Signals and Systems: Chapter 5 Laplace Transform

– Frequency domain analysis with Fourier transform is extremely useful for the studies of signals and LTI system.. – Problem: many signals do not have Fourier transform 0 , exp – Lapl

ELEG 3124 SYSTEMS AND SIGNALS

Ch 5 Laplace Transform

Dr Jingxian Wu

wuj@uark.edu

(These slides are taken from Dr Jingxian Wu, University of Arkansas, 2020.)

EE 2000 SIGNALS AND SYSTEMS

• Introduction

• Laplace Transform

• Properties of Laplace Transform

• Inverse Laplace Transform

• Applications of Laplace Transform

• Why Laplace transform?

– Frequency domain analysis with Fourier transform is extremely

useful for the studies of signals and LTI system

• Convolution in time domain ➔ Multiplication in frequency domain

– Problem: many signals do not have Fourier transform

0 ),

( ) exp(

)

– Laplace transform can solve this problem

• It exists for most common signals

• Follow similar property to Fourier transform

• It doesn’t have any physical meaning; just a mathematical tool

to facilitate analysis

– Fourier transform gives us the frequency domain representation of signal

• Introduction

• Laplace Transform

• Properties of Laplace Transform

• Inverse Lapalace Transform

• Applications of Fourier Transform

LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM

• Bilateral Laplace transform (two-sided Laplace transform)

, ) exp(

) ( )

– : a function of time t → x(t) is called the time domain signal

– a function of s → is called the s-domain signal

– S-domain is also called as the complex frequency domain



s = +

) ( )

( s L x t

)

(t x

: )

(s

• Time domain v.s S-domain

LAPLACE TRANSFORM

• Time domain v.s s-domain

– : a function of time t → x(t) is called the time domain signal

– a function of s → is called the s-domain signal

• S-domain is also called the complex frequency domain

– By converting the time domain signal into the s-domain, we can

usually greatly simplify the analysis of the LTI system

– S-domain system analysis:

• 1 Convert the time domain signals to the s-domain with the Laplace transform

• 2 Perform system analysis in the s-domain

• 3 Convert the s-domain results back to the time-domain

6

)

(t x

: )

(s

• Example

– Find the Bilateral Laplace transform of x ( t ) = exp( − at ) u ( t )

• Region of Convergence (ROC)

– The range of s that the Laplace transform of a signal converges.

– The Laplace transform always contains two components

• The mathematical expression of Laplace transform

• ROC

LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM

• Example

– Find the Laplace transform of x ( t ) = − exp( − at ) u ( − t )

LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM

• Example

– Find the Laplace transform of x ( t ) = 3 exp( − 2 t ) u ( t ) + 4 exp( t ) u ( − t )

LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM

LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM

• Unilateral Laplace transform (one-sided Laplace transform)

=

0 ( ) exp( ) )

X

– :The value of x(t) at t = 0 is considered.

– Useful when we dealing with causal signals or causal systems

X

• Example: find the unilateral Laplace transform of the

LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM

• Introduction

• Laplace Transform

• Properties of Laplace Transform

• Inverse Lapalace Transform

• Applications of Fourier Transform

1 t X s

) ( )

( )

( )

16PROPERTIES: TIME SHIFTING

) exp(

) ( )

( )

PROPERTIES: SHIFTING IN THE s DOMAIN

• Shifting in the s domain

Re(s

) ( )

) Re(

) Re(s  + s0

18PROPERTIES: TIME SCALING

• Time scaling

– If

– Then

) ( )

1

} Re{ s  a 

at

1

} Re{ s  

• Example

– Find the Laplace transform of x ( t ) = u ( at )

PROPERTIES: DIFFERENTIATION IN TIME DOMAIN

• Differentiation in time domain

– If

– Then

) ( )

) 0 ( )

( )

• Example

– Find the Laplace transform of g ( t ) = sin2 t  u ( t ),

) 0 ( )

0 ( )

0 ( )

(

)

(  n − n−1 − − − (n−2) − − (n−1) −n

n

g sg

g s s

G

s dt

t g d



0 ) 0 ( − =

g

) 0 ( ' )

0 ( )

( )

y '( 0−) = 1

y

PROPERTIES: DIFFERENTIATION IN S DOMAIN

• Differentiation in s domain

– If

– Then

) ( )

n

n n

ds

s X d t

22PROPERTIES: CONVOLUTION

) ( ) ( )

( )

) ( ) ( s H s X

PROPERTIES: INTEGRATION IN TIME DOMAIN

• Integration in time domain

– If

– Then

) ( )

) (

1 )

(

s

d x

24PROPERTIES: CONVOLUTION

a t rect

2 2

PROPERTIES: CONVOLUTION

• Example

– For a LTI system, the input is , and the

output of the system is

) ( ) 2 exp(

26PROPERTIES: CONVOLUTION

• Example

– Find the Laplace transform of the impulse response of the LTI

system described by the following differential equation

) ( )

( ' 3 ) ( )

( ' 3 ) ( '

assume the system was initially relaxed ( )(n)( 0 ) = (n)( 0 ) = 0

x y

)(t 0t X s j0 X s j0

 ( ) ( )

2

)sin(

)(t 0t j X s j0 X s j0

28PROPERTIES: MODULATION

• Example

– Find the Laplace transform of x ( t ) = exp( − at ) sin( 0t ) u ( t )

PROPERTIES: INITIAL VALUE THEOREM

• Initial value theorem

– If the signal is infinitely differentiable on an interval around

) 0

x

s→  + =

– The behavior of x(t) for small t is determined by the behavior of

X(s) for large s



=

30PROPERTIES: INITIAL VALUE THEOREM

• Example

– The Laplace transform of x(t) is

Find the value of ( ) (s a)(s b)

d cs s

x

PROPERTIES: FINAL VALUE THEOREM

• Final value theorem

– If

– Then:

) ( )

) ( lim

) (

lim

0 sX s t

x

s

t→   →

• Example

– The input is applied to a system with transfer

function , find the value of

0

=

) ( )

c b

s s

c s

H

+ +

=

) (

)

32

• Introduction

• Laplace Transform

• Properties of Laplace Transform

• Inverse Lapalace Transform

• Applications of Fourier Transform

34INVERSE LAPLACE TRANSFORM

• Inverse Laplace transform

0 1

1 1

0 1

1 1

)

(

a s a s

a s

a

b s b s

b s

b s

n

n n

m m m

m

+ +

+ +

+ +

+ +

– Evaluation of the above integral requires the use of contour

integration in the complex plan ➔ difficult

• Inverse Laplace transform: special case

– In many cases, the Laplace transform can be expressed as a

rational function of s

– Procedure of Inverse Laplace Transform

• 1 Partial fraction expansion of X(s)

• 2 Find the inverse Laplace transform through Laplace transform table

 −+

j X s st ds j

(

INVERSE LAPLACE TRANSFORM

• Review: Partial Fraction Expansion with non-repeated

linear factors

3 2

1

)

(

a s

C a

s

B a

s

A s

2

) ( ) ( s a2 X s s a

3

) ( ) ( s a3 X s s a

• Example

– Find the inverse Laplace transform of

s s

s

s s

X

4 3

1 2

)

− +

+

=

36INVERSE LAPLACE TRANSFORM

• Example

– Find the Inverse Laplace transform of

2 3

2 )

2

+ +

=

s s

s s

X

• If the numerator polynomial has order higher than or equal to the order

of denominator polynomial, we need to rearrange it such that the

denominator polynomial has a higher order

INVERSE LAPLACE TRANSFORM

• Partial Fraction Expansion with repeated linear factors

B a

s

A a

s

A b

s a

s

s X

) (

) (

1 )

(

( )

a ss X a s

s X a

s ds

d A

=

−

1 B = ( s − b ) X ( s ) s=b

38INVERSE LAPLACE TRANSFORM

• High-order repeated linear factors

b s

B a

s

A a

s

A a

s

A b

s a

) (

) (

) (

1 )

N k

N

k N

ds

d k

• Introduction

• Laplace Transform

• Properties of Laplace Transform

• Inverse Lapalace Transform

• Applications of Laplace Transform

APPLICATION: LTI SYSTEM REPRESENTATION

• LTI system

– System equation: a differential equation describes the input output

relationship of the system

)()

()

()

()

()

()

)

(

t x b t

x b t

x b t

y a t

y a t

y a t

n

n n

N

t x b t

y a t

y

0

) ( 1

0

) ( )

(

)()

()

(

– S-domain representation

)()

b s

Y s a s

M

m

m m N

n

n n

)()

n

n n N

M

m

m m

s a s

s b s

X

s Y s

H

APPLICATION: LTI SYSTEM REPRESENTATION

• Simulation diagram (first canonical form)

Simulation diagram

23

)

2

++

s

s s

S H

APPLICATION: COMBINATIONS OF SYSTEMS

• Combination of systems

– Cascade of systems

– Parallel systems

)()()

(S H1 s H2 s

)()

()

(S H1 s H2 s

23

)

2

++

s

s s

S H

APPLICATION: LTI SYSTEM REPRESENTATION

• Example:

– Find the transfer function of the system

LTI system

APPLICATION: LTI SYSTEM REPRESENTATION

• Poles and zeros

)(

))(

(

)(

))(

()(

1 1

1 1

p s p

s p

s

z s z

s z

s s

H

N N

M M

N

p p

p1, 2,,

APPLICATION: STABILITY

• Review: BIBO Stable

– Bounded input always leads to bounded output





−+| h(t)| dt

• The positions of poles of H(s) in the s-domain

determine if a system is BIBO stable.

N

N m

s s

A s

s

A s

s

A s

)(

2

2 1

1

– Simple poles: the order of the pole is 1, e.g

– Multiple-order poles: the poles with higher order E.g

1

2

s

48APPLICATION: STABILITY

• Case 1: simple poles in the left half plane

)()sin(

)exp(

1)

(

1

k k

• If all the poles of the system are on the left half plane,

then the system is stable.

Impulse response

k k

)exp(

1)

• If at least one pole of the system is on the right half

plane, then the system is unstable.

Impulse response

50APPLICATION: STABILITY

• Case 3: Simple poles on the imaginary axis

)()sin(

1)

1

k k

APPLICATION: STABILITY

• Case 4: multiple-order poles in the left half plane

)()sin(

)exp(

1)

)exp(

1)

1)

52APPLICATION: STABILITY

• Example:

– Check the stability of the following system

13 6

2 3 )

+ +

+

=

s s

s s

H