra-ket_notation bra-ket] notation as
: <math> <v(t) | v(t)> \!</math> or <math> <v|v> \!</math>
By [[Rayliegh's Theroem]],
: <math> <v|v> = \int_{-\infty}^{\infty} |V(f)|^2\,df </math>
This implies that the energy of a signal can be found by integrating the square of the fourier transform of the signal,
: <math> W = \int_{-\infty}^{\infty} |V(f)|^2\,df </math>
This page is far from complete please feel free to pick up where it has been left off.

Energy in a signal

2006-10-11T04:19:10Z

Nathan: /* Energy of a Signal */

*[[Signals and systems|Signals and Systems]]
===Definition of Energy===
Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,
: <math> W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}</math>

Power represents a change in energy.
: <math> P(t) = \frac{dW}{dt} </math>

This means we can also write energy as
: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>

===Energy of a Signal===
From circuit analysis we know that the power generated by a voltage source is,
: <math>P(t) = {v^2(t) \over R}</math>
Assuming that R is 1 then the total energy is just,
: <math>W = \int_{-\infty}^\infty |v|^2(t) \, dt</math>
This can be written using [http://en.wikipedia.org/wiki/Bra-ket_notation bra-ket] notation as
: <math> <v(t) | v(t)> \!</math> or <math> <v|v> \!</math>
By [[Rayliegh's Theroem]],
: <math> <v|v> = \int_{-\infty}^{\infty} |V(f)|^2\,df </math>
This page is far from complete please feel free to pick up where it has been left off.

Energy in a signal

2006-10-11T04:18:41Z

Nathan: /* Energy of a Signal */

*[[Signals and systems|Signals and Systems]]
===Definition of Energy===
Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,
: <math> W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}</math>

Power represents a change in energy.
: <math> P(t) = \frac{dW}{dt} </math>

This means we can also write energy as
: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>

===Energy of a Signal===
From circuit analysis we know that the power generated by a voltage source is,
: <math>P(t) = {v^2(t) \over R}</math>
Assuming that R is 1 then the total energy is just,
: <math>W = \int_{-\infty}^\infty |v|^2(t) \, dt</math>
This can be written using [http://en.wikipedia.org/wiki/Bra-ket_notation bra-ket] notation as
: <math> <v(t) | v(t)> \!</math> or <math> <v|v> \!</math>
By [Rayliegh's Theroem],
: <math> <v|v> = \int_{-\infty}^{\infty} |V(f)|^2\,df </math>
This page is far from complete please feel free to pick up where it has been left off.

Energy in a signal

2006-10-11T04:14:02Z

Nathan: /* Energy of a Signal */

*[[Signals and systems|Signals and Systems]]
===Definition of Energy===
Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,
: <math> W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}</math>

Power represents a change in energy.
: <math> P(t) = \frac{dW}{dt} </math>

This means we can also write energy as
: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>

===Energy of a Signal===
From circuit analysis we know that the power generated by a voltage source is,
: <math>P(t) = {v^2(t) \over R}</math>
Assuming that R is 1 then the total energy is just,
: <math>W = \int_{-\infty}^\infty |v|^2(t) \, dt</math>
This can be written using [http://en.wikipedia.org/wiki/Bra-ket_notation bra-ket] notation as
: <math> <v(t) | v(t)> \!</math> or <math> <v|v> \!</math>

This page is far from complete please feel free to pick up where it has been left off.

Energy in a signal

2006-10-11T04:13:48Z

Nathan: /* Energy of a Signal */

*[[Signals and systems|Signals and Systems]]
===Definition of Energy===
Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,
: <math> W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}</math>

Power represents a change in energy.
: <math> P(t) = \frac{dW}{dt} </math>

This means we can also write energy as
: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>

===Energy of a Signal===
From circuit analysis we know that the power generated by a voltage source is,
: <math>P(t) = {v^2(t) \over R}</math>
Assuming that R is 1 then the total energy is just,
: <math>W = \int_{-\infty}^\infty |v|^2(t) \, dt</math>
This can be written using [http://en.wikipedia.org/wiki/Bra-ket_notation bra-ket] notation as
: <math> <v(t) | v(t)> \!</math> or <math> <v|v> /!</math>

This page is far from complete please feel free to pick up where it has been left off.

Energy in a signal

2006-10-11T04:13:20Z

Nathan: /* Energy of a Signal */

*[[Signals and systems|Signals and Systems]]
===Definition of Energy===
Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,
: <math> W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}</math>

Power represents a change in energy.
: <math> P(t) = \frac{dW}{dt} </math>

This means we can also write energy as
: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>

===Energy of a Signal===
From circuit analysis we know that the power generated by a voltage source is,
: <math>P(t) = {v^2(t) \over R}</math>
Assuming that R is 1 then the total energy is just,
: <math>W = \int_{-\infty}^\infty |v|^2(t) \, dt</math>
This can be written using [http://en.wikipedia.org/wiki/Bra-ket_notation bra-ket] notation as
: <math> <v(t) | v(t)> \!</math>

This page is far from complete please feel free to pick up where it has been left off.

Energy in a signal

2006-10-11T04:13:08Z

Nathan: /* Energy of a Signal */

*[[Signals and systems|Signals and Systems]]
===Definition of Energy===
Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,
: <math> W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}</math>

Power represents a change in energy.
: <math> P(t) = \frac{dW}{dt} </math>

This means we can also write energy as
: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>

===Energy of a Signal===
From circuit analysis we know that the power generated by a voltage source is,
: <math>P(t) = {v^2(t) \over R}</math>
Assuming that R is 1 then the total energy is just,
: <math>W = \int_{-\infty}^\infty |v|^2(t) \, dt</math>
This can be written using [http://en.wikipedia.org/wiki/Bra-ket_notation bra-ket] notation as
<math> <v(t)|v(t)> \!</math>

This page is far from complete please feel free to pick up where it has been left off.

Energy in a signal

2006-10-11T04:10:06Z

Nathan: /* Energy of a Signal */

*[[Signals and systems|Signals and Systems]]
===Definition of Energy===
Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,
: <math> W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}</math>

Power represents a change in energy.
: <math> P(t) = \frac{dW}{dt} </math>

This means we can also write energy as
: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>

===Energy of a Signal===
From circuit analysis we know that the power generated by a voltage source is,
: <math>P(t) = {v^2(t) \over R}</math>
Assuming that R is 1 then the total energy is just,
: <math>W = \int_{-\infty}^\infty |v|^2(t) \, dt</math>
This can be written using [http://en.wikipedia.org/wiki/Bra-ket_notation bra-ket] notation as
<math> <v(t)|v(t)> </math>

This page is far from complete please feel free to pick up where it has been left off.

Energy in a signal

2006-10-11T04:05:54Z

Nathan: /* Energy of a Signal */

Energy in a signal

2006-10-11T04:05:34Z

Nathan: /* Energy of a Signal */

Energy in a signal

2006-10-11T04:00:17Z

Nathan: /* Energy of a Signal */

Energy in a signal

2006-10-11T03:59:58Z

Nathan: /* Energy of a Signal */

*[[Signals and systems|Signals and Systems]]
===Definition of Energy===
Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,
: <math> W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}</math>

Power represents a change in energy.
: <math> P(t) = \frac{dW}{dt} </math>

This means we can also write energy as
: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>

===Energy of a Signal===
From circuit analysis we know that the power generated by a voltage source is,
: <math>P(t) = {\mathbf{V}^2(t) \over R}</math>
Assuming that R is 1 then the total energy is just,
: <math>W = \int_{-\infty}^\infty |\mathbf{V}|^2(t) \, \mathrm{d}\mathbf{t}</math>

This page is far from complete please feel free to pick up where it has been left off.

Energy in a signal

2006-10-11T03:56:51Z

Nathan: /* Energy of a Signal */

*[[Signals and systems|Signals and Systems]]
===Definition of Energy===
Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,
: <math> W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}</math>

Power represents a change in energy.
: <math> P(t) = \frac{dW}{dt} </math>

This means we can also write energy as
: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>

===Energy of a Signal===
From circuit analysis we know that the power generated by a voltage source is,
: <math>P(t) = {\mathbf{V}^2(t) \over R}</math>
Assuming that R is 1 then the total energy is just,
: <math>W = \int_{-\infty}^\infty |\mathbf{V}|^2(t) \mathrm{d}\mathbf{t}</math>

This page is far from complete please feel free to pick up where it has been left off.

Energy in a signal

2006-10-11T03:56:28Z

Nathan: /* Energy of a signal */

*[[Signals and systems|Signals and Systems]]
===Definition of Energy===
Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,
: <math> W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}</math>

Power represents a change in energy.
: <math> P(t) = \frac{dW}{dt} </math>

This means we can also write energy as
: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>

===Energy of a Signal===
From circuit analysis we know that the power generated by voltage source is,
: <math>P(t) = {\mathbf{V}^2(t) \over R}</math>
Assuming that R is 1 then the total energy is just,
: <math>W = \int_{-\infty}^\infty |\mathbf{V}|^2(t) \mathrm{d}\mathbf{t}</math>

This page is far from complete please feel free to pick up where it has been left off.

Energy in a signal

2006-10-11T03:56:16Z

Nathan: /* Energy of a signal */

*[[Signals and systems|Signals and Systems]]
===Definition of Energy===
Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,
: <math> W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}</math>

Power represents a change in energy.
: <math> P(t) = \frac{dW}{dt} </math>

This means we can also write energy as
: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>

===Energy of a signal===
From circuit analysis we know that the power generated by voltage source is,
: <math>P(t) = {\mathbf{V}^2(t) \over R}</math>
Assuming that R is 1 then the total energy is just,
: <math>W = \int_{-\infty}^\infty |\mathbf{V}|^2(t) \mathrm{d}\mathbf{t}</math>

This page is far from complete please feel free to pick up where it has been left off.

Energy in a signal

2006-10-11T03:55:09Z

Nathan: /* Energy of a signal */

*[[Signals and systems|Signals and Systems]]
===Definition of Energy===
Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,
: <math> W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}</math>

Power represents a change in energy.
: <math> P(t) = \frac{dW}{dt} </math>

This means we can also write energy as
: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>

===Energy of a signal===
From circiut analysis we know that the power generated by voltage source is,
: <math>P(t) = {\mathbf{V}^2(t) \over R}</math>
Assuming that R is 1 then the total energy is just,
: <math>W = \int_{-\infty}^\infty |\mathbf{V}|^2(t) \mathrm{d}\mathbf{t}</math>

This page is far from complete please feel free to pick up where it has been left off.

Energy in a signal

2006-10-11T03:46:51Z

Nathan: /* Energy of a signal */

*[[Signals and systems|Signals and Systems]]
===Definition of Energy===
Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,
: <math> W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}</math>

Power represents a change in energy.
: <math> P(t) = \frac{dW}{dt} </math>

This means we can also write energy as
: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>

===Energy of a signal===
From circiut analysis we know that the power generated by voltage source is,
: <math>P(t) = {\mathbf{V}^2(t) \over R}</math>
Assuming that R is 1 then the total energy is just,
: <math>W = \int_{-\infty}^\infty |\mathbf{V}|^2(t) \mathrm{d}\mathbf{t}</math>

This page is far from complete please feel free to pick up where it has been left off.

Energy in a signal

2006-10-11T03:45:06Z

Nathan: /* Energy of a signal */

Energy in a signal

2006-10-11T03:44:48Z

Nathan: /* Energy of a signal */

Energy in a signal

2006-10-11T03:40:01Z

Nathan: /* Definition of Energy */

Orthogonal functions

2006-10-04T03:08:17Z

Nathan: /* Vectors */

==Introduction==
In this article we will examine another viewpoint for functions than that traditionally taken. Normally we think of a function <math> f(t) </math> as a complicated entity - <math> f() </math> in a simple environment (one dimension, or along the t axis). Now we want to think of a function as a vector or point (a simple thing) in a very complicated environment (possibly an infinite dimensional space).

==Vectors==
Recall that vectors consist of an ordered set of numbers. Often the numbers are Real numbers, but we shall allow them to be Complex for our purposes. The numbers represent the amount of the vector in the direction denoted by the position of the number in the list. Each position in the list is associated with a direction. For example, the vector
<math> \vec \bold v = <1, 4, 3> </math> means that the vector <math> \vec \bold v </math> is one unit in the first direction (often the x direction), four units in the second direction (often the y direction), and three units in the last direction (often the z direction). We say the component of <math> \vec \bold v </math> in the second direction is 4. This is often written as <math> v_y = 4 </math>.
====Vector notation====
We don't have to use x, y, and z as the direction names; we can use numbers, like 1, 2, and 3 instead. The advantage of this is that it leads to more compact notation, and extends to more than three dimensions much better. For example we could say <math> v_2 = 4 </math> instead of <math> v_y = 4 </math>. Instead of writing <math> \vec \bold v = <1, 4, 3> </math> we can write <math> \vec \bold v = \sum_{k=1}^3 v_k \hat \bold a_k </math> where the <math>\hat \bold a_k </math> denotes a basis vector in the kth direction, <math>v_1 = 1,</math> <math> v_2 = 4, </math> and <math> v_3 = 3</math>. The idea of basis vectors was implicit in the notation <math> \vec \bold v = <1, 4, 3> </math>.

===Inner products for vectors===
When vectors are real, inner products (sometimes called dot products) give the component of one vector in another vector's direction, scaled by the magnitude (length) of the second vector. Inner products are useful to find components of vectors. We commonly use a dot as the symbol for inner product. For example, the inner product of <math> \vec \bold v </math> and <math> \vec \bold a_n </math> is written:

<math> \vec \bold v \bullet \vec \bold a_n </math>

====Orthogonality for vectors====
It is quite handy to pick the directions used so that they are perpendicular (or orthogonal). With this arrangement the basis vectors have no components in each other's directions, which means that

<math>\vec \bold a_k \bullet \vec \bold a_n = w_k \delta_{k,n} </math>

where the <math> w_k </math> is the square of the length of <math> \vec \bold a_k </math> and the symbol <math> \delta_{k,n} </math>, known as the [http://en.wikipedia.org/wiki/Kronecker_delta Kronecker delta], is one when k = n and zero otherwise.
=====Normalization=====
When the <math> w_k = 1</math> we have an orthonormal basis set, meaning that it is both orthogonal and that the basis vectors are normalized to unity (or have length one). Orthonormal vector systems are very popular. In fact they are the most common vector systems you will find. The reason they are so handy is each direction is uncoupled from the others.

For example, to find <math> v_n </math>, we take the inner product of the vector <math> \vec \bold v </math> with a unit vector in the nth direction, <math> \vec \bold a_n </math>. We write this operation like this:

<math> \vec \bold v \bullet \vec \bold a_n = \sum_{k=1}^3 v_k \vec \bold a_k \bullet \vec \bold a_n = \sum_{k=1}^3 v_k \delta_{k,n} = v_n </math>

Suppose we have two vectors from an orthonormal system, <math> \vec \bold u </math> and <math> \vec \bold v </math>. Taking the inner product of these vectors, we get

<math> \vec \bold u \bullet \vec \bold v = \sum_{k=1}^3 u_k \vec \bold a_k \bullet \sum_{m=1}^3 v_m \vec \bold a_m = \sum_{k=1}^3 u_k \sum_{m=1}^3 v_m \vec \bold a_k \bullet \vec \bold a_m = \sum_{k=1}^3 u_k \sum_{m=1}^3 v_m \delta_{k,m} = \sum_{k=1}^3 v_k u_k </math>

This shows that when we have an orthonormal vector space, inner products boil down to summing the products of like components. Also note that if we take the inner product of <math> \vec \bold v </math> with itself, we get

<math> \vec \bold v \bullet \vec \bold v = \sum_{k=1}^3 v_k \vec \bold a_k \bullet \sum_{m=1}^3 v_m \vec \bold a_m = \sum_{k=1}^3 v_k \sum_{m=1}^3 v_m \vec \bold a_k \bullet \vec \bold a_m = \sum_{k=1}^3 v_k \sum_{m=1}^3 v_m \delta_{k,m} = \sum_{k=1}^3 v_k^2</math>

which is the magnitude of the vector <math> \vec \bold v </math> squared (<math> | \vec \bold v |^2 </math>) from the Pythagorean Theorem.

====Changing vector basis sets====
Sometimes in our studies we find it useful to change basis sets. For example, when solving a physics problem with cylindrical symmetry, it is often easier to use cylindrical coordinates, and the basis vectors that go with that system, rather than the more usual Cartesian coordinates and basis vectors.
=====So, how do I change the basis set?=====
If the new basis set is orthonormal, it is really pretty simple. You need to project the vector you want changed onto each of the new basis vectors. This means that the new components are just the inner product of the vector and the appropriate basis function. If the new basis set is not orthonormal, and if there are n dimensions in each basis set, you will have n linear coupled equations in n unknowns to solve.

===More vector questions===

[[Complex vector inner products|What if the vectors have complex components?]]

[[Vector weighting functions|What if not all components of the vectors have the same units?]]

[[Multiple dimensional vectors|What if there are more than three dimensions?]]

==Functions and vectors, an analogy==
We may think of the number of the direction, <math> k </math>, as the independent variable of a vector and the component in that direction, <math> v_k </math> as the dependent variable of the vector <math> \vec \bold v </math> in a similar way to the way we think of t as the independent variable of a function f(), where f(t) is the dependent variable of f. Probably the biggest difference here is that t often takes on real values from <math> - \infty </math> to <math> \infty </math>, and <math> k \in {1, 2, 3} </math>. Using this analogy, we may think of a function as a vector having an uncountably infinite number of dimensions.

====Can we write functions in an analogous way to the way we write vectors?====

Remember we wrote <math> \vec \bold v = \sum_{k=1}^3 v_k \hat \bold a_k </math>. Can we write something similar for a function, f(t) defined for t an element of the reals? Well maybe.... If the sum over the dummy index k becomes an integral over the dummy variable, x, and the unit vectors <math> \vec \bold a_k </math> are replaced with something like <math> \delta(x-t) </math>, the [http://en.wikipedia.org/wiki/Delta_function Dirac delta function]. The result would look something like this:

<math> f(t) = \int_{- \infty}^\infty f(x) \delta (x-t) dx </math>.

This works! The Dirac delta functions, playing the roll of the basis vectors, are called basis functions. The function f(x) plays the roll of the vector coefficients <math>v_k</math>. This gives us another way to think of the function f().

===Inner products for functions===
[[Orthogonal functions#Inner products for vectors|Above]] we found that a vector inner product between <math>\vec \bold u </math> and <math>\vec \bold v </math> could be written as <math> \vec \bold u \bullet \vec \bold v = \sum_{k=1}^3 u_k v_k </math>. If we follow our above analogy, we should be able to replace the sum over k with an integral over x. There is one little notational problem, and that is we don't want to confuse the functional inner product with a simple muliply, so we need some new notation to denote this new inner product. In [http://en.wikipedia.org/wiki/Quantum_mechanics quantum mechanics], physicists use the [http://en.wikipedia.org/wiki/Bra-ket_notation bra-ket] notation. Let's borrow that.

<math> <u|v> = \int_{-\infty}^\infty u^*(x) v(x) dx </math>

Note the complex conjugate on the function u(x). That is in case u(x) is a complex valued function. For the analogous case with vectors see [[Complex vector inner products]].
====Orthogonality for functions====
Two functions, <math>u(t)</math> and <math>v(t)</math> are said to be orthogonal on the interval <math> (a,b) </math> with respect to the weighting function <math> w(t) </math> if and only if
<math>\int_a^b w(x) u^*(x) v(x) dx = 0 </math>.
The weighting function is often unity, but it is included so that different values of t can be weighted appropriately in analogy to the way the <math>w_k</math> weight was used when the vector basis set was orthogonal, but not orthonormal (that is, different basis vectors had different numerical lengths), as we discussed [[Vector weighting functions|here]]. Unless otherwise noted we will use <math> w(t) = 1 </math>, so that the defining relation for orthogonality of functions <math> u </math> and <math> v </math> becomes
<math>\int_a^b u^*(x) v(x) dx = 0 </math>.

====Changing basis sets with functions====
====Examples====
*[[Fourier series]]
*[[Reconstructing bandlimited signals from sample points]]

==Other resources on orthogonality==
[http://en.wikipedia.org/wiki/Inner_product Wikipedia inner product]

Principle author of this page: [[User:Frohro|Rob Frohne]]

Orthogonal functions

2006-10-04T03:01:15Z

Nathan: /* Introduction */

==Introduction==
In this article we will examine another viewpoint for functions than that traditionally taken. Normally we think of a function <math> f(t) </math> as a complicated entity - <math> f() </math> in a simple environment (one dimension, or along the t axis). Now we want to think of a function as a vector or point (a simple thing) in a very complicated environment (possibly an infinite dimensional space).

==Vectors==
Recall that vectors consist of an ordered set of numbers. Often the numbers are Real numbers, but we shall allow them to be from the Complex numbers for our purposes. The numbers represent the amount of the vector in the direction denoted by the position of the number in the list. Each position in the list is associated with a direction. For example, the vector
<math> \vec \bold v = <1, 4, 3> </math> means that the vector <math> \vec \bold v </math> is one unit in the first direction (often the x direction), four units in the second direction (often the y direction), and three units in the last direction (often the z direction). We say the component of <math> \vec \bold v </math> in the second direction is 4. This is often written as <math> v_y = 4 </math>.
====Vector notation====
We don't have to use x, y, and z as the direction names; we can use numbers, like 1, 2, and 3 instead. The advantage of this is that it leads to more compact notation, and extends to more than three dimensions much better. For example we could say <math> v_2 = 4 </math> instead of <math> v_y = 4 </math>. Instead of writing <math> \vec \bold v = <1, 4, 3> </math> we can write <math> \vec \bold v = \sum_{k=1}^3 v_k \hat \bold a_k </math> where the <math>\hat \bold a_k </math> denotes a basis vector in the kth direction, <math>v_1 = 1,</math> <math> v_2 = 4, </math> and <math> v_3 = 3</math>. The idea of basis vectors was implicit in the notation <math> \vec \bold v = <1, 4, 3> </math>.

===Inner products for vectors===
When vectors are real, inner products, sometimes called dot products give the component of one vector in another vector's direction, scaled by the magnitude (length) of the second vector. Inner products are useful to find components of vectors. We commonly use a dot as the symbol for inner product. For example, the inner product of <math> \vec \bold v </math> and <math> \vec \bold a_n </math> is written:

<math> \vec \bold v \bullet \vec \bold a_n </math>

====Orthogonality for vectors====
It is quite handy to pick the directions used so that they are perpendicular (or orthogonal). With this arrangement the basis vectors have no components in each other's directions, which means that

<math>\vec \bold a_k \bullet \vec \bold a_n = w_k \delta_{k,n} </math>

where the <math> w_k </math> is the square of the length of <math> \vec \bold a_k </math> and the symbol <math> \delta_{k,n} </math>, known as the [http://en.wikipedia.org/wiki/Kronecker_delta Kronecker delta], is one when k = n and zero otherwise.
=====Normalization=====
When the <math> w_k = 1</math> we have an orthonormal basis set, meaning that it is both orthogonal and that the basis vectors are normalized to unity (or have length one). Orthonormal vector systems are very popular. In fact they are the most common vector systems you will find. The reason they are so handy is each direction is uncoupled from the others.

For example, to find <math> v_n </math>, we take the inner product of the vector <math> \vec \bold v </math> with a unit vector in the nth direction, <math> \vec \bold a_n </math>. We write this operation like this:

<math> \vec \bold v \bullet \vec \bold a_n = \sum_{k=1}^3 v_k \vec \bold a_k \bullet \vec \bold a_n = \sum_{k=1}^3 v_k \delta_{k,n} = v_n </math>

Suppose we have two vectors from an orthonormal system, <math> \vec \bold u </math> and <math> \vec \bold v </math>. Taking the inner product of these vectors, we get

<math> \vec \bold u \bullet \vec \bold v = \sum_{k=1}^3 u_k \vec \bold a_k \bullet \sum_{m=1}^3 v_m \vec \bold a_m = \sum_{k=1}^3 u_k \sum_{m=1}^3 v_m \vec \bold a_k \bullet \vec \bold a_m = \sum_{k=1}^3 u_k \sum_{m=1}^3 v_m \delta_{k,m} = \sum_{k=1}^3 v_k u_k </math>

This shows that when we have an orthonormal vector space, inner products boil down to summing the products of like components. Also note that if we take the inner product of <math> \vec \bold v </math> with itself, we get

<math> \vec \bold v \bullet \vec \bold v = \sum_{k=1}^3 v_k \vec \bold a_k \bullet \sum_{m=1}^3 v_m \vec \bold a_m = \sum_{k=1}^3 v_k \sum_{m=1}^3 v_m \vec \bold a_k \bullet \vec \bold a_m = \sum_{k=1}^3 v_k \sum_{m=1}^3 v_m \delta_{k,m} = \sum_{k=1}^3 v_k^2</math>

which is the magnitude of the vector <math> \vec \bold v </math> squared (<math> | \vec \bold v |^2 </math>) from the Pythagorean Theorem.

====Changing vector basis sets====
Sometimes in our studies we find it useful to change basis sets. For example, when solving a physics problem with cylindrical symmetry, it is often easier to use cylindrical coordinates, and the basis vectors that go with that system, rather than the more usual Cartesian coordinates and basis vectors.
=====So, how do I change the basis set?=====
If the new basis set is orthonormal, it is really pretty simple. You need to project the vector you want changed onto each of the new basis vectors. This means that the new components are just the inner product of the vector and the appropriate basis function. If the new basis set is not orthonormal, and if there are n dimensions in each basis set, you will have n linear coupled equations in n unknowns to solve.

===More vector questions===

[[Complex vector inner products|What if the vectors have complex components?]]

[[Vector weighting functions|What if not all components of the vectors have the same units?]]

[[Multiple dimensional vectors|What if there are more than three dimensions?]]

==Functions and vectors, an analogy==
We may think of the number of the direction, <math> k </math>, as the independent variable of a vector and the component in that direction, <math> v_k </math> as the dependent variable of the vector <math> \vec \bold v </math> in a similar way to the way we think of t as the independent variable of a function f(), where f(t) is the dependent variable of f. Probably the biggest difference here is that t often takes on real values from <math> - \infty </math> to <math> \infty </math>, and <math> k \in {1, 2, 3} </math>. Using this analogy, we may think of a function as a vector having an uncountably infinite number of dimensions.

====Can we write functions in an analogous way to the way we write vectors?====

Remember we wrote <math> \vec \bold v = \sum_{k=1}^3 v_k \hat \bold a_k </math>. Can we write something similar for a function, f(t) defined for t an element of the reals? Well maybe.... If the sum over the dummy index k becomes an integral over the dummy variable, x, and the unit vectors <math> \vec \bold a_k </math> are replaced with something like <math> \delta(x-t) </math>, the [http://en.wikipedia.org/wiki/Delta_function Dirac delta function]. The result would look something like this:

<math> f(t) = \int_{- \infty}^\infty f(x) \delta (x-t) dx </math>.

This works! The Dirac delta functions, playing the roll of the basis vectors, are called basis functions. The function f(x) plays the roll of the vector coefficients <math>v_k</math>. This gives us another way to think of the function f().

===Inner products for functions===
[[Orthogonal functions#Inner products for vectors|Above]] we found that a vector inner product between <math>\vec \bold u </math> and <math>\vec \bold v </math> could be written as <math> \vec \bold u \bullet \vec \bold v = \sum_{k=1}^3 u_k v_k </math>. If we follow our above analogy, we should be able to replace the sum over k with an integral over x. There is one little notational problem, and that is we don't want to confuse the functional inner product with a simple muliply, so we need some new notation to denote this new inner product. In [http://en.wikipedia.org/wiki/Quantum_mechanics quantum mechanics], physicists use the [http://en.wikipedia.org/wiki/Bra-ket_notation bra-ket] notation. Let's borrow that.

<math> <u|v> = \int_{-\infty}^\infty u^*(x) v(x) dx </math>

Note the complex conjugate on the function u(x). That is in case u(x) is a complex valued function. For the analogous case with vectors see [[Complex vector inner products]].
====Orthogonality for functions====
Two functions, <math>u(t)</math> and <math>v(t)</math> are said to be orthogonal on the interval <math> (a,b) </math> with respect to the weighting function <math> w(t) </math> if and only if
<math>\int_a^b w(x) u^*(x) v(x) dx = 0 </math>.
The weighting function is often unity, but it is included so that different values of t can be weighted appropriately in analogy to the way the <math>w_k</math> weight was used when the vector basis set was orthogonal, but not orthonormal (that is, different basis vectors had different numerical lengths), as we discussed [[Vector weighting functions|here]]. Unless otherwise noted we will use <math> w(t) = 1 </math>, so that the defining relation for orthogonality of functions <math> u </math> and <math> v </math> becomes
<math>\int_a^b u^*(x) v(x) dx = 0 </math>.

====Changing basis sets with functions====
====Examples====
*[[Fourier series]]
*[[Reconstructing bandlimited signals from sample points]]

==Other resources on orthogonality==
[http://en.wikipedia.org/wiki/Inner_product Wikipedia inner product]

Principle author of this page: [[User:Frohro|Rob Frohne]]

Discrete Fourier transform

2006-10-04T02:49:32Z

Nathan:

==Discrete Fourier Transform==

User:Nathan

2006-10-02T03:53:33Z

Nathan:

Welcome to my humble page.

==My Family==
[[Image:FerchFamily.jpg]]

Signals and Systems

2006-10-02T03:53:06Z

Nathan: /* 2006-2007 contributors */

[http://www.wwc.edu/~frohro/ClassNotes/engr455index.htm Class notes for Signals & Systems]

== Topics ==
*[[Orthogonal functions]]
*[[Fourier series]]
*[[Fourier transform]]
*[[Sampling]]
*[[Discrete Fourier transform]]
*[[Fourier series - by Ray Betz|Signals and Systems - by Ray Betz]]
*[[FIR Filter Example]]
*[[2005-2006 Assignments]]

I couldn't figure out how to get to others Users pages easily so I decided to start posting them here, please add yours:

[[User:Frohro|Rob Frohne]]

==2004-2005 contributors==

[[User:Barnsa|Sam Barnes]]

[[User:Santsh|Shawn Santana]]

[[User:Goeari|Aric Goe]]

[[User:Caswto|Todd Caswell]]

[[User:Andeda|David Anderson]]

[[User:Guenan|Anthony Guenterberg]]

==2005-2006 contributors==

[[User:GabrielaV|Gabriela Valdivia]]

[[User:SDiver|Raymond Betz]]

[[User:chrijen|Jenni Christensen]]

[[User:wonoje|Jeffrey Wonoprabowo]]

[[User:wilspa|Paul Wilson]]

==2006-2007 contributors==

[[User:Smitry|Ryan J Smith]]

[[User:Nathan|Nathan Ferch]]

Signals and Systems

2006-10-02T03:33:25Z

Nathan: /* 2006-2007 contributors */

Signals and Systems

2006-10-02T03:33:07Z

Nathan: /* Topics */

Signals and Systems

2006-10-02T03:32:50Z

Nathan: /* Topics */

[http://www.wwc.edu/~frohro/ClassNotes/engr455index.htm Class notes for Signals & Systems]

== Topics ==
*[[Orthogonal functions]]
*[[Fourier series]]
*[[Fourier transform]]
*[[Sampling]]
*[[Discrete Fourier transform]]
*[[Fourier series - by Ray Betz|Signals and Systems - by Ray Betz]]
*[[FIR Filter Example]]
*[[2005-2006 Assignments]]

I couldn't figure out how to get to others Users pages easily so I decided to start posting them here, please add yours:

[[User:Frohro|Rob Frohne]]

[[User:Nathan|Nathan Ferch]]

==2004-2005 contributors==

[[User:Barnsa|Sam Barnes]]

[[User:Santsh|Shawn Santana]]

[[User:Goeari|Aric Goe]]

[[User:Caswto|Todd Caswell]]

[[User:Andeda|David Anderson]]

[[User:Guenan|Anthony Guenterberg]]

==2005-2006 contributors==

[[User:GabrielaV|Gabriela Valdivia]]

[[User:SDiver|Raymond Betz]]

[[User:chrijen|Jenni Christensen]]

[[User:wonoje|Jeffrey Wonoprabowo]]

[[User:wilspa|Paul Wilson]]

==2006-2007 contributors==

[[User:Smitry|Ryan J Smith]]

User:Nathan

2006-10-02T03:31:10Z

Nathan:

Welcome to my humble page.

'''Me with my family:'''
[[Image:FerchFamily.jpg]]

User:Nathan

2006-10-02T03:30:55Z

Nathan:

Welcome to my humble page.

'''My Family and I:'''
[[Image:FerchFamily.jpg]]

User:Nathan

2006-10-02T03:30:39Z

Nathan:

Welcome to my humble page.

'''My Family:'''
[[Image:FerchFamily.jpg]]

User:Nathan

2006-10-02T03:29:24Z

Nathan:

Welcome to my humble page.

[[Image:FerchFamily.jpg]]