# 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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Digital signal processing and digital filter design (draft). OpenStax CNX. Nov 17, 2012 Download for free at http://cnx.org/content/col10598/1.6
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 By Marriyam Rana By OpenStax By Janet Forrester By By Rhodes By OpenStax By P. Wynn Norman By OpenStax By OpenStax By Robert Morris