# 3.1 Complex fourier series

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 $\frac{1}{T}$ . The complex Fourier series expresses the signal as a superposition ofcomplex exponentials having frequencies $\frac{k}{T}$ , $k=\{\text{…}, -1, 0, 1, \text{…}\}$ .

$s(t)=\sum_{k=()}$ c k 2 k t T
with ${c}_{k}=\frac{1}{2}({a}_{k}-i{b}_{k})$ . 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 $\{e^{i\frac{2\pi kt}{T}}\}$ 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,

$\sin (2\pi ft)=\frac{1}{2i}e^{i\times 2\pi ft}-\frac{1}{2i}e^{-(i\times 2\pi ft)}$
Thus, ${c}_{1}=\frac{1}{2i}$ , ${c}_{-1}=-\left(\frac{1}{2i}\right)$ , and the other coefficients are zero.

To find the Fourier coefficients, we note the orthogonality property

$\int_{0}^{T} e^{i\frac{2\pi kt}{T}}e^{-i\frac{2\pi lt}{T}}\,d t=\begin{cases}T & \text{if k=l}\\ 0 & \text{if k\neq l}\end{cases}$
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 $e^{-(i\times 2\pi lt)}$ and integrate over the interval $\left[0,T\right]$ .
$\begin{array}{l}{c}_{k}=\frac{1}{T}\int_{0}^{T} s(t)e^{-(i\frac{2\pi kt}{T})}\,d t\\ {c}_{0}=\frac{1}{T}\int_{0}^{T} s(t)\,d t\end{array}$

Finding the Fourier series coefficients for the square wave ${\mathrm{sq}}_{T}(t)$ is very simple. Mathematically, this signal can be expressed as ${\mathrm{sq}}_{T}(t)=\begin{cases}1 & \text{if 0< t< \frac{T}{2}}\\ -1 & \text{if \frac{T}{2}< t< T}\end{cases}$ The expression for the Fourier coefficients has the form

${c}_{k}=\frac{1}{T}\int_{0}^{\frac{T}{2}} e^{-(i\frac{2\pi kt}{T})}\,d t-\frac{1}{T}\int_{\frac{T}{2}}^{T} e^{-(i\frac{2\pi kt}{T})}\,d t$
When integrating an expression containing $i$ , treat it just like any other constant.
The two integrals are very similar, one equaling the negative of theother. The final expression becomes
${c}_{k}=\frac{-2}{i\times 2\pi k}(-1^{k}-1)=\begin{cases}\frac{2}{i\pi k} & \text{if k\text{odd}}\\ 0 & \text{if k\text{even}}\end{cases}$
$\mathrm{sq}(t)=\sum_{k\in \{\dots , -3, -1, 1, 3, \dots \}} \frac{2}{i\pi k}e^{(i)\frac{2\pi kt}{T}}$
Consequently, the square wave equals a sum of complex exponentials, but only those having frequencies equal to odd multiples of thefundamental frequency $\frac{1}{T}$ . The coefficients decay slowly as the frequency index $k$ increases. This index corresponds to the $k$ -th harmonic of the signal's period.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
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
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
revolt
da
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
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
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 a peer
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!