Signals and Systems
Fall 2003
Lecture #17
4 November 2003
1,Motivation and Definition of the (Bilateral)
Laplace Transform
2,Examples of Laplace Transforms and Their
Regions of Convergence (ROCs)
3,Properties of ROCs
CT Fourier transform enables us to do a lot of things,e.g.
— Analyze frequency response of LTI systems
— Sampling
— Modulation
#
Motivation for the Laplace Transform
In particular,Fourier transform cannot handle large (and important)
classes of signals and unstable systems,i.e,when
Why do we need yet another transform?
One view of Laplace Transform is as an extension of the Fourier
transform to allow analysis of broader class of signals and systems
Motivation for the Laplace Transform (continued)
How do we analyze such signals/systems?
Recall from Lecture #5,eigenfunction property of LTI systems:
— e
st
is an eigenfunction of any LTI system
— s = σ + jω can be complex in general
In many applications,we do need to deal with unstable systems,e.g.
— Stabilizing an inverted pendulum
— Stabilizing an airplane or space shuttle
#
— Instability is desired in some applications,e.g,oscillators and
lasers
(2) A critical issue in dealing with Laplace transform is convergence:
— X(s) generally exists only for some values of s,
located in what is called the region of convergence (ROC)
The (Bilateral) Laplace Transform
(3) If s = jω is in the ROC (i.e,σ = 0),then
Basic ideas:
(1)
absolute
integrability
condition
absolute integrability needed
Example #1:
This converges only if Re(s+a) > 0,i.e,Re(s) > -Re(a)
Unstable:
no Fourier Transform
but Laplace Transform
exists
Example #2:
This converges only if Re(s+a) < 0,i.e,Re(s) < -Re(a)
Key Point (and key difference from FT),Need both X(s) and ROC to
uniquely determine x(t),No such an issue for FT.
Graphical Visualization of the ROC
Example #2Example #1
Rational Transforms
Many (but by no means all) Laplace transforms of interest to us
are rational functions of s (e.g.,Examples #1 and #2; in general,
impulse responses of LTI systems described by LCCDEs),where
Roots of N(s) = zeros of X(s)
Roots of D(s) = poles of X(s)
Any x(t) consisting of a linear combination of complex
exponentials for t > 0 and for t < 0 (e.g.,as in Example #1 and #2)
has a rational Laplace transform.
Example #3
Notation:
× — pole
°
— zero
Q,Does x(t) have FT?
BOTH required →
ROC intersection
Laplace Transforms and ROCs
X(s) is defined only in ROC; we don’t allow impulses in LTs
Some signals do not have Laplace Transforms (have no ROC)
(a)
(b)
Properties of the ROC
1) The ROC consists of a collection of lines parallel to the jω-axis in
the s-plane (i.e,the ROC only depends on σ).
Why?
The ROC can take on only a small number of different forms
2) If X(s) is rational,then the ROC does not contain any poles.
Why?
Poles are places where D(s) = 0
X(s) =
N(s)
D(s)
=∞ Not convergent.
More Properties
3) If x(t) is of finite duration and is absolutely integrable,then the ROC
is the entire s-plane.
ROC Properties that Depend on Which Side You Are On - I
4) If x(t) is right-sided (i.e,if it is zero before some time),and if
Re(s) = σ
o
is in the ROC,then all values of s for which
Re(s) > σ
o
are also in the ROC.
ROC is a right half plane (RHP)
ROC Properties that Depend on Which Side You Are On - II
5) If x(t) is left-sided (i.e,if it is zero after some time),and if
Re(s) = σ
o
is in the ROC,then all values of s for which
Re(s) < σ
o
are also in the ROC.
ROC is a left half plane (LHP)
Still More ROC Properties
6) If x(t) is two-sided and if the line Re(s) = σ
o
is in the ROC,
then the ROC consists of a strip in the s-plane that includes
the line Re(s) = σ
o
.
ROC is
RHP
ROC is
LHP
Strip =
RHP ∩ LHP
Example:
Intuition?
Okay,multiply by
constant (e
0t
) and
will be integrable
Looks bad,no e
σ t
will dampen both
sides
Example (continued):
What if b < 0 No overlap? No Laplace Transform
Properties,Properties
7) If X(s) is rational,then its ROC is bounded by poles or extends to
infinity,In addition,no poles of X(s) are contained in the ROC.
(a) (b) (c)
8) Suppose X(s) is rational,then
(a) If x(t) is right-sided,the ROC is to the right of the rightmost pole.
(b) If x(t) is left-sided,the ROC is to the left of the leftmost pole.
9) If ROC of X(s) includes the jω-axis,then FT of x(t) exists.
Example:
9) If ROC of X(s) includes the jω-axis,then FT of x(t) exists.
Three possible ROCs
x(t) is right-sided
ROC:
x(t) is left-sided
ROC:
x(t) extends for all time
Fourier
Transform
exists?
No
No
ROC,Yes
III
I
II
Fall 2003
Lecture #17
4 November 2003
1,Motivation and Definition of the (Bilateral)
Laplace Transform
2,Examples of Laplace Transforms and Their
Regions of Convergence (ROCs)
3,Properties of ROCs
CT Fourier transform enables us to do a lot of things,e.g.
— Analyze frequency response of LTI systems
— Sampling
— Modulation
#
Motivation for the Laplace Transform
In particular,Fourier transform cannot handle large (and important)
classes of signals and unstable systems,i.e,when
Why do we need yet another transform?
One view of Laplace Transform is as an extension of the Fourier
transform to allow analysis of broader class of signals and systems
Motivation for the Laplace Transform (continued)
How do we analyze such signals/systems?
Recall from Lecture #5,eigenfunction property of LTI systems:
— e
st
is an eigenfunction of any LTI system
— s = σ + jω can be complex in general
In many applications,we do need to deal with unstable systems,e.g.
— Stabilizing an inverted pendulum
— Stabilizing an airplane or space shuttle
#
— Instability is desired in some applications,e.g,oscillators and
lasers
(2) A critical issue in dealing with Laplace transform is convergence:
— X(s) generally exists only for some values of s,
located in what is called the region of convergence (ROC)
The (Bilateral) Laplace Transform
(3) If s = jω is in the ROC (i.e,σ = 0),then
Basic ideas:
(1)
absolute
integrability
condition
absolute integrability needed
Example #1:
This converges only if Re(s+a) > 0,i.e,Re(s) > -Re(a)
Unstable:
no Fourier Transform
but Laplace Transform
exists
Example #2:
This converges only if Re(s+a) < 0,i.e,Re(s) < -Re(a)
Key Point (and key difference from FT),Need both X(s) and ROC to
uniquely determine x(t),No such an issue for FT.
Graphical Visualization of the ROC
Example #2Example #1
Rational Transforms
Many (but by no means all) Laplace transforms of interest to us
are rational functions of s (e.g.,Examples #1 and #2; in general,
impulse responses of LTI systems described by LCCDEs),where
Roots of N(s) = zeros of X(s)
Roots of D(s) = poles of X(s)
Any x(t) consisting of a linear combination of complex
exponentials for t > 0 and for t < 0 (e.g.,as in Example #1 and #2)
has a rational Laplace transform.
Example #3
Notation:
× — pole
°
— zero
Q,Does x(t) have FT?
BOTH required →
ROC intersection
Laplace Transforms and ROCs
X(s) is defined only in ROC; we don’t allow impulses in LTs
Some signals do not have Laplace Transforms (have no ROC)
(a)
(b)
Properties of the ROC
1) The ROC consists of a collection of lines parallel to the jω-axis in
the s-plane (i.e,the ROC only depends on σ).
Why?
The ROC can take on only a small number of different forms
2) If X(s) is rational,then the ROC does not contain any poles.
Why?
Poles are places where D(s) = 0
X(s) =
N(s)
D(s)
=∞ Not convergent.
More Properties
3) If x(t) is of finite duration and is absolutely integrable,then the ROC
is the entire s-plane.
ROC Properties that Depend on Which Side You Are On - I
4) If x(t) is right-sided (i.e,if it is zero before some time),and if
Re(s) = σ
o
is in the ROC,then all values of s for which
Re(s) > σ
o
are also in the ROC.
ROC is a right half plane (RHP)
ROC Properties that Depend on Which Side You Are On - II
5) If x(t) is left-sided (i.e,if it is zero after some time),and if
Re(s) = σ
o
is in the ROC,then all values of s for which
Re(s) < σ
o
are also in the ROC.
ROC is a left half plane (LHP)
Still More ROC Properties
6) If x(t) is two-sided and if the line Re(s) = σ
o
is in the ROC,
then the ROC consists of a strip in the s-plane that includes
the line Re(s) = σ
o
.
ROC is
RHP
ROC is
LHP
Strip =
RHP ∩ LHP
Example:
Intuition?
Okay,multiply by
constant (e
0t
) and
will be integrable
Looks bad,no e
σ t
will dampen both
sides
Example (continued):
What if b < 0 No overlap? No Laplace Transform
Properties,Properties
7) If X(s) is rational,then its ROC is bounded by poles or extends to
infinity,In addition,no poles of X(s) are contained in the ROC.
(a) (b) (c)
8) Suppose X(s) is rational,then
(a) If x(t) is right-sided,the ROC is to the right of the rightmost pole.
(b) If x(t) is left-sided,the ROC is to the left of the leftmost pole.
9) If ROC of X(s) includes the jω-axis,then FT of x(t) exists.
Example:
9) If ROC of X(s) includes the jω-axis,then FT of x(t) exists.
Three possible ROCs
x(t) is right-sided
ROC:
x(t) is left-sided
ROC:
x(t) extends for all time
Fourier
Transform
exists?
No
No
ROC,Yes
III
I
II
