5.4 Discrete-time signals and systems

 Page 1 / 1
(Blank Abstract)

Mathematically, analog signals are functions having as their independent variables continuous quantities, such as space andtime. Discrete-time signals are functions defined on the integers; they are sequences. As with analog signals, we seekways of decomposing discrete-time signals into simpler components. Because this approach leads to a betterunderstanding of signal structure, we can exploit that structure to represent information (create ways of representinginformation with signals) and to extract information (retrieve the information thus represented). For symbolic-valued signals,the approach is different: We develop a common representation of all symbolic-valued signals so that we can embody theinformation they contain in a unified way. From an 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 representinformation so that it can be extracted later.

Real- and complex-valued signals

A discrete-time signal is represented symbolically as $s(n)$ , where $n=\{\dots , -1, 0, 1, \dots \}$ . The discrete-time cosine signal is plotted as a stem plot. Can you find the formula for this signal? We usually draw discrete-time signals as stem plots to emphasize the fact they are functions defined only on theintegers. We can delay a discrete-time signal by an integer just as with analog ones. A signal delayed by $m$ samples has the expression $s(n-m)$ .

Complex exponentials

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

$s(n)=e^{i\times 2\pi fn}$
Note that the frequency variable $f$ is dimensionless and that adding an integer to the frequency of the discrete-time complex exponential has no effect on the signal's value.
$e^{i\times 2\pi (f+m)n}=e^{i\times 2\pi fn}e^{i\times 2\pi mn}=e^{i\times 2\pi fn}$
This derivation follows because the complex exponential evaluated at an integer multiple of $2\pi$ equals one. Thus, we need only consider frequency to have a value in some unit-length interval.

Sinusoids

Discrete-time sinusoids have the obvious form $s(n)=A\cos (2\pi fn+\phi )$ . As opposed to analog complex exponentials and sinusoids that can have their frequencies be any real value,frequencies of their discrete-time counterparts yield unique waveforms only when $f$ lies in the interval $\left(-\left(\frac{1}{2}\right) , \frac{1}{2}\right]$ . This choice of frequency interval is arbitrary; we can also choose the frequency to lie in the interval $\left[0 , 1\right)$ . How to choose a unit-length interval for a sinusoid's frequency will become evident later.

Unit sample

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

$\delta (n)=\begin{cases}1 & \text{if n=0}\\ 0 & \text{otherwise}\end{cases}$ 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 $\delta (n-m)$ , we can decompose any signal as a sum of unit samples delayed to the appropriate location andscaled by the signal value.
$s(n)=\sum_{m=()}$ s m δ n m
This kind of decomposition is unique to discrete-time signals, and will prove useful subsequently.

Unit step

The unit step in discrete-time is well-defined at the origin, as opposed to the situation with analog signals.

$u(n)=\begin{cases}1 & \text{if n\ge 0}\\ 0 & \text{if n< 0}\end{cases}$

Symbolic signals

An interesting aspect of discrete-time signals is that their values do not need to be real numbers. We do have real-valueddiscrete-time signals like the sinusoid, but we also have signals that denote the sequence of characters typed on thekeyboard. Such characters certainly aren't real numbers, and as a collection of possible signal values, they have littlemathematical structure other than that they are members of a set. More formally, each element of the symbolic-valued signal $s(n)$ takes on one of the values $\{{a}_{1}, \dots , {a}_{K}\}$ 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.

Discrete-time systems

Discrete-time systems can act on discrete-time signals in ways similar to those found in analog signals and systems. Becauseof the role of software in discrete-time systems, many more different systems can be envisioned and "constructed" withprograms than can be with analog signals. In fact, a special class of analog signals can be converted into discrete-timesignals, processed with software, and converted back into an analog signal, all without the incursion of error. For suchsignals, systems can be easily produced in software, with equivalent analog realizations difficult, if not impossible,to design.

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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good By By  By By Edward Biton By    By By Rhodes 