# System classifications and properties

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Describes various classifications of systems.

## Introduction

In this module some of the basic classifications of systems will be briefly introduced and the most important propertiesof these systems are explained. As can be seen, the properties of a system provide an easy way to distinguish onesystem from another. Understanding these basic differences between systems, and their properties, will be a fundamentalconcept used in all signal and system courses. Once a set of systems can be identified as sharing particular properties, one no longer hasto reprove a certain characteristic of a system each time, but it can simply be known due to the the systemclassification.

## Continuous vs. discrete

One of the most important distinctions to understand is the difference between discrete time and continuous time systems. A system in which the input signal and output signal both have continuous domains is said to be a continuous system. One in which the input signal and output signal both have discrete domains is said to be a discrete system. Of course, it is possible to conceive of signals that belong to neither category, such as systems in which sampling of a continuous time signal or reconstruction from a discrete time signal take place.

## Linear vs. nonlinear

A linear system is any system that obeys the properties of scaling (first order homogeneity) and superposition (additivity) further described below. A nonlinear system is any system that does not have at least one of these properties.

To show that a system $H$ obeys the scaling property is to show that

$H(kf(t))=kH(f(t))$

To demonstrate that a system $H$ obeys thesuperposition property of linearity is to show that

$H({f}_{1}(t)+{f}_{2}(t))=H({f}_{1}(t))+H({f}_{2}(t))$

It is possible to check a system for linearity in a single (though larger) step. To do this, simply combine the firsttwo steps to get

$H({k}_{1}(){f}_{1}(t)+{k}_{2}(){f}_{2}(t))={k}_{1}()H({f}_{1}(t))+{k}_{2}()H({f}_{2}(t))$

## Time invariant vs. time varying

A system is said to be time invariant if it commutes with the parameter shift operator defined by ${S}_{T}\left(f\left(t\right)\right)=f\left(t-T\right)$ for all $T$ , which is to say

$H{S}_{T}={S}_{T}H$

for all real $T$ . Intuitively, that means that for any input function that produces some output function, any time shift of that input function will produce an output function identical in every way except that it is shifted by the same amount. Any system that does not have this property is said to be time varying. This block diagram shows what the condition for time invariance. The output is the same whether the delay is puton the input or the output.

## Causal vs. noncausal

A causal system is one in which the output depends only on current or past inputs, but not future inputs. Similarly, an anticausal system is one in which the output depends only on current or future inputs, but not past inputs. Finally, a noncausal system is one in which the output depends on both past and future inputs. All "realtime" systems must be causal, since they can not have future inputs available to them.

One may think the idea of future inputs does not seem to make much physical sense; however, we have only beendealing with time as our dependent variable so far, which is not always the case. Imagine rather that we wanted to doimage processing. Then the dependent variable might represent pixel positions to the left and right (the "future") of the currentposition on the image, and we would not necessarily have a causal system.

## Stable vs. unstable

There are several definitions of stability, but the one that will be used most frequently in this course will be bounded input, bounded output (BIBO) stability. In this context, a stable system is one in which the output is bounded if the input is also bounded. Similarly, an unstable system is one in which at least one bounded input produces an unbounded output.

Representing this mathematically, a stable system must have the following property, where $x(t)$ is the input and $y(t)$ is the output. The output must satisfy the condition

$\left|y(t)\right|\le {M}_{y}()$
whenever we have an input to the system that satisfies
$\left|x(t)\right|\le {M}_{x}()$
${M}_{x}$ and ${M}_{y}$ both represent a set of finite positive numbers and these relationships hold for all of $t$ . Otherwise, the system is unstable.

## System classifications summary

This module describes just some of the many ways in which systems can be classified. Systems can be continuous time, discrete time, or neither. They can be linear or nonlinear, time invariant or time varying, and stable or unstable. We can also divide them based on their causality properties. There are other ways to classify systems, such as use of memory, that are not discussed here but will be described in subsequent modules.

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what are the products of Nano chemistry?
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
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
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I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
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Rafiq
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Damian
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What is STMs full form?
LITNING
scanning tunneling microscope
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Rafiq
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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
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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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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