



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 here using thenotation that the Fourier series of a real valued function
$x(t)$ over
$\{0\le t\le T\}$ is given by
$\mathcal{F}\{x(t)\}=c(k)$ and
$\tilde{x}(t)$ denotes the periodic extensions of
$x(t)$ .
 Linear:
$\mathcal{F}\{x+y\}=\mathcal{F}\{x\}+\mathcal{F}\{y\}$ Idea
of superposition. Also scalability:
$\mathcal{F}\{ax\}=a\mathcal{F}\{x\}$
 Extensions of
$x(t)$ :
$\tilde{x}(t)=\tilde{x}(t+T)$
$\tilde{x}(t)$ is periodic.
 Even and Odd Parts:
$x(t)=u(t)+jv(t)$ and
$C(k)=A(k)+jB(k)=C(k){e}^{j\theta (k)}$
$u$
$v$
$A$
$B$
$C$
$\theta $ even
0even
0even
0odd
00
oddeven
00
even0
eveneven
$\pi /2$ 0
oddodd
0even
$\pi /2$
 Convolution: If continuous cyclic convolution is definedby
$y(t)=h(t)\circ x(t)={\int}_{0}^{T}\tilde{h}(t\tau )\tilde{x}(\tau )d\tau $ then
$\mathcal{F}\{h(t)\circ x(t)\}=\mathcal{F}\{h(t)\}\mathcal{F}\{x(t)\}$
 Multiplication: If discrete convolution is definedby
$e(n)=d(n)*c(n)=\sum _{m=\infty}^{\infty}d(m)c(nm)$ then
$\mathcal{F}\{h(t)x(t)\}=\mathcal{F}\{h(t)\}*\mathcal{F}\{x(t)\}$ This
property is the inverse of property 4 and vice versa.
 Parseval:
$\frac{1}{T}{\int}_{0}^{T}{x(t)}^{2}dt=\sum _{k=\infty}^{\infty}{C(k)}^{2}$ This
property says the energy calculated in the time domain is the same as thatcalculated in the frequency (or Fourier) domain.
 Shift:
$\mathcal{F}\{\tilde{x}(t{t}_{0})\}=C(k){e}^{j2\pi {t}_{0}k/T}$ A
shift in the time domain results in a linear phase shift in the frequencydomain.
 Modulate:
$\mathcal{F}\{x(t){e}^{j2\pi Kt/T}\}=C(kK)$ Modulation
in the time domain results in a shift in the frequency domain. This propertyis the inverse of property 7.
 Orthogonality of basis functions:
$${\int}_{0}^{T}{e}^{j2\pi mt/T}{e}^{j2\pi nt/T}dt=T\delta (nm)=\{\begin{array}{ll}T& {\text{if\hspace{0.5em}}}n=m\\ 0& {\text{if\hspace{0.5em}}}n\ne m\text{.}\end{array}$$
Orthogonality
allows the calculation of coefficients using inner products. It also allowsParseval's Theorem in property 6. A relaxed version of orthogonality is called
"tight frames" and is important in overspecified systems, especially inwavelets.
Questions & Answers
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
What is meant by 'nano scale'?
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
nano basically means 10^(9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:
OpenStax, Principles of digital communications. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10805/1.1
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