##### Document Text Contents

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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/

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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

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