<< Chapter < Page Chapter >> Page >
The module looks at decomposing signals through orthonormal basis expansion to provide an alternative representation. The module presents many examples of solving these problems and looks at them in several spaces and dimensions.

Main idea

When working with signals many times it is helpful to break up a signal into smaller, more manageable parts. Hopefully bynow you have been exposed to the concept of eigenvectors and there use in decomposing a signal into one of its possible basis.By doing this we are able to simplify our calculations of signals and systems through eigenfunctions of LTI systems .

Now we would like to look at an alternative way to represent signals, through the use of orthonormal basis . We can think of orthonormal basis as a set of building blockswe use to construct functions. We will build up the signal/vector as a weighted sum of basis elements.

The complex sinusoids 1 T ω 0 n t for all n form an orthonormal basis for L 2 0 T .

In our Fourier series equation, f t n c n ω 0 n t , the c n are just another representation of f t .

Got questions? Get instant answers now!

For signals/vectors in a Hilbert Space , the expansion coefficients are easy to find.

Alternate representation

Recall our definition of a basis : A set of vectors b i in a vector space S is a basis if

  1. The b i are linearly independent.
  2. The b i span S . That is, we can find α i , where α i (scalars) such that
    x x S x i α i b i
    where x is a vector in S , α is a scalar in , and b is a vector in S .

Condition 2 in the above definition says we can decompose any vector in terms of the b i . Condition 1 ensures that the decomposition is unique (think about this at home).

The α i provide an alternate representation of x .

Let us look at simple example in 2 , where we have the following vector: x 1 2 Standard Basis: e 0 e 1 1 0 0 1 x e 0 2 e 1 Alternate Basis: h 0 h 1 1 1 1 -1 x 3 2 h 0 -1 2 h 1

Got questions? Get instant answers now!

In general, given a basis b 0 b 1 and a vector x 2 , how do we find the α 0 and α 1 such that

x α 0 b 0 α 1 b 1

Finding the coefficients

Now let us address the question posed above about finding α i 's in general for 2 . We start by rewriting [link] so that we can stack our b i 's as columns in a 2×2 matrix.

x α 0 b 0 α 1 b 1
x b 0 b 1 α 0 α 1

Here is a simple example, which shows a little more detail about the above equations.

x 0 x 1 α 0 b 0 0 b 0 1 α 1 b 1 0 b 1 1 α 0 b 0 0 α 1 b 1 0 α 0 b 0 1 α 1 b 1 1
x 0 x 1 b 0 0 b 1 0 b 0 1 b 1 1 α 0 α 1

Got questions? Get instant answers now!

Simplifying our equation

To make notation simpler, we define the following two itemsfrom the above equations:

  • Basis Matrix : B b 0 b 1
  • Coefficient Vector : α α 0 α 1
This gives us the following, concise equation:
x B α
which is equivalent to x i 1 0 α i b i .

Given a standard basis, 1 0 0 1 , then we have the following basis matrix: B 0 1 1 0

Got questions? Get instant answers now!

To get the α i 's, we solve for the coefficient vector in [link]

α B x
Where B is the inverse matrix of B .


Let us look at the standard basis first and try to calculate α from it. B 1 0 0 1 I Where I is the identity matrix . In order to solve for α let us find the inverse of B first (which is obviously very trivial in this case): B 1 0 0 1 Therefore we get, α B x x

Got questions? Get instant answers now!

Let us look at a ever-so-slightly more complicated basis of 1 1 1 -1 h 0 h 1 Then our basis matrix and inverse basis matrix becomes: B 1 1 1 -1 B 1 2 1 2 1 2 -1 2 and for this example it is given that x 3 2 Now we solve for α α B x 1 2 1 2 1 2 -1 2 3 2 2.5 0.5 and we get x 2.5 h 0 0.5 h 1

Got questions? Get instant answers now!

Now we are given the following basis matrix and x : b 0 b 1 1 2 3 0 x 3 2 For this problem, make a sketch of the bases and then represent x in terms of b 0 and b 1 .

In order to represent x in terms of b 0 and b 1 we will follow the same steps we used in the above example. B 1 2 3 0 B 0 1 2 1 3 -1 6 α B x 1 2 3 And now we can write x in terms of b 0 and b 1 . x b 0 2 3 b 1 And we can easily substitute in our known values of b 0 and b 1 to verify our results.

Got questions? Get instant answers now!

A change of basis simply looks at x from a "different perspective." B transforms x from the standard basis to our new basis, b 0 b 1 . Notice that this is a totally mechanical procedure.

Extending the dimension and space

We can also extend all these ideas past just 2 and look at them in n and n . This procedure extends naturally to higher (>2) dimensions. Given a basis b 0 b 1 b n 1 for n , we want to find α 0 α 1 α n 1 such that

x α 0 b 0 α 1 b 1 α n 1 b n 1
Again, we will set up a basis matrix B b 0 b 1 b 2 b n 1 where the columns equal the basis vectors and it will alwaysbe an n×n matrix (although the above matrix does not appear to be square since we left terms in vector notation).We can then proceed to rewrite [link] x b 0 b 1 b n 1 α 0 α n 1 B α and α B x

Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where is the latest information on a no technology how can I find it
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now

Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?