<< Chapter < Page Chapter >> Page >
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

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
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
characteristics of micro business
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Pdf generation test course. OpenStax CNX. Dec 16, 2009 Download for free at http://legacy.cnx.org/content/col10278/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Pdf generation test course' conversation and receive update notifications?