<< Chapter < Page Chapter >> Page >

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

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
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 ?
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
can nanotechnology change the direction of the face of the world
Prasenjit Reply
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
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, Brief notes on signals and systems. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10565/1.7
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Brief notes on signals and systems' conversation and receive update notifications?