# 1.2 Discrete-time systems

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In the context of discussing signal processing, the most general definition of a system is similar to that of a function. A system is adevice, formula, rule, or some process that assigns an output signal from some given class to each possible input signal chosen from some allowedclass. From this definition one can pose three interesting and practical problems.

1. Analysis : If the input signal and the system are given,find the output signal.
2. Control : If the system and the output signal are given,find the input signal.
3. Synthesis : If the input signal and output signal are given,find the system.

The definition of input and output signal can be quite diverse. They could be scalars, vectors, functions, functionals, or other objects.

All three of these problems are important, but analysis is probably the most basic and its study usually precedes that of the other two. Analysisusually results in a unique solution. Control is often unique but there are some problems where several inputs would give the same output.Synthesis is seldom unique. There are usually many possible systems that will give the same output for a given input.

In order to develop tools for analysis, control, and design of discrete-time systems, specific definitions, restrictions, andclassifications must be made. It is the explicit statement of what a system is, not what it isn't, that allows a descriptive theory and designmethods to be developed.

## Classifications

The basic classifications of signal processing systems are defined and listed here. We will restrict ourselves to discrete-time systems thathave ordered sequences of real or complex numbers as inputs and outputs and will denote the input sequence by $x\left(n\right)$ and the output sequence by $y\left(n\right)$ and show the process of the system by $x\left(n\right)\to y\left(n\right)$ . Although the independent variable $n$ could represent any physical variable, our most common usages causes us to generically call it time butthe results obtained certainly are not restricted to this interpretation.

1. Linear, A system is classified as linear if two conditions are true.
• If $x\left(n\right)\to y\left(n\right)$ then $a\phantom{\rule{0.166667em}{0ex}}x\left(n\right)\to a\phantom{\rule{0.166667em}{0ex}}y\left(n\right)$ for all $a$ . This property is called homogeneity or scaling.
• If ${x}_{1}\left(n\right)\to {y}_{1}\left(n\right)$ and ${x}_{2}\left(n\right)\to {y}_{2}\left(n\right)$ , then $\left({x}_{1}\left(n\right)+{x}_{2}\left(n\right)\right)\to \left({y}_{1}\left(n\right)+{y}_{2}\left(n\right)\right)$ for all ${x}_{1}$ and ${x}_{2}$ . This property is called superposition or additivity.
If a system does not satisfy both of these conditions for all inputs, it is classified as nonlinear. For most practical systems, one of theseconditions implies the other. Note that a linear system must give a zero output for a zero input.
2. Time Invariant , also called index invariant or shift invariant. A system is classified as time invariant if $x\left(n+k\right)\to y\left(n+k\right)$ for any integer $k$ . This states that the system responds the same way regardless of when the input is applied. In most cases, the systemitself is not a function of time.
3. Stable . A system is called bounded-input bounded-output stable if for all bounded inputs, the corresponding outputs are bounded. This means that the output must remain bounded even for inputs artificiallyconstructed to maximize a particular system's output.
4. Causal . A system is classified as causal if the output of a system does not precede the input. For linear systems this means that theimpulse response of a system is zero for time before the input. This concept implies the interpretation of $n$ as time even though it may not be. A system is semi-causal if after a finite shift in time, the impulseresponse is zero for negative time. If the impulse response is nonzero for $n\to -\infty$ , the system is absolutely non-causal. Delays are simple to realize in discrete-time systems and semi-causal systems canoften be made realizable if a time delay can be tolerated.
5. Real-Time . A discrete-time system can operate in “real-time" if an output value in the output sequence can be calculated by the systembefore the next input arrives. If this is not possible, the input and output must be stored in blocks and the system operates in “batch" mode.In batch mode, each output value can depend on all of the input values and the concept of causality does not apply.

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
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
characteristics of micro business
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
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Cesar
I'm interested in nanotube
Uday
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Berger describes sociologists as concerned with
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