# 2.5 Simple systems

 Page 1 / 1
Systems manipulate signals. There are a few simple systems which will perform simple functions upon signals. Examples include amplification (or attenuation),time-reversal, delay, and differentiation/integration.

Systems manipulate signals, creating output signals derived from their inputs. Why the following are categorized as "simple" willonly become evident towards the end of the course.

## Sources

Sources produce signals without having input. We like to think of these as having controllable parameters, like amplitude andfrequency. Examples would be oscillators that produce periodic signals like sinusoids and square waves and noise generatorsthat yield signals with erratic waveforms (more about noise subsequently). Simply writing an expression for the signalsthey produce specifies sources. A sine wave generator might be specified by $y(t)=A\sin (2\pi {f}_{0}t)u(t)$ , which says that the source was turned on at $t=0$ to produce a sinusoid of amplitude $A$ and frequency ${f}_{0}$ .

## Amplifiers

An amplifier multiplies its input by a constant known as the amplifier gain .

$y(t)=Gx(t)$

The gain can be positive or negative (if negative, we would say that the amplifier inverts its input) and its magnitude can be greater than one or less than one. If less than one, the amplifieractually attenuates . A real-world example of an amplifier is your home stereo. You control the gain by turningthe volume control.

## Delay

A system serves as a time delay when the output signal equals the input signal at an earlier time.

$y(t)=x(t-\tau )$

Here, $\tau$ is the delay. The way to understand this system is to focus on the time origin: The output at time $t=\tau$ equals the input at time $t=0$ . Thus, if the delay is positive, the output emerges later thanthe input, and plotting the output amounts to shifting the input plot to the right. The delay can be negative, in whichcase we say the system advances its input. Such systems are difficult to build (they would have toproduce signal values derived from what the input will be ), but we will have occasion to advance signals in time.

## Time reversal

Here, the output signal equals the input signal flipped aboutthe time origin.

$y(t)=x(-t)$

Again, such systems are difficult to build, but the notion of time reversal occurs frequently in communications systems.

Mentioned earlier was the issue of whether the ordering of systems mattered. In other words, if we have two systems in cascade, does theoutput depend on which comes first? Determine if the ordering matters for the cascade of an amplifier and a delay and for the cascade of atime-reversal system and a delay.

In the first case, order does not matter; in the second it does. "Delay" means $t\to t-\tau$ . "Time-reverse" means $t\to -t$

Case 1 $y(t)=Gx(t-\tau )$ , and the way we apply the gain and delay the signalgives the same result.

Case 2 Time-reverse then delay: $y(t)=x(-(t-\tau ))=x(-t+\tau )$ . Delay then time-reverse: $y(t)=x(-t-\tau )$ .

## Derivative systems and integrators

Systems that perform calculus-like operations on their inputs can produce waveforms significantly different than present inthe input. Derivative systems operate in a straightforward way: A first-derivative system would have the input-outputrelationship $y(t)=\frac{d x(t)}{d t}}$ . Integral systems have the complication that the integral'slimits must be defined. It is a signal theory convention that the elementary integral operation have a lower limit of $()$ , and that the value of all signals at $t=()$ equals zero. A simple integrator would have input-output relation

$y(t)=\int_{()} \,d \alpha$ t x α

## Linear systems

Linear systems are a class of systems rather than having a specific input-output relation. Linearsystems form the foundation of system theory, and are the most important class of systems in communications. They have theproperty that when the input is expressed as a weighted sum of component signals, the output equals the same weighted sum ofthe outputs produced by each component. When $S(·)$ is linear,

$S({G}_{1}{x}_{1}(t)+{G}_{2}{x}_{2}(t))={G}_{1}S({x}_{1}(t))+{G}_{2}S({x}_{2}(t))$
for all choices of signals and gains.

This general input-output relation property can be manipulated to indicate specific properties shared by all linear systems.

• $S(Gx(t))=GS(x(t))$ The colloquialism summarizing this property is "Double the input, you double the output." Note that this property isconsistent with alternate ways of expressing gain changes: Since $2x(t)$ also equals $x(t)+x(t)$ , the linear system definition provides the same output nomatter which of these is used to express a given signal.
• $S(0)=0$ If the input is identically zero for all time , the output of a linear system must be zero. This property follows from the simple derivation $S(0)=S(x(t)-x(t))=S(x(t))-S(x(t))=0$ .
Just why linear systems are so important is related not only to their properties, which are divulged throughout thiscourse, but also because they lend themselves to relatively simple mathematical analysis. Said another way, "They'rethe only systems we thoroughly understand!"

We can find the output of any linear system to a complicated input by decomposing the input into simple signals. The equation above says that when a system is linear, its output to a decomposedinput is the sum of outputs to each input. For example, if $x(t)=e^{-t}+\sin (2\pi {f}_{0}t)$ the output $S(x(t))$ of any linear system equals $y(t)=S(e^{-t})+S(\sin (2\pi {f}_{0}t))$

## Time-invariant systems

Systems that don't change their input-output relation with time are said to be time-invariant. The mathematical way ofstating this property is to use the signal delay concept described in Simple Systems .

$(y(t)=S(x(t)))\implies (y(t-\tau )=S(x(t-\tau )))$
If you delay (or advance) the input, the output is similarly delayed (advanced). Thus, a time-invariant system responds toan input you may supply tomorrow the same way it responds to the same input applied today; today's output is merely delayedto occur tomorrow.

The collection of linear, time-invariant systems are the most thoroughly understood systems. Much of the signal processing and system theorydiscussed here concentrates on such systems. For example, electric circuits are, for the most part, linear andtime-invariant. Nonlinear ones abound, but characterizing them so that you can predict their behavior for any input remainsan unsolved problem.

Linear, time-invariant table
Input-Output Relation Linear Time-Invariant
$y(t)=\frac{d x}{d t}}$ yes yes
$y(t)=\frac{d^{2}x}{dt^{2}}$ yes yes
$y(t)=\frac{d x}{d t}}^{2}$ no yes
$y(t)=\frac{d x}{d t}}+x$ yes yes
$y(t)={x}_{1}()+{x}_{2}()$ yes yes
$y(t)=x(t-\tau )$ yes yes
$y(t)=\cos (2\pi ft)x(t)$ yes no
$y(t)=x(-t)$ yes no
$y(t)={x}^{2}(t)$ no yes
$y(t)=\left|x(t)\right|$ no yes
$y(t)=mx(t)+b$ no yes

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
learn
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
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good