# Introduction and motivation

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Introduction and motivation for a graduate electrical engineering course on Signal Theory.

## Introduction and motivation

Most areas of electrical and computer engineering (beyond signal processing) deal with signals. Communications is about transmitting, receiving, and interpreting signals. Signals are used to probe and model systems in control and circuit design. The images acquired by radar systems and biomedical devices are signals that change in space and time, respectively. Signals are used in microelectronic devices to convey digital information or send instructions to processors.

This course will provide a mathematical framework to handle signals and operations on signals. Some of the questions that will be answered in this course include:

• What is a signal? How do we represent it?
• How do we represent operations on signals?
• What does it mean for signals to be similar/different from each other?
• When is a candidate signal a good/bad approximation (i.e., a simplified version) of a target signal?
• When is a signal “interesting” or “boring”?
• How can we characterize groups of signals?
• How do we find the best approximation of a target signal in a group of candidates?

## Signal theory

The signal theory presented in this course has three main components:

• Signal representations and signal spaces , which provide a framework to talk about sets of signal and to define signal approximations.
• Distances and norms to evaluate and compare signals. Norms provide a measure of strength, amplitude, or “interestingness” of a signal, and distances provide a measure of similarity between signals.
• Projection theory and signal estimation to work with signals that have been distorted, aiming to recover the best approximation in a defined set.

## Operator theory

Operators are mathematical representations of systems that manipulate a signal. The operator theory presented in this course has three main components:

• Operator properties that allow us to characterize their effect on signals in a simple fashion.
• Operator characterization that allow us to model their effect on arbitrary inputs.
• Operator operations (no pun intended) that allow us to create new systems and reverse the effect of a system on a signal.

## Optimization theory

Optimization is an area of applied mathematics that, in the context of our course, will allow us to determine the best signal output for a given problem using defined metrics, such as signal denoising or compression, codebook design, and radar pulse shaping. The optimization theory presented in this course has three main components:

• Optimization guarantees that rely on properties of the metrics and signal sets we search over to formally ensure that the optimal signal can be found.
• Unconstrained optimization , where we search for the optimum over an entire signal space.
• Constrained optimization , where the optimal signal must meet additional specific requirements.

## Example

As an example, consider the following communications channel:

A mathematical formulation of this channel requires us to:

• establish which signals $x$ can be input into the transmitter;
• how the transmitter $F$ , the channel $H$ , and the receiver $G$ are characterized;
• how the concatenation of the blocks $F$ and $H$ is expressed;
• how the noise addition operation is formulated;
• how we measure whether the decoded message $\stackrel{^}{x}$ is a good approximation of the input $x$ ;
• how is the receiver $G$ designed to be optimal for all the choices above.

For this example, by the end of the course, you will be able to solve the problem of selecting the transmitter/receiver pair $F,G$ that minimizes the power of the error $e=\stackrel{^}{x}-x$ while meeting maximum transmission power constraints $\frac{\mathrm{power}\left(F\left(x\right)\right)}{\mathrm{power}\left(x\right)}<{P}_{max}$ .

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
Do somebody tell me a best nano engineering book for beginners?
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
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
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
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