<< Chapter < Page Chapter >> Page >

Normalized dtft as an operator

Note that by taking the DTFT of a sequence we get a function defined on [ - π , π ] . In vector space notation we can view the DTFT as an operator (transformation). In this context it is useful to consider the normalizedDTFT

F ( x ) : = X ( e j ω ) = 1 2 π n = - x [ n ] e - j ω n .

One can show that the summation converges for any x 2 ( π ) , and yields a function X ( e j ω ) L 2 [ - π , π ] . Thus,

F : 2 ( Z ) L 2 [ - π , π ]

can be viewed as a linear operator!

Note: It is not at all obvious that F can be defined for all x 2 ( Z ) . To show this, one can first argue that if x 1 ( Z ) , then

X ( e j ω ) 1 2 π n = - x [ n ] e - j ω n 1 2 π n = - x [ n ] e - j w n = 1 2 π n = - x [ n ] <

For an x 2 ( Z ) 1 ( Z ) , one must show that it is always possible to construct a sequence x k 2 ( Z ) 1 ( Z ) such that

lim k x k - x 2 = 0 .

This means { x k } is a Cauchy sequence, so that since 2 ( Z ) is a Hilbert space, the limit exists (and is x ). In this case

X ( e j ω ) = lim k X k ( e j ω ) .

So for any x 2 ( Z ) , we can define F ( x ) = X ( e j ω ) , where X ( e j ω ) L 2 [ - π , π ] .

Can we always get the original x back? Yes, the DTFT is invertible

F - 1 ( X ) = 1 2 π - π π X ( e j ω ) · e j ω n d ω

To verify that F - 1 ( F ( x ) ) = x , observe that

1 2 π - π π 1 2 π k = - x [ k ] e - j ω k e j ω n d ω = 1 2 π k = - x [ k ] - π π e - j ω ( k - n ) d ω = 1 2 π k = - x [ k ] · 2 π δ [ n - k ] = x [ n ]

One can also show that for any X L 2 [ - π , π ] , F ( F - 1 ( X ) ) = X .

Operators that satisfy this property are called unitary operators or unitary transformations . Unitary operators are nice! In fact, if A = X Y is a unitary operator between two Hilbert spaces, then one can show that

x 1 , x 2 = A x 1 , A x 2 x 1 , x 2 X ,

i.e., unitary operators obey Plancherel's and Parseval's theorems!

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
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
Good
Berger describes sociologists as concerned with
Mueller Reply
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, Digital signal processing. OpenStax CNX. Dec 16, 2011 Download for free at http://cnx.org/content/col11172/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Digital signal processing' conversation and receive update notifications?

Ask