ECE
8443
– PatternContinuous
Recognition
EE
3512
– Signals:
and Discrete
LECTURE 23: THE LAPLACE TRANSFORM
• Objectives:
Motivation
The Bilateral Transform
Region of Convergence (ROC)
Properties of the ROC
Rational Transforms
• Resources:
MIT 6.003: Lecture 17
Wiki: Laplace Transform
Wiki: Bilateral Transform
Wolfram: Laplace Transform
IntMath: Laplace Transform
CNX: Region of Convergence
URL:
Motivation for the Laplace Transform
• The CT Fourier transform enables us to:
 Solve linear constant coefficient differential equations (LCCDE);
 Analyze the frequency response of LTI systems;
 Analyze and understand the implications of sampling;
 Develop communications systems based on modulation principles.
• Why do we need another transform?

 The Fourier transform cannot handle unstable signals:
 x(t ) dt  

(Recall this is related to the fact that the eigenfunction, ejt, has unit
amplitude, |ejt| = 1.
 There are many problems in which we desire to analyze and control unstable
systems (e.g., the space shuttle, oscillators, lasers).
• Consider the simple unstable system:
• This is an unstable causal system.
x(t )
• We cannot analyze its behavior using:
   X e H e 
Ye
j
j
j
 Note however we can use time domain techniques
such as convolution and differential equations.
EE 3512: Lecture 23, Slide 1
y (t )
CT LTI
h(t )
h(t )  e t u (t )
t
The Bilateral (Two-sided) Laplace Transform
• Recall the eigenfunction property of an
LTI system:
• est is an eigenfunction of ANY
LTI system.
• s can be complex: s    j
x(t )  e st
y(t )  H ( s)e st
CT LTI
h(t )

H (s) 
 st
h
(
t
)
e
dt (assuming convergenc e)


• We can define the bilateral, or two-sided, Laplace transform:

x(t )  X ( s ) 
 st
x
(
t
)
e
dt  Lx(t )


• Several important observations are:
X (  j ) 
 Can be viewed as a generalization
of the Fourier transform:
 X(s) generally exists for a certain range of
values of s. We refer to this as the region
of convergence (ROC). Note that this
only depends on  and not .

 (  j  ) t
x
(
t
)
e
dt





 x(t )e e
 t
 jt

dt  F x(t )e t





t
ROC    j   x(t )e dt   



 If s = j is in the ROC (i.e.,  = 0), then:
Lx(t ) s  j  X ( s) s  j  Fx(t )
and there is a clear relationship between the Laplace and Fourier transforms.
EE 3512: Lecture 23, Slide 2
Example: A Right-Sided Signal
• Example: x1 t   e  at u(t ) where a
is an arbitrary real or complex
number.
• Solution:

X 1 ( s) 


( s  a ) t
e
u
(
t
)
e
dt

e
e
dt

e
dt



 at
 st

 1 ( s  a ) t

e
sa
 at
 st
0


0
0


 1 ( s  a ) 
e
 1  ???
sa
This converges only if:
Re{s  a}  0  Re{s}   Re{a}
and we can write:
1
X 1 (s) 
when
sa
Re{s}   Re{a}
ROC
• The ROC can be visualized using s-plane plot shown above. The shaded
region defines the values of s for which the Laplace transform exists. The
ROC is a very importance property of a two-sided Laplace transform.
EE 3512: Lecture 23, Slide 3
Example: A Left-Sided Signal
• Example: x2 t   e  at u (t )
• Solution:

X 2 (s) 
e
 at
u (t )e  st dt

0
0
   e e dt    e ( s  a )t dt
 at
 st

1 ( s  a )t

e
sa

0




1
1  e ( s  a )   ???
sa
This converges only if:
Re{ s  a}  0  Re{ s}   Re{a}
and we can write:
1
X ( s) 
when
sa
Re{ s}   Re{a}
ROC
• The transform is the same but the ROC is different. This is a major difference
from the Fourier transform – we need both the transform and the ROC to
uniquely specify the signal. The FT does not have an ROC issue.
EE 3512: Lecture 23, Slide 4
Rational Transforms
• Many transforms of interest to us will be ratios of polynomials in s, which we
refer to as a rational transform:
N ( s)
H ( s) 
where N ( s)  b0  b1 s  b2 s 2  ... D( s)  1  a1 s  a 2 s 2  ...
D( s )
• The zeroes of the polynomial, N(s), are called zeroes of H(s).
• The zeroes of the polynomial, D(s), are called poles of H(s).
• Any signal that is a sum of (complex) one-sided exponentials can be
expressed as a rational transform. Examples include circuit analysis.
• Example: xt   3e 2t u(t )  2e t u(t )
zero
poles

X ( s )   (3e 2t  2e t )e  st dt
0


0
0
 3 e ( s  2 ) t dt  2  e ( s 1) t dt


3
2


s  2 0 s 1 0
1st term : Re{ s}  2
2 nd term : Re{ s}  1
3
2
s7



Re{ s}  2
s  2 s  1 ( s  2)( s  1)
EE 3512: Lecture 23, Slide 5
• Does this signal have a
Fourier transform?
Properties of the ROC
• There are some signals, particularly two-sided signals such as x(t )  e and
x(t )  e j0t that do not have Laplace transforms.
t
• The ROC typically assumes a few simple shapes. It is usually the intersection
of lines parallel to the imaginary axis. Why?
• For rational transforms, the ROC does not include any poles. Why?
• If x(t ) is of finite duration and absolutely
integrable, its ROC is the complete s-plane.

X 2 ( s) 
 x(t )u(t )e
 st
dt

T2
  x(t )e dt   if
 st
T1
T2
 x(t ) dt  
T1
• If x(t ) is right-sided, and if
0 is in the ROC, all points
to the right of 0 are in
the ROC.
• If x(t ) is left-sided, points to
the left of 0 are in the ROC.
EE 3512: Lecture 23, Slide 6
More Properties of the ROC
• If x(t ) is two-sided, then the ROC consists of the intersection of a left-sided
and right-sided version of x(t ) , which is a strip in the s-plane:
EE 3512: Lecture 23, Slide 7
Example of a Two-sided Signal
x(t )  e
bt
bt
 e u(t )  e u(t )
bt
1st term : Res  b
1
1
X (s)  

s  b s  b 2 nd term : Res  b

 2b
s2  b2
(b  0) with ROC below
• If b < 0, the Laplace transform does not exist.
• Hence, the ROC plays an integral role in the Laplace transform.
EE 3512: Lecture 23, Slide 8
ROC for Rational Transforms
• Since the ROC cannot include poles, the ROC is bounded by the poles for a
rational transform.
• If x(t) is right-sided, the ROC begins to the right of the rightmost pole. If x(t) is
left-sided, the ROC begins to the left of the leftmost pole. If x(t) is doublesided, the ROC will be the intersection of these two regions.
• If the ROC includes the j-axis, then the Fourier transform of x(t) exists. Hence,
the Fourier transform can be considered to be the evaluation of the Laplace
transform along the j-axis.
EE 3512: Lecture 23, Slide 9
Example
( s  3)
( s  1)( s  2)
• Can we uniquely determine the original signal, x(t)?
• Consider the Laplace transform: X ( s) 
• There are three possible ROCs:
• ROC III: only if x(t) is right-sided.
• ROC I: only if x(t) is left-sided.
• ROC II: only if x(t) has a Fourier transform.
EE 3512: Lecture 23, Slide 10
Summary
• Introduced the bilateral Laplace transform and discussed its merits relative to
the Fourier transform.
• Demonstrated calculation of the double-sided transform for simple
exponential signals.
• Discussed the important role that the Region of Convergence (ROC) plays in
this transform.
• Introduced the concept of a rational transform as a ratio of two polynomials.
• Explored how some basic properties of a signal influence the ROC.
• Next:
 The Inverse Laplace Transform
 Properties of the Laplace Transform
EE 3512: Lecture 23, Slide 11