EE 64EE 64Linear System TheoryLinear System Theory

M. R. Gustafson II

Adjunct Assistant Professor

Duke University

Introduction (Education)Introduction (Education)

BSE in Electrical EngineeringBSE, MS, and PhD in Mechanical

Engineering and Materials ScienceStarting my twelfth year at Duke

Introduction (Military)Introduction (Military)

Duke NROTC 1989-1993Lieutenant, U.S. Naval Reserve

Engineering Duty OfficerNaval Research Laboratories

Science and Technology Unit, Raleigh, NC

Class ObjectivesClass Objectives

To learn the fundamental engineering mathematics of signal representations, linear system responses, convolution, and correlation,

To understand Fourier series, Fourier transforms, transfer functions, Laplace transforms, state variables, transfer functions, and stability,

To see discrete-time signals, z transforms, discrete-time Fourier transforms, and the fast Fourier transform, and

To meet other people in engineering.

Introductions & Roll CallIntroductions & Roll Call

Resources (Books)Resources (Books)

Signals & Systems, Alan V. Oppenheimer and Alan S. Willsky

Linear System Theory Lecture Notes, Dean McCumber

Resources (Web)Resources (Web)

OIT Guide– http://www.oit.duke.edu– http://www.oit.duke.edu/unix-manual

Class Web page– http://kepler.egr.duke.edu/EE64F00

• Syllabus, grading, assignment information, policies

Resources (Newsgroup)Resources (Newsgroup)

duke.courses.ee64The newsgroup will be used to post

announcements and answer questions.Use this to post items that are of interest to

the rest of the class.Students are allowed to answer questions as

long as the answers are correct and do not violate the honor code!

Resources (Public Clusters)Resources (Public Clusters)

MAPLE, MATLAB, and SIMULINK will run on all acpub machines. They will also run over xwin32 and eXodus.

Public UNIX machines are in Teer (new!), Hudson Hall, Soc-Psych, Bio-Sci, Carr, West Duke, and Trent.

Check the OIT schedule to make sure there is no lab before entering - respect other people's lab times.

Assignments and GradingAssignments and Grading

Breakdown:– (15%) Homework– (5%) Correlation Project– (10%) Stabilization Project– (15%) Radio Project– (15%) Test I– (15%) Test II– (25%) Final Exam

HomeworkHomework

Homework will be assigned each week and turned in the following week. Homework will consist of problems from the texts as well as some problems written up by the instructor.

ProjectsProjects

Correlation Project– Detect the presence of a sequence in a

noisy signal using correlationRadio Project

– Build a working AM/FM radio and understand its major components

Analysis and Stabilization Project– Model a dynamic system and stabilize it

analytically

TestsTests

There will be three tests in this class -- two during the semester and one final exam.

Course Web PageCourse Web Page

kepler.egr.duke.edu/EE64F00 Netscape on acpub

– Web crawlers• Yahoo• Hotbot• Google

– Unregulated information! Even less trustworthy than regular press :)

demonstration

Course NewsgroupCourse Newsgroup

duke.classes.ee64tin program

– Finding groups– Posting messages– Saving messages– Mailing messages

demonstration

SignalsSignals

What is a signal?What is the difference between a

continuous and a discrete signal?– What is "Xeno's Paradox?"

Signal PowerSignal Power

Signal power is calculated assuming that the signal is a voltage on a 1 resistor. Assuming you have a signal x(t), the power is:

22 |)(|)(1

)( txtvR

tp

Average Signal PowerAverage Signal Power

Given that definition, the average power of a signal x(t) between times t1 and t2 is:

The average power over all time is:

dttxtt

Pt

tttav

2

121

2

12, |)(|

1

dttxT

PT

TT 2|)(|2

1lim

Signal EnergySignal Energy

Signal energy is found by recalling that power is the rate of change of energy. Energy, therefore, is the integral of power, so:

The total signal energy is:

dttxEt

ttt 2

121

2|)(|

dttxET

TT 2|)(|lim

Power / Energy SignalsPower / Energy Signals

A power signal is a signal that has infinite total energy

An energy signal is a signal that has finite total energy and thus zero average power

T

EP

Tav 2

lim,

Signal TransformationsSignal Transformations

Given a signal x(t), a signal y(t) can be written based on x(t) using scaling and shifting

See scale_shift.mws for examples (all programs from class are in ~mrg/public/EE64F00)

0)()( ttxtxty

Signal PropertiesSignal Properties

A signal x(t) is periodic with period T if it has the property that there is some positive T for which x(t)=x(t+T)

A signal x(t) is even if x(t)=x(-t)A signal x(t) is odd if x(t)=-x(-t)

Even / Odd PartsEven / Odd Parts

The even part of a signal is given by

The odd part of a signal is given by

)()(2

1)()}({ txtxtxtx e

)()(2

1)()}({ txtxtxtx o

Next TimeNext Time

Complex exponentialsUnit impulse and step functionsSystems and system properties

Assignment for WednesdayAssignment for Wednesday

Check out the class web pageCheck out the class newsgroup

Read Chapter 1 in Oppenheim & Willsky

Class Feedback SystemClass Feedback System

Four class members to present informal “status report” on how class and lab are going

Volunteers?

Questions??Questions??

?