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Signals can be represented by discrete quantities instead of as a function of a continuous variable. These discrete time signals do notnecessarily have to take real number values. Many properties of continuous valued signals transfer almost directly to the discretedomain.

So far, we have treated what are known as analog signals and systems. Mathematically, analog signals are functions having continuous quantities as theirindependent variables, such as space and time. Discrete-time signals are functions defined on the integers; they are sequences. One ofthe fundamental results of signal theory will detail conditions under which an analog signal can be converted into a discrete-time one andretrieved without error . This result is important because discrete-time signals can be manipulated bysystems instantiated as computer programs. Subsequent modules describe how virtually all analog signal processing can beperformed with software.

As important as such results are, discrete-time signals are more general, encompassing signals derived fromanalog ones and signals that aren't. For example, the characters forming a text file form a sequence,which is also a discrete-time signal. We must deal with such symbolic valued signals and systems as well.

As with analog signals, we seek ways of decomposing real-valueddiscrete-time signals into simpler components. With this approach leading to a better understanding of signal structure,we can exploit that structure to represent information (create ways of representing information with signals) and to extractinformation (retrieve the information thus represented). For symbolic-valued signals, the approach is different: We develop acommon representation of all symbolic-valued signals so that we can embody the information they contain in a unified way. Froman information representation perspective, the most important issue becomes, for both real-valued and symbolic-valued signals,efficiency; What is the most parsimonious and compact way to represent information so that it can be extracted later.

Real- and complex-valued signals

A discrete-time signal is represented symbolically as s n , where n -1 0 1 . We usually draw discrete-time signals as stem plots toemphasize the fact they are functions defined only on the integers. We can delay a discrete-time signal by an integerjust as with analog ones. A delayed unit sample has the expression δ n m , and equals one when n m .

Discrete-time cosine signal

The discrete-time cosine signal is plotted as a stem plot. Can you find the formula for this signal?

Complex exponentials

The most important signal is, of course, the complex exponential sequence .

s n 2 f n

Sinusoids

Discrete-time sinusoids have the obvious form s n A 2 f n φ . As opposed to analog complex exponentials and sinusoids thatcan have their frequencies be any real value, frequencies of their discrete-time counterparts yield unique waveforms only when f lies in the interval 1 2 1 2 . This property can be easily understood by noting that addingan integer to the frequency of the discrete-time complex exponential has no effect on the signal's value.

2 f m n 2 f n 2 m n 2 f n
This derivation follows because the complex exponential evaluated at an integer multiple of 2 equals one.

Unit sample

The second-most important discrete-time signal is the unit sample , which is defined to be

δ n 1 n 0 0

Unit sample

The unit sample.

Examination of a discrete-time signal's plot, like that of the cosine signal shown in [link] , reveals that all signals consist of a sequence of delayed andscaled unit samples. Because the value of a sequence at each integer m is denoted by s m and the unit sample delayed to occur at m is written δ n m , we can decompose any signal as a sum of unit samples delayed to the appropriate location and scaled bythe signal value.

s n m s m δ n m
This kind of decomposition is unique to discrete-time signals, and will prove useful subsequently.

Discrete-time systems can act on discrete-time signals in wayssimilar to those found in analog signals and systems. Because of the role of software in discrete-time systems, many moredifferent systems can be envisioned and “constructed” with programs than can be with analog signals. In fact, a specialclass of analog signals can be converted into discrete-time signals, processed with software, and converted back into ananalog signal, all without the incursion of error. For such signals, systems can be easily produced in software, withequivalent analog realizations difficult, if not impossible, to design.

Symbolic-valued signals

Another interesting aspect of discrete-time signals is thattheir values do not need to be real numbers. We do have real-valued discrete-time signals like the sinusoid, but wealso have signals that denote the sequence of characters typed on the keyboard. Such characters certainly aren't realnumbers, and as a collection of possible signal values, they have little mathematical structure other than that they aremembers of a set. More formally, each element of the symbolic-valued signal s n takes on one of the values a 1 a K which comprise the alphabet A . This technical terminology does not mean we restrict symbols to being members of the Englishor Greek alphabet. They could represent keyboard characters, bytes (8-bit quantities), integers that convey dailytemperature. Whether controlled by software or not, discrete-time systems are ultimately constructed from digitalcircuits, which consist entirely of analog circuit elements. Furthermore, the transmission andreception of discrete-time signals, like e-mail, is accomplished with analog signals and systems. Understandinghow discrete-time and analog signals and systems intertwine is perhaps the main goal of this course.

Questions & Answers

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Damian Reply
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Daniel
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it is a goid question and i want to know the answer as well
Maciej
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Abigail
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 ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
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what is system testing?
AMJAD
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
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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Source:  OpenStax, Fundamentals of electrical engineering i. OpenStax CNX. Aug 06, 2008 Download for free at http://legacy.cnx.org/content/col10040/1.9
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