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This module contains the definition of the Plancharel theorem and Parseval's theorem along with proofs and examples.

Parseval's theorem

Continuous Time Fourier Series preserves signal energy


0 T | f ( t ) | 2 d t = T n = - | C n | 2 with unnormalized basis e j 2 π T n t 0 T | f ( t ) | 2 d t = n = - | C n ' | 2 with unnormalized basis e j 2 π T n t T | | f | | 2 2 L 2 [ 0 , T ) e n e r g y = | | C n ' | | 2 2 l 2 ( Z ) e n e r g y

Prove: plancherel theorem

Given f ( t ) C T F S c n g ( t ) C T F S d n Then 0 T f ( t ) g * ( t ) d t = T n = - c n d n * with unnormalized basis e j 2 π T n t 0 T f ( t ) g * ( t ) d t = n = - c n ' ( d n ' ) * with normalized basis e j 2 π T n t T f , g L 2 ( 0 , T ] = c , d l 2 ( Z )

Periodic signals power

Energy = | | f | | 2 = - | f ( t ) | 2 d t = Power = lim T Energy in [ 0 , T ) T = lim T T n | c n | 2 T = n Z | c n | 2 (unnormalized FS)

Fourier series of square pulse iii -- compute the energy

f ( t ) = n = - c n e j 2 π T n t FS c n = 1 2 sin π 2 n π 2 n energy in time domain: | | f | | 2 2 = 0 T | f ( t ) | 2 d t = T 2 apply Parseval's Theorem: T n | c n | 2 = T 4 n sin π 2 n π 2 n 2 = T 4 4 π 2 n sin π 2 n 2 n 2 = T π 2 π 2 4 + n odd 1 n 2 π 2 4 = T 2

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Plancharel theorem

Plancharel theorem

The inner product of two vectors/signals is the same as the 2 inner product of their expansion coefficients.

Let b i be an orthonormal basis for a Hilbert Space H . x H , y H x i α i b i y i β i b i then x y H i α i β i

Applying the Fourier Series, we can go from f t to c n and g t to d n t 0 T f t g t n c n d n inner product in time-domain = inner product of Fourier coefficients.

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x i α i b i y j β j b j x y H i α i b i j β j b j i α i b i j β j b j i α i j β j b i b j i α i β i by using inner product rules

b i b j 0 when i j and b i b j 1 when i j

If Hilbert space H has a ONB, then inner products are equivalent to inner products in 2 .

All H with ONB are somehow equivalent to 2 .

square-summable sequences are important

Plancharels theorem demonstration

Interact (when online) with a Mathematica CDF demonstrating Plancharels Theorem visually. To Download, right-click and save target as .cdf.

Parseval's theorem: a different approach

Parseval's theorem

Energy of a signal = sum of squares of its expansion coefficients

Let x H , b i ONB

x i α i b i Then H x 2 i α i 2

Directly from Plancharel H x 2 x x H i α i α i i α i 2

Fourier Series 1 T w 0 n t f t 1 T n c n 1 T w 0 n t t 0 T f t 2 n c n 2

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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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
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.
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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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