<< Chapter < Page Chapter >> Page >
This module describes the continuous time Fourier Series (CTFS). It is based on the following modules:Fourier Series: Eigenfunction Approach at http://cnx.org/content/m10496/latest/ by Justin Romberg, Derivation of Fourier Coefficients Equation at http://cnx.org/content/m10733/latest/ by Michael Haag,Fourier Series and LTI Systems at http://cnx.org/content/m10752/latest/ by Justin Romberg, and Fourier Series Wrap-Up at http://cnx.org/content/m10749/latest/ by Michael Haag and Justin Romberg.

Introduction

In this module, we will derive an expansion for continuous-time, periodic functions, and in doing so, derive the Continuous Time Fourier Series (CTFS).

Since complex exponentials are eigenfunctions of linear time-invariant (LTI) systems , calculating the output of an LTI system given s t as an input amounts to simple multiplication, where H s is the eigenvalue corresponding to s. As shown in the figure, a simple exponential input would yield the output

y t H s s t

Simple LTI system.

Using this and the fact that is linear, calculating y t for combinations of complex exponentials is also straightforward.

c 1 s 1 t c 2 s 2 t c 1 H s 1 s 1 t c 2 H s 2 s 2 t n c n s n t n c n H s n s n t

The action of H on an input such as those in the two equations above is easy to explain. independently scales each exponential component s n t by a different complex number H s n . As such, if we can write a function f t as a combination of complex exponentials it allows us to easily calculate the output of a system.

Fourier series synthesis

Joseph Fourier demonstrated that an arbitrary f t can be written as a linear combination of harmonic complex sinusoids

f t n c n j ω 0 n t
where ω 0 2 T is the fundamental frequency. For almost all f t of practical interest, there exists c n to make [link] true. If f t is finite energy ( f t L 0 T 2 ), then the equality in [link] holds in the sense of energy convergence; if f t is continuous, then [link] holds pointwise. Also, if f t meets some mild conditions (the Dirichlet conditions), then [link] holds pointwise everywhere except at points of discontinuity.

The c n - called the Fourier coefficients - tell us "how much" of the sinusoid j ω 0 n t is in f t . The formula shows f t as a sum of complex exponentials, each of which is easily processed by an LTI system (since it is an eigenfunction of every LTI system). Mathematically, it tells us that the set ofcomplex exponentials n n j ω 0 n t form a basis for the space of T-periodic continuous time functions.

We know from Euler's formula that cos ( ω t ) + sin ( ω t ) = 1 - j 2 e j ω t + 1 + j 2 e - j ω t .

Got questions? Get instant answers now!

Synthesis with sinusoids demonstration

timeshiftDemo
Interact(when online) with a Mathematica CDF demonstrating sinusoid synthesis. To download, right click and save as .cdf.

Guitar oscillations on an iphone

Fourier series analysis

Finding the coefficients of the Fourier series expansion involves some algebraic manipulation of the synthesis formula. First of all we will multiply both sides of the equation by j ω 0 k t , where k .

f t j ω 0 k t n c n j ω 0 n t j ω 0 k t
Now integrate both sides over a given period, T :
t T 0 f t j ω 0 k t t T 0 n c n j ω 0 n t j ω 0 k t
On the right-hand side we can switch the summation andintegral and factor the constant out of the integral.
t T 0 f t j ω 0 k t n c n t T 0 j ω 0 n k t
Now that we have made this seemingly more complicated, let us focus on just the integral, t T 0 j ω 0 n k t , on the right-hand side of the above equation. For this integral we will need to consider two cases: n k and n k . For n k we will have:
n n k t T 0 j ω 0 n k t T
For n k , we will have:
n n k t T 0 j ω 0 n k t t T 0 ω 0 n k t j t T 0 ω 0 n k t
But ω 0 n k t has an integer number of periods, n k , between 0 and T . Imagine a graph of the cosine; because it has an integer number of periods, there areequal areas above and below the x-axis of the graph. This statement holds true for ω 0 n k t as well. What this means is
t T 0 ω 0 n k t 0
which also holds for the integral involving the sine function. Therefore, we conclude the following about our integral ofinterest:
t T 0 j ω 0 n k t T n k 0
Now let us return our attention to our complicated equation, [link] , to see if we can finish finding an equation for our Fourier coefficients. Using thefacts that we have just proven above, we can see that the only time [link] will have a nonzero result is when k and n are equal:
n n k t T 0 f t j ω 0 n t T c n
Finally, we have our general equation for the Fourier coefficients:
c n 1 T t T 0 f t j ω 0 n t

Consider the square wave function given by

x ( t ) = 1 / 2 t 1 / 2 - 1 / 2 t > 1 / 2

on the unit interval t Z [ 0 , 1 ) .

c k = 0 1 x ( t ) e - j 2 π k t d t = 0 1 / 2 1 2 e - j 2 π k t d t - 1 / 2 1 1 2 e - j 2 π k t d t = j - 1 + e j π k 2 π k

Thus, the Fourier coefficients of this function found using the Fourier series analysis formula are

c k = - j / π k k odd 0 k even .
Got questions? Get instant answers now!

Fourier series summary

Because complex exponentials are eigenfunctions of LTI systems, it is often useful to represent signals using a set of complex exponentials as a basis. The continuous time Fourier series synthesis formula expresses a continuous time, periodic function as the sum of continuous time, discrete frequency complex exponentials.

f t n c n j ω 0 n t
The continuous time Fourier series analysis formula gives the coefficients of the Fourier series expansion.
c n 1 T t T 0 f t j ω 0 n t
In both of these equations ω 0 2 T is the fundamental frequency.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask