



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
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
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
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
what is biological synthesis of nanoparticles
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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