1. Signal and Systems Chapter 9: Laplace Transform
Motivation and Definition of the (Bilateral) Laplace Transform
Examples of Laplace Transforms and Their Regions of Convergence (ROCs)
Properties of ROCs
Inverse Laplace Transforms
Laplace Transform Properties
The System Function of an LTI System
Geometric Evaluation of Laplace Transforms and Frequency Responses

2. Motivation for the Laplace Transform
Book Chapter#: Section#
Motivation for the Laplace Transform
CT Fourier transform enables us to do a lot of things, e.g.
Analyze frequency response of LTI systems
Sampling
Modulation
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
In particular, Fourier transform cannot handle large (and important) classes of signals and unstable systems, i.e. when −∞ ∞ |𝑥(𝑡)| 𝑑𝑡=∞
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3. Motivation for the Laplace Transform (continued)
Book Chapter#: Section#
Motivation for the Laplace Transform (continued)
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
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
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4. The (Bilateral) Laplace Transform
Book Chapter#: Section#
The (Bilateral) Laplace Transform
𝑥(𝑡)↔𝑋(𝑠)= −∞ ∞ 𝑥(𝑡) 𝑒 −𝑠𝑡 𝑑𝑡= 𝐿{𝑥(𝑡)
s = σ+ jω is a complex variable – Now we explore the full range of 𝑠
Basic ideas:
𝑋(𝑠)=𝑋(𝜎+𝑗𝜔)= −∞ ∞ 𝑥(𝑡) 𝑒 −𝜎𝑡 ] 𝑒 −𝑗𝜔𝑡 𝑑𝑡= 𝐹{𝑥(𝑡) 𝑒 −𝜎𝑡
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): 𝑅𝑂𝐶={𝑠=𝜎+𝑗𝜔 so that −∞ ∞ |𝑥 𝑡 𝑒 −𝜎𝑡 |𝑑𝑡<∞
If 𝑠 = 𝑗𝜔 is in the ROC (i.e. σ= 0), then
𝑋(𝑠) | 𝑠=𝑗ω =𝐹{𝑥(𝑡)
absolute integrability needed
absolute integrability condition
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5. Book Chapter#: Section#
Example #1:
𝑥 1 (𝑡)= 𝑒 −𝑎𝑡 𝑢(𝑡 (a – an arbitrary real or complex number)
𝑋 1 (𝑠)= −∞ ∞ 𝑒 −𝑎𝑡 𝑢(𝑡) 𝑒 −𝑠𝑡 𝑑𝑡 = 0 ∞ 𝑒 −(𝑠+𝑎)𝑡 =− 1 𝑠+𝑎 [ 𝑒 −(𝑠+𝑎)∞ −1 𝑋 1 (𝑠)= −∞ ∞ 𝑒 −𝑎𝑡 𝑢(𝑡) 𝑒 −𝑠𝑡 𝑑𝑡 = 0 ∞ 𝑒 −(𝑠+𝑎)𝑡 =− 1 𝑠+𝑎 [ 𝑒 −(𝑠+𝑎)∞ −1
This converges only if Re(s+a) > 0, i.e. Re(s) > -Re(a)
𝑋 1 𝑠 = 1 𝑠+𝑎 ,ℜ𝑒 𝑠 >−ℜ𝑒{𝑎
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6. Book Chapter#: Section#
Example #2:
𝑥 2 (𝑡)=− 𝑒 −𝑎𝑡 𝑢(−𝑡 𝑋 2 (𝑠)=− −∞ ∞ 𝑒 −𝑎𝑡 𝑢(−𝑡) 𝑒 −𝑠𝑡 𝑑𝑡 =− −∞ 0 𝑒 −(𝑠+𝑎)𝑡 𝑑𝑡 = 1 𝑠+𝑎 𝑒 −(𝑠+𝑎)𝑡 | −∞ 0 = 1 𝑠+𝑎 [1− 𝑒 (𝑠+𝑎)∞
This converges only if Re(s+a) < 0, i.e. Re(s) < -Re(a)
𝑋 2 (𝑠)= 1 𝑠+𝑎 ,ℜ𝑒{𝑠}<−ℜ𝑒{𝑎 Same as 𝑋 1 (s), but different ROC
Key Point (and key difference from FT): Need both X(s) and ROC to uniquely determine x(t). No such an issue for FT.
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7. Graphical Visualization of the ROC
Book Chapter#: Section#
Graphical Visualization of the ROC
Example1: 𝑋 1 (𝑠)= 1 𝑠+𝑎 ,ℜ𝑒{𝑠}>−ℜ𝑒{𝑎 𝑥 1 (𝑡)= 𝑒 −𝑎𝑡 𝑢(𝑡)→𝑟𝑖𝑔ℎ𝑡−𝑠𝑖𝑑𝑒𝑑
Example2: 𝑋 2 (𝑠)= 1 𝑠+𝑎 ,ℜ𝑒{𝑠}<−ℜ𝑒{𝑎 𝑥 2 (𝑡)=− 𝑒 −𝑎𝑡 𝑢(−𝑡)→𝑙𝑒𝑓𝑡−𝑠𝑖𝑑𝑒𝑑
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8. Book Chapter#: Section#
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
X(s) = N(s)/D(s), N(s),D(s) – polynomials in s
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.
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9. Book Chapter#: Section#
Example #3
𝑥(𝑡)=3 𝑒 2𝑡 𝑢(𝑡)−2 𝑒 −𝑡 𝑢(𝑡 𝑋(𝑠)= 0 ∞ 3 𝑒 2𝑡 −2 𝑒 −𝑡 ] 𝑒 −𝑠𝑡 𝑑𝑡 𝑋(𝑠)=3 0 ∞ 𝑒 −(𝑠−2)𝑡 𝑑𝑡−2 0 ∞ 𝑒 −(𝑠+1)𝑡 𝑑𝑡 𝑋(𝑠)= 3 𝑠−2 − 2 𝑠+1 = 𝑠+7 𝑠 2 −𝑠−2 ,ℜ𝑒{𝑠}>2
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10. Laplace Transforms and ROCs
Book Chapter#: Section#
Laplace Transforms and ROCs
Some signals do not have Laplace Transforms (have no ROC)
𝑎)𝑥(𝑡)=𝐶 𝑒 −𝑡 for all t since −∞ ∞ |𝑥(𝑡) 𝑒 −𝜎𝑡 |𝑑𝑡=∞ for all 𝜎
𝑏)𝑥(𝑡)= 𝑒 𝑗 𝜔 0 𝑡 for all t 𝐹𝑇:𝑋(𝑗ω)=2𝜋𝛿(𝜔− 𝜔 0
−∞ ∞ |𝑥(𝑡) 𝑒 −𝜎𝑡 |𝑑𝑡= −∞ ∞ 𝑒 −𝜎𝑡 𝑑𝑡=∞ for all 𝜎
X(s) is defined only in ROC; we don’t allow impulses in LTs
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11. Book Chapter#: Section#
Properties of the ROC
The ROC can take on only a small number of different forms
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? −∞ ∞ |𝑥(𝑡) 𝑒 −𝑠𝑡 |𝑑𝑡= −∞ ∞ |𝑥(𝑡) 𝑒 −𝜎𝑡 | 𝑑𝑡<∞ depends only on 𝜎=ℜ𝑒{𝑠
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.
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12. Book Chapter#: Section#
More Properties
If x(t) is of finite duration and is absolutely integrable, then the ROC is the entire s-plane.
𝑋(𝑠)= −∞ ∞ 𝑥(𝑡) 𝑒 −𝑠𝑡 𝑑𝑡= 𝑇 1 𝑇 2 𝑥(𝑡) 𝑒 −𝑠𝑡 𝑑𝑡<∞  𝑖𝑓  𝑇 1 𝑇 2 |𝑥(𝑡)| 𝑑𝑡<∞
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13. ROC Properties that Depend on Which Side You Are On - I
Book Chapter#: Section#
ROC Properties that Depend on Which Side You Are On - I
If x(t) is right-sided (i.e. if it is zero before some time), and if Re(s) = 𝜎 0 is in the ROC, then all values of s for which Re(s) > 𝜎 0 are also in the ROC.
ROC is a right half plane (RHP)
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14. ROC Properties that Depend on Which Side You Are On -II
Book Chapter#: Section#
ROC Properties that Depend on Which Side You Are On -II
If x(t) is left-sided (i.e. if it is zero after some time), and if Re(s) = 𝜎 0 is in the ROC, then all values of s for which Re(s) < 𝜎 0 are also in the ROC.
ROC is a left half plane (LHP)
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15. Still More ROC Properties
Book Chapter#: Section#
Still More ROC Properties
If x(t) is two-sided and if the line Re(s) = 𝜎 0 is in the ROC, then the ROC consists of a strip in the s-plane
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16. Example: 𝑥(𝑡)= 𝑒 −𝑏|𝑡| Intuition?
Book Chapter#: Section#
Example:
𝑥(𝑡)= 𝑒 −𝑏|𝑡|
Intuition?
Okay: multiply by constant ( 𝑒 𝜎𝑡 ) and will be integrable
Looks bad: no 𝑒 𝜎𝑡 will dampen both sides
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17. Example (continued): Overlap if 𝑏>0⇒𝑋 𝑠 = −2𝑏 𝑠 2 − 𝑏 2 , with ROC:
Book Chapter#: Section#
Example (continued):
𝑥(𝑡)= 𝑒 𝑏𝑡 𝑢(−𝑡)+ 𝑒 −𝑏𝑡 𝑢(𝑡)⇒− 1 𝑠−𝑏 ,ℜ𝑒{𝑠}<𝑏+ 1 𝑠+𝑏 ,ℜ𝑒{𝑠}>−𝑏
Overlap if 𝑏>0⇒𝑋 𝑠 = −2𝑏 𝑠 2 − 𝑏 2 , with ROC:
What if b < 0? ⇒No overlap ⇒ No Laplace Transform
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18. Properties, Properties
Book Chapter#: Section#
Properties, Properties
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.
Suppose X(s) is rational, then
If x(t) is right-sided, the ROC is to the right of the rightmost pole.
If x(t) is left-sided, the ROC is to the left of the leftmost pole.
If ROC of X(s) includes the jω-axis, then FT of x(t) exists.
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19. Example: Three possible ROCs x(t) is right-sided ROC: III No
Book Chapter#: Section#
Example:
Three possible ROCs
x(t) is right-sided ROC: III No
x(t) is left-sided ROC: I No
x(t) extends for all time ROC: II Yes
Fourier Transform exists?
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20. Inverse Laplace Transform
Book Chapter 9 : Section 2
Inverse Laplace Transform
Fix σ ∈ ROC and apply the inverse Fourier
But s = σ + jω (σ fixed)⇒ ds =jdω
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21. Inverse Laplace Transforms Via Partial Fraction
Book Chapter 9 : Section 2
Inverse Laplace Transforms Via Partial Fraction
Expansion and Properties
Example:
Three possible ROC’s — corresponding to three different signals
Recall
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22. ROC I: — Left-sided signal.
Book Chapter 9 : Section 2
ROC I: — Left-sided signal.
ROC II: — Two-sided signal, has Fourier Transform.
ROC III: — Right-sided signal.
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23. Properties of Laplace Transforms
Book Chapter 9 : Section 2
Properties of Laplace Transforms
Many parallel properties of the CTFT, but for Laplace transforms we need to determine implications for the ROC
For example:
Linearity
ROC at least the intersection of ROCs of X1(s) and X2(s)
ROC can be bigger (due to pole-zero cancellation)
⇒ ROC entire s-
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24. Time Shift Book Chapter 9 : Section 2
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25. s-Domain Differentiation
Book Chapter 9 : Section 2
Time-Domain Differentiation
ROC could be bigger than the ROC of X(s), if there is pole-zero cancellation. E.g.,
s-Domain Differentiation
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26. x(t)=etu(t),and h(t)=-e-tu(-t)
Book Chapter 9 : Section 2
Convolution Property
x(t)
y(t)=h(t)＊x(t)
h(t)
For
Then
ROC of Y(s) = H(s)X(s): at least the overlap of the ROCs of H(s) & X(s)
ROC could be empty if there is no overlap between the two ROCs
E.g.
x(t)=etu(t),and h(t)=-e-tu(-t)
ROC could be larger than the overlap of the two.
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27. x(t) The System Function of an LTI System
Book Chapter 9 : Section 2
The System Function of an LTI System
x(t)
y(t)
h(t)
The system function characterizes the system ⇓
System properties correspond to properties of H(s) and its ROC
A first example:
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28. Geometric Evaluation of Rational Laplace Transforms
Book Chapter 9 : Section 2
Geometric Evaluation of Rational Laplace Transforms
Example #1: A first-order zero
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29. Example #2: A first-order pole
Book Chapter 9 : Section 2
Example #2: A first-order pole
Still reason with vector, but remember to "invert" for poles
Example #3: A higher-order rational Laplace transform
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30. First-Order System Graphical evaluation of H(jω)
Book Chapter 9 : Section 2
First-Order System
Graphical evaluation of H(jω)
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31. Bode Plot of the First-Order System
Book Chapter 9 : Section 2
Bode Plot of the First-Order System
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32. 2 poles on negative real axis
Book Chapter 9 : Section 2
Second-Order System
complex poles
— Underdamped
double pole at s = −ωn
— Critically damped
2 poles on negative real axis
— Overdamped
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33. Book Chapter 9 : Section 2
Demo Pole-zero diagrams, frequency response, and step response of first-order and second-order CT causal systems
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34. Bode Plot of a Second-Order System
Book Chapter 9 : Section 2
Bode Plot of a Second-Order System
Top is flat when
ζ= 1/√2 = 0.707
⇒a LPF for
ω < ωn
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35. Unit-Impulse and Unit-Step Response of a Second- Order System
Book Chapter 9 : Section 2
Unit-Impulse and Unit-Step Response of a Second- Order System
No oscillations when
ζ ≥ 1
⇒ Critically (=) and over (>) damped.
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36. First-Order All-Pass System
Book Chapter 9 : Section 2
First-Order All-Pass System
Two vectors have the same lengths
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37. x(t) CT System Function Properties Question:
y(t)
h(t)
H(s) = “system function”
1) System is stable   ROC of H(s) includes jω axis
2) Causality => h(t) right-sided signal => ROC of H(s) is a right-half plane
Question:
If the ROC of H(s) is a right-half plane, is the system causal?
E.x.
Non-causal

38. Properties of CT Rational System Functions
a) However, if H(s) is rational, then
The system is causal ⇔ The ROC of H(s) is to the right of the rightmost pole
b) If H(s) is rational and is the system function of a causal system, then
The system is stable ⇔ jω-axis is in ROC
⇔ all poles are in the LHP

39. Checking if All Poles Are In the Left-Half Plane
Poles are the roots of D(s)=sn+an-1sn-1+…+a1s+a0
Method #1: Calculate all the roots and see!
Method #2: Routh-Hurwitz – Without having to solve for roots.

40. Initial- and Final-Value Theorems
If x(t) = 0 for t < 0 and there are no impulses or higher order discontinuities at the origin, then
Initial value
If x(t) = 0 for t < 0 and x(t) has a finite limit as t → ∞, then
Final value

41. Applications of the Initial- and Final-Value Theorem
For
n-order of polynomial N(s), d – order of polynomial D(s)
• Initial value：
• Final value：

42. LTI Systems Described by LCCDEs
roots of numerator ⇒ zeros
roots of denominator ⇒ poles
ROC =? Depends on: 1) Locations of all poles.
2) Boundary conditions, i.e.
right-, left-, two-sided signals.

43. System Function Algebra
Example: A basic feedback system consisting of causal blocks
More on this later in feedback
ROC: Determined by the roots of 1+H1(s)H2(s), instead of H1(s)

44. Block Diagram for Causal LTI Systems with Rational System Functions
Example：
— Can be viewed as cascade of two systems.

45. Example (continued)
Instead of
We can construct H(s) using:
Notation: /s—an integrator

46. Note also that
Lesson to be learned：There are many different ways to construct a system that performs a certain function.

47. The Unilateral Laplace Transform
(The preferred tool to analyze causal CT systems described by LCCDEs with initial conditions)
Note:
If x(t) = 0 for t < 0,
Unilateral LT of x(t) = Bilateral LT of x(t)u(t-)
For example, if h(t) is the impulse response of a causal LTI system then,
Convolution property: If x1(t) = x2(t) = 0 for t < 0,
Same as Bilateral Laplace transform

48. Differentiation Property for Unilateral Laplace Transform
Initial condition!
integration by parts
∫f．dg=fg－∫g．df
Derivation：
Note：

49. Use of ULTs to Solve Differentiation Equations with Initial Conditions
Example：
Take ULT：
ZIR — Response for zero input x(t)=0
ZSR — Response for zero state,
β=γ=0, initially at rest

50. Example (continued)
Response for LTI system initially at rest (β = γ = 0 )
Response to initial conditions alone (α = 0). For example: