Download The Laplace Transform PDF

TitleThe Laplace Transform
LanguageEnglish
File Size937.7 KB
Total Pages84
Table of Contents
                            Introduction
Introduction
Introduction
Introduction
Introduction
Introduction: Complex Exponential e-st
Introduction: the complex s-plane
Outline
Laplace Transform
Laplace Transform
Laplace Transform
Laplace Transform
Laplace Transform:Order of X(s)
Laplace Transform:Poles
Laplace Transform: Zeros
Laplace Transform:Visualization
Laplace Transform: Example
Laplace Transform: Example
Laplace Transform: Example
Laplace Transform: Example
Laplace Transform: Example
Laplace Transform: Example
Laplace Transform: Example
Outline
Region of Convergence for Laplace Transform
Region of Convergence for Laplace Transform
Region of Convergence for Laplace Transform
Region of Convergence for Laplace Transform: Example
Common Laplace Transforms Pairs
Common Laplace Transforms Pairs
Outline
Review:Partial Fraction Expansion
Review:Partial Fraction Expansion
Review:Partial Fraction Expansion: Different terms of 1st degree
Review: Partial Fraction Expansion: Repeated terms of 1st degree
Review: Partial Fraction Expansion: Different quadratic terms
Review: Partial Fraction Expansion: Repeated quadratic terms
Outline
The Inverse Laplace Transform
The Inverse Laplace Transform: Solving Using Tables
The Inverse Laplace Transform: Example
The Inverse Laplace Transform: Example
Outline
Properties of the Laplace Transform
Properties of the Laplace Transform
Properties of the Laplace Transform
Properties of the Laplace Transform
Properties of the Laplace Transform: Examples
Properties of the Laplace Transform
Properties of the Laplace Transform
Properties of the Laplace Transform:Examples
Properties of the Laplace Transform: Examples
Properties of the Laplace Transform: Examples
Properties of the Laplace Transform: summary
Properties of the Laplace Transform: Summary
Outline
Analysis and Characterization of LTI Systems Using the Laplace Transform
Transfer Function of an LTI system
Transfer Function of an LTI system
Analysis and Characterization of LTI Systems Using the Laplace Transform
Analysis and Characterization of LTI Systems Using the Laplace Transform: Example
Analysis and Characterization of LTI Systems Using the Laplace Transform: Example
Analysis and Characterization of LTI Systems Using the Laplace Transform:Example
Analysis and Characterization of LTI Systems Using the Laplace Transform
Analysis and Characterization of LTI Systems Using the Laplace Transform:Example
Unstable Behavior
Outline
LTI Systems Characterized by Linear Constant-Coefficient DE
LTI Systems Characterized by Linear Constant-Coefficient DE
LTI Systems Characterized by Linear Constant-Coefficient DE
LTI Systems Characterized by Linear Constant-Coefficient DE: Example
LTI Systems Characterized by Linear Constant-Coefficient DE: Example
LTI Systems Characterized by Linear Constant-Coefficient DE: Example
LTI Systems Characterized by Linear Constant-Coefficient DE: Solution Procedure
LTI Systems Characterized by Linear Constant-Coefficient DE: Solution Procedure
LTI Systems Characterized by Linear Constant-Coefficient DE: Solution Procedure
LTI Systems Characterized by Linear Constant-Coefficient DE: Solution Procedure
LTI Systems Characterized by Linear Constant-Coefficient DE: Solution Procedure
Outline
Summary
Summary: a quiz
Summary: quiz solution
Summary: quiz solution
                        
Document Text Contents
Page 1

1

ELEC361: Signals And Systems

Topic 9:
The Laplace Transform

o Introduction
o Laplace Transform & Examples
o Region of Convergence of the Laplace Transform
o Review: Partial Fraction Expansion
o Inverse Laplace Transform & Examples
o Properties of the Laplace Transform & Examples
o Analysis and Characterization of LTI Systems Using the

Laplace Transform
o LTI Systems Characterized by Linear Constant-Coefficient DE
o SummaryDr. Aishy Amer

Concordia University
Electrical and Computer Engineering

Figures and examp les in these course slides are taken from the follo wing sources:

•A. Oppenhei m, A.S. Will sky and S.H. Nawab, Signals and Sy stem s, 2nd Edi tion, Prenti ce-Hall, 1997

•M.J. Rober ts, Sig nals and Systems, McGraw Hill, 2004

•J. McClellan, R . Schafer, M. Yoder, Signal Proces sing First , Prent ice Hall, 2003

•Web Site of Dr. Wm. Hug h Blanton, http://fa culty .etsu. edu/b lanton/

Page 2

2

Introduction
Transforms: Mathematical conversion from
one way of thinking to another to make a
problem easier to solve

Reduces complexity of the original problem

Laplace
transform

solution
in

s domain

inverse
Laplace

transform

solution
in time
domain

problem
in time
domain

• Other transforms
• Fourier Transform
• z-transform

s = +j

Page 42

42

The Inverse Laplace
Transform: Example

Page 43

43

The Inverse Laplace
Transform: Example

Page 83

83

Summary: quiz solution

A function of the form
can be seen as the combination of the two
exponentials
Since we know that
and
Therefore, we know that there will be a pole at
s = –β and a pole at s = 3
Knowing that the ROC is the intersection of the
individual ROC’s

)(
5
3

)(
4
1

)( 3 tuetueth tt

}Re{,)()()( s
s

A
sHtuAeth t

3}Re{,
3

)()( 3 s
s

B
tuBeth t

Page 84

84

Summary: quiz solution

a) > 0 (not including 0), such that – is in the left half of the s-
plane. This way, the ROC (common intersection) will include
the j -axis, thus implying that the Fourier transform exists
further implying that the system is stable

b) For the system to be causal, the ROC must extend outward to
positive infinity. Furthermore, since an ROC cannot contain a
pole, Re{s} > 3, for the system to be causal

c) This system can only be causal or stable but not both. This is
because, in order to be causal and stable, all poles must lie in
the left half of the s-plane such that the ROC can possibly
extend from the rightmost pole to + infinity and include the j -
axis. However, due to the pole at s = 3. The ROC cannot
extend toward infinity and include the j -axis. Therefore, the
system cannot be both stable and causal

Similer Documents