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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?

Course overview

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.


As an example, consider the following communications channel:

Communications channel
Block diagram for a 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 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 = x ^ - x while meeting maximum transmission power constraints power ( F ( x ) ) power ( x ) < P max .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Signal theory. OpenStax CNX. Oct 18, 2013 Download for free at http://legacy.cnx.org/content/col11542/1.3
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