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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 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 to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
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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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