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Although the function to be expanded is defined only over a specific finite region, the series converges to a function that is defined over thereal line and is periodic. It is equal to the original function over the region of definition and is a periodic extension outside of the region.Indeed, one could artificially extend the given function at the outset and then the expansion would converge everywhere.

A geometric view

It can be very helpful to develop a geometric view of the Fourier series where x ( t ) is considered to be a vector and the basis functions are the coordinate or basis vectors. The coefficients become the projections of x ( t ) on the coordinates. The ideas of a measure of distance, size, and orthogonality are important and the definition of error is easy topicture. This is done in [link] , [link] , [link] using Hilbert space methods.

Properties of the fourier series

The properties of the Fourier series are important in applying it to signal analysis and to interpreting it. The main properties are given hereusing the notation that the Fourier series of a real valued function x ( t ) over { 0 t T } is given by F { x ( t ) } = c ( k ) and x ˜ ( t ) denotes the periodic extensions of x ( t ) .

  1. Linear: F { x + y } = F { x } + F { y }
    Idea of superposition. Also scalability: F { a x } = a F { x }
  2. Extensions of x ( t ) : x ˜ ( t ) = x ˜ ( t + T )
    x ˜ ( t ) is periodic.
  3. Even and Odd Parts: x ( t ) = u ( t ) + j v ( t ) and C ( k ) = A ( k ) + j B ( k ) = | C ( k ) | e j θ ( k )
    u v A B | C | θ
    even 0 even 0 even 0
    odd 0 0 odd even 0
    0 even 0 even even π / 2
    0 odd odd 0 even π / 2
  4. Convolution: If continuous cyclic convolution is defined by
    y ( t ) = h ( t ) x ( t ) = 0 T h ˜ ( t - τ ) x ˜ ( τ ) d τ

    then F { h ( t ) x ( t ) } = F { h ( t ) } F { x ( t ) }
  5. Multiplication: If discrete convolution is defined by
    e ( n ) = d ( n ) * c ( n ) = m = - d ( m ) c ( n - m )

    then F { h ( t ) x ( t ) } = F { h ( t ) } * F { x ( t ) }
    This property is the inverse of property 4 and vice versa.
  6. Parseval: 1 T 0 T | x ( t ) | 2 d t = k = - | C ( k ) | 2
    This property says the energy calculated in the time domain is the same as that calculated in the frequency (or Fourier) domain.
  7. Shift: F { x ˜ ( t - t 0 ) } = C ( k ) e - j 2 π t 0 k / T
    A shift in the time domain results in a linear phase shift in the frequency domain.
  8. Modulate: F { x ( t ) e j 2 π K t / T } = C ( k - K )
    Modulation in the time domain results in a shift in the frequency domain. This property is the inverse of property 7.
  9. Orthogonality of basis functions:
    0 T e - j 2 π m t / T e j 2 π n t / T d t = T δ ( n - m ) = T if n = m 0 if n m .
    Orthogonality allows the calculation of coefficients using inner products in [link] and [link] . It also allows Parseval's Theorem in property 6 . A relaxed version of orthogonality is called “tight frames" and is importantin over-specified systems, especially in wavelets.

Examples

  • An example of the Fourier series is the expansion of a square wave signal with period 2 π . The expansion is
    x ( t ) = 4 π [ sin ( t ) + 1 3 sin ( 3 t ) + 1 5 sin ( 5 t ) ] .
    Because x ( t ) is odd, there are no cosine terms (all a ( k ) = 0 ) and, because of its symmetries, there are no even harmonics (even k terms are zero). The function is well defined and bounded; its derivative is not,therefore, the coefficients drop off as 1 k .
  • A second example is a triangle wave of period 2 π . This is a continuous function where the square wave was not. The expansion of thetriangle wave is
    x ( t ) = 4 π [ sin ( t ) - 1 3 2 sin ( 3 t ) + 1 5 2 sin ( 5 t ) + ] .
    Here the coefficients drop off as 1 k 2 since the function and its first derivative exist and are bounded.

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
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Source:  OpenStax, Digital signal processing and digital filter design (draft). OpenStax CNX. Nov 17, 2012 Download for free at http://cnx.org/content/col10598/1.6
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