<< 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 .

Examples

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

what is economic
Charles Reply
what are the type of economic
Charles
macroeconomics,microeconomics,positive economics and negative economics
Gladys
what are the factors of production
Gladys
process of production
Mutia
Basically factors of production are four (4) namely: 1. Entrepreneur 2. Capital 3. Labour and; 4. Land but there has been a new argument to include an addition one to the the numbers to 5 which is "Technology"
Elisha
what is land as a factor of production
Gladys
what is Economic
Abu
economics is how individuals bussiness and governments make the best decisions to get what they want and how these choices interact in the market
Nandisha
Economics as a social science, which studies human behaviour as a relationship between ends and scarce means, which have alternative uses.
Yhaar
how will a country's population be equal to it's labour force
Hope Reply
what is the meaning of ppf
Obeng Reply
What is Economic
Governor Reply
economic
Nwosu
Economics is the social science that deals with the unlimited human wants in the face of scarce (limited in supply) resources.
Azka
what is market
Gift Reply
marker is the interaction of buying and selling
David
market refers to the interaction of the processes of buying and selling of commodities between the buyer and the seller.
stephen
market is a place where two parties gather to facilitate exchange of goods and services.
Yhaar
what are some good sources of information to find trends in various Industries
James
how do on know that marketing is going on
Mutia
what is consumption
Raj
Using revenue
Prince
What is stock market
Prince
What are the marmet function
Odirile Reply
price elasticity of demand is the degree of responsiveness of a quantity demanded to the change in price of the commodity in question.
Gladys Reply
What does elasticity mean
Prince
Elasticity means change in demand with the change in price. It is elastic if the demand changes with the price change whereas it is inelastic if the demand is not affected due to change in price
Devesh
Okay
Olatunde
meaning
KP
okay
Binta
I have a question
Binta
what is the importance of learning economics?
Thelma Reply
it helps to make the correct choice
Gladys
it helps firm to produce products that will bring more profit
Gladys
the difference between needs and wants
londiwe Reply
needs are things that we basically can't live without wants are just luxury things
Thelma
needs are things without them we can't live but want are things without we can live
KP
what is education
KP
it's a process in which we give or receiving methodical instructions
Thelma
what is mixed economy
Amex
what is a deadweight loss? how monopoly creates a deadweight loss?
Ashraf Reply
who are u?
Lamine
haha
Cleaford
scarm
nura
what it this
Cleaford
hi y'all
Dope
how does group chat help y'all 🤔
Dope
hi y'all
Dope
how does group chat help y'all 🤔
Dope
how does group chat help y'all 🤔
Dope
to learn from one another
Lamine
oh okay
Dope
😟
Creative
Yes
Lamine
what is type of economic
taiwo Reply
how to understand basics of economics
Aarif Reply
what is demand schedle
Princess Reply
When you make a Scedule of the demand you made
Rodeen
this is helpful for rbi grade b
Prema Reply
What is macroeconomics
Kauna Reply
It's one of the two branches of Economics that deal with the aggregate economy.
Mayen
it's about inflation, occupation, gdp and so on
alberto
What is differences between Microeconomics and Macroeconomic?
Bethrand
microeconomics focuses on the action of individual agents in the economy such as businesses, workers and household. while macroeconomics looks at the economy as a whole. it focuses on broad issues in the economy such as government deficit, economy growth, levels of exports and imports, and
Thelma
inflationary increase in prices
Thelma
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





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?

Ask