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Definition of the complex Fourier series.

In an earlier module , we showed that a square wave could be expressed as a superposition of pulses. As useful asthis decomposition was in this example, it does not generalize well to other periodic signals:How can a superposition of pulses equal a smooth signal like a sinusoid?Because of the importance of sinusoids to linear systems, you might wonder whether they could be added together to represent alarge number of periodic signals. You would be right and in good company as well. Euler and Gauss in particular worried about this problem, and Jean Baptiste Fourier got the credit even though tough mathematical issues were notsettled until later. They worked on what is now known as the Fourier series : representing any periodic signal as a superposition of sinusoids.

But the Fourier series goes well beyond being another signal decomposition method.Rather, the Fourier series begins our journey to appreciate how a signal can be described in either the time-domain or the frequency-domain with no compromise. Let s t be a periodic signal with period T . We want to show that periodic signals, even those that haveconstant-valued segments like a square wave, can be expressed as sum of harmonically related sine waves: sinusoids having frequencies that are integer multiples of the fundamental frequency . Because the signal has period T , the fundamental frequency is 1 T . The complex Fourier series expresses the signal as a superposition ofcomplex exponentials having frequencies k T , k 1 0 1 .

s t k c k 2 k t T
with c k 1 2 a k b k . The real and imaginary parts of the Fourier coefficients c k are written in this unusual way for convenience in defining the classic Fourier series.The zeroth coefficient equals the signal's average value and is real- valued for real-valued signals: c 0 a 0 . The family of functions 2 k t T are called basis functions and form the foundation of the Fourier series. No matter what theperiodic signal might be, these functions are always present and form the representation's building blocks. They depend on thesignal period T , and are indexed by k .
Assuming we know the period, knowing the Fourier coefficientsis equivalent to knowing the signal. Thus, it makes no difference if we have a time-domain or a frequency-domain characterization of the signal.

What is the complex Fourier series for a sinusoid?

Because of Euler's relation,

2 f t 1 2 2 f t 1 2 2 f t
Thus, c 1 1 2 , c 1 1 2 , and the other coefficients are zero.

To find the Fourier coefficients, we note the orthogonality property

t 0 T 2 k t T 2 l t T T k l 0 k l
Assuming for the moment that the complex Fourier series "works," we can find a signal's complex Fourier coefficients, its spectrum , by exploiting the orthogonality properties of harmonically related complexexponentials. Simply multiply each side of [link] by 2 l t and integrate over the interval 0 T .
c k 1 T t 0 T s t 2 k t T c 0 1 T t 0 T s t

Finding the Fourier series coefficients for the square wave sq T t is very simple. Mathematically, this signal can be expressed as sq T t 1 0 t T 2 1 T 2 t T The expression for the Fourier coefficients has the form

c k 1 T t 0 T 2 2 k t T 1 T t T 2 T 2 k t T
When integrating an expression containing , treat it just like any other constant.
The two integrals are very similar, one equaling the negative of theother. The final expression becomes
c k 2 2 k 1 k 1 2 k k odd 0 k even
sq t k k -3 -1 1 3 2 k 2 k t T
Consequently, the square wave equals a sum of complex exponentials, but only those having frequencies equal to odd multiples of thefundamental frequency 1 T . The coefficients decay slowly as the frequency index k increases. This index corresponds to the k -th harmonic of the signal's period.

Questions & Answers

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
what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, Pdf generation test course. OpenStax CNX. Dec 16, 2009 Download for free at http://legacy.cnx.org/content/col10278/1.5
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