<< Chapter < Page Chapter >> Page >
Development of ideas of vector expansion

Most people with technical backgrounds are familiar with the ideas of expansion vectors or basis vectors and of orthogonality; however, therelated concepts of biorthogonality or of frames and tight frames are less familiar but also important. In the study of wavelet systems, we find thatframes and tight frames are needed and should be understood, at least at a superficial level. One can find details in [link] , [link] , [link] , [link] , [link] . Another perhaps unfamiliar concept is that of an unconditional basis usedby Donoho, Daubechies, and others [link] , [link] , [link] to explain why wavelets are good for signal compression, detection, and denoising [link] , [link] . In this chapter, we will very briefly define and discuss these ideas. At this point, you may want to skip thesesections and perhaps refer to them later when they are specifically needed.

Bases, orthogonal bases, and biorthogonal bases

A set of vectors or functions f k ( t ) spans a vector space F (or F is the Span of the set) if any element of that space can be expressed as a linear combination of members of thatset, meaning: Given the finite or infinite set of functions f k ( t ) , we define Span k { f k } = F as the vector space with all elements of the space of the form

g ( t ) = k a k f k ( t )

with k Z and t , a R . An inner product is usually defined for this space and is denoted f ( t ) , g ( t ) . A norm is defined and is denoted by f = f , f .

We say that the set f k ( t ) is a basis set or a basis for a given space F if the set of { a k } in [link] are unique for any particular g ( t ) F . The set is called an orthogonal basis if f k ( t ) , f ( t ) = 0 for all k . If we are in three dimensional Euclidean space, orthogonal basis vectors are coordinate vectors that are at right (90 o ) angles to each other. We say the set is an orthonormal basis if f k ( t ) , f ( t ) = δ ( k - ) i.e. if, in addition to being orthogonal, the basis vectors are normalized to unity norm: f k ( t ) = 1 for all k .

From these definitions it is clear that if we have an orthonormal basis, we can express any element in the vector space, g ( t ) F , written as [link] by

g ( t ) = k g ( t ) , f k ( t ) f k ( t )

since by taking the inner product of f k ( t ) with both sides of [link] , we get

a k = g ( t ) , f k ( t )

where this inner product of the signal g ( t ) with the basis vector f k ( t ) “picks out" the corresponding coefficient a k . This expansion formulation or representation is extremely valuable. It expresses [link] as an identity operator in the sense that the inner product operates on g ( t ) to produce a set of coefficients that, when used to linearly combine the basis vectors, gives back the original signal g ( t ) . It is the foundation of Parseval's theorem which says the norm or energycan be partitioned in terms of the expansion coefficients a k . It is why the interpretation, storage, transmission, approximation, compression, andmanipulation of the coefficients can be very useful. Indeed, [link] is the form of all Fourier type methods.

Although the advantages of an orthonormal basis are clear, there are cases where the basis system dictated by the problem is not and cannot (orshould not) be made orthogonal. For these cases, one can still have the expression of [link] and one similar to [link] by using a dual basis set f ˜ k ( t ) whose elements are not orthogonal to each other, but to the corresponding element of the expansion set

Questions & Answers

What is diseconomic
Alixe Reply
how can price determination be the central problem of micro economics
simon Reply
marginal cost formula
Nandu Reply
you should differentiate the total cost function in order to get marginal cost function then you can get marginal cost from it
boniphace
What about total cost
Foday
ok
Foday
how can price determination be the central problem if micro economics
simon
formula of cross elasticity of demand
Theresia Reply
what is ceteris paribus
Priyanka Reply
what is ceteris parabus
Priyanka
Ceteris paribus - Literally, "other things being equal"; usually used in economics to indicate that all variables except the ones specified are assumed not to change.
Abdullah
What is broker
scor
land is natural resources that is made by nature
scor
What is broker
scor
what is land
kafui
What is broker
scor
land is natural resources that is made by nature
scor
whats poppina nigga turn it up for a minute get it
amarsyaheed Reply
what is this?
Philo
am from nigeria@ pilo
Frank
am from nigeria@ pilo
Frank
so
owusu
what is production possibility frontier
owusu
it's a summary of opportunity cost depicted on a curve.
okhiria
please help me solve this question with the aid of appropriate diagrams explain how each of the following changes will affect the market price and quantity of bread 1. A
Manuela Reply
please l need past question about economics
Prosper Reply
ok let me know some of the questions please.
Effah
ok am not wit some if den nw buh by tommorow I shall get Dem
adepojurafiu
Hi guys can I get Adam Smith's WEALTH OF NATIONS fo sale?
Ukpen
hello I'm Babaisa alhaji Mustapha. I'm studying Economics in the university of Maiduguri
Babaisa
okay
Humaira
my name is faisal Yahaya. i studied economics at Kaduna state university before proceeding to West African union university benin republic for masters
Faisal
Hi guys..I am from Bangladesh..
Mannan
Wat d meaning of management
igwe Reply
disaster management cycle
Gogul Reply
cooperate social responsibility
igwe
Fedric Wilson Taylor also define management as the act of knowing what to do and seeing that it is done in the best and cheapest way
OLANIYI
difference between microeconomics and macroeconomic
Ugyen Reply
microeconomics is the study of individual units, firm and government while macroeconomics is the study of the economic aggregates.
okhiria
The classical theory of full employment
Lovely
what is monopoli power
Adzaho Reply
the situation that prevails when economic forces balance so that economic variables neither increase nor decrease
Bombey
what is equilibrium
Kabir
what are the important of economic to accounting students with references
salihu Reply
Economics is important because it helps people understand how a variety of factors work with and against each other to control how resources such as labor and capital get used, and how inflation, supply, demand, interest rates and other factors determine how much you pay for goods and services.
Muhammad
explain the steps taken by the government in developing rural market?
Azeem Reply
government provide good road for than
Abigailb
government should provide good agricultural project and it should also provide good road so that the the product that will come out of the farm will be easy transport to the market
ALIMAMY
farming equipments should be provided to farmers to help them improve in farming
Agbor
improving the transport systems providing enterpreneur edecation to the mass living in rural zones enforcment of loans and capital for the people rising awareness on the advantages of rural areas
abdul
contribution of Adam smith in economics
abel Reply
I will join
Dexter
I will join
Patrick
Hey
Fatima
Hey
Amir
Hello
AS
hey
Umarou
I love this book and i need extra Economic book
Amir
Hey
Amir
what's happening here
AS
I love this book and i need extra Economic book
Amir
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, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Wavelets and wavelet transforms' conversation and receive update notifications?

Ask