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Signals occur in a wide range of physical phenomenon. They might be human speech, blood pressure variations with time, seismic waves,radar and sonar signals, pictures or images, stress and strain signals in a building structure, stock market prices, a city'spopulation, or temperature across a plate. These signals are often modeled or represented by a real or complex valued mathematicalfunction of one or more variables. For example, speech is modeled by a function representing air pressure varying with time. Thefunction is acting as a mathematical analogy to the speech signal and, therefore, is called an analog signal. For these signals, the independent variable is time and it changescontinuously so that the term continuous-time signal is also used. In our discussion, we talk of the mathematical function asthe signal even though it is really a model or representation of the physical signal.

The description of signals in terms of their sinusoidal frequency content has proven to be one of the most powerful tools ofcontinuous and discrete-time signal description, analysis, and processing. For that reason, we will start the discussion ofsignals with a development of Fourier transform methods. We will first review the continuous-time methods of the Fourier series (FS),the Fourier transform or integral (FT), and the Laplace transform (LT). Next the discrete-time methods will be developed in moredetail with the discrete Fourier transform (DFT) applied to finite length signals followed by the discrete-time Fourier transform(DTFT) for infinitely long signals and ending with the Z-transform which allows the powerful tools of complex variable theory to beapplied.

More recently, a new tool has been developed for the analysis of signals. Wavelets and wavelet transforms [link] , [link] , [link] , [link] , [link] are another more flexible expansion system that also can describe continuousand discrete-time, finite or infinite duration signals. We will very briefly introduce the ideas behind wavelet-based signal analysis.

The fourier series

The problem of expanding a finite length signal in a trigonometric series was posed and studied in the late 1700's by renowned mathematicians suchas Bernoulli, d'Alembert, Euler, Lagrange, and Gauss. Indeed, what we now call the Fourier series and the formulas for the coefficients were used byEuler in 1780. However, it was the presentation in 1807 and the paper in 1822 by Fourier stating that an arbitrary function could be represented bya series of sines and cosines that brought the problem to everyone's attention and started serious theoretical investigations and practicalapplications that continue to this day [link] , [link] , [link] , [link] , [link] , [link] . The theoretical work has been at the center of analysis and the practical applications havebeen of major significance in virtually every field of quantitative science and technology. For these reasons and others, the Fourier seriesis worth our serious attention in a study of signal processing.

Definition of the fourier series

We assume that the signal x ( t ) to be analyzed is well described by a real or complex valued function of a real variable t defined over a finite interval { 0 t T } . The trigonometric series expansion of x ( t ) is given by

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
for teaching engĺish at school how nano technology help us
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
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?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Brief notes on signals and systems. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10565/1.7
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