<< 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

show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
what is the meaning
explain and give four Example hyperbolic function
Lukman Reply
⅗ ⅔½
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
on number 2 question How did you got 2x +2
combine like terms. x + x + 2 is same as 2x + 2
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
how do I set up the problem?
Harshika Reply
what is a solution set?
find the subring of gaussian integers?
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
hi mam
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
yes i wantt to review
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
please help me prove quadratic formula
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
may God blessed u for that. Please I want u to help me in sets.
what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Need help solving this problem (2/7)^-2
Simone Reply
what is the coefficient of -4×
Mehri Reply
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
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?