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

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=∞ Computer Engineering Department, Signal and Systems

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 Computer Engineering Department, Signal and Systems

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 Computer Engineering Department, Signal and Systems

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 Computer Engineering Department, Signal and Systems

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. Computer Engineering Department, Signal and Systems

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 Computer Engineering Department, Signal and Systems

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. Computer Engineering Department, Signal and Systems

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 Computer Engineering Department, Signal and Systems

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 Computer Engineering Department, Signal and Systems

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. Computer Engineering Department, Signal and Systems

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<∞ Computer Engineering Department, Signal and Systems

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) Computer Engineering Department, Signal and Systems

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) Computer Engineering Department, Signal and Systems

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 Computer Engineering Department, Signal and Systems

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 Computer Engineering Department, Signals and Systems

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 Computer Engineering Department, Signal and Systems

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. Computer Engineering Department, Signal and Systems

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? Computer Engineering Department, Signal and Systems

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 Computer Engineering Department, Signal and Systems Computer Engineering Department, Signal and Systems 20

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 Computer Engineering Department, Signal and Systems

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. Computer Engineering Department, Signal and Systems

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- Computer Engineering Department, Signal and Systems

Time Shift Book Chapter 9 : Section 2 Computer Engineering Department, Signal and Systems

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 Computer Engineering Department, Signal and Systems

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. Computer Engineering Department, Signal and Systems

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: Computer Engineering Department, Signal and Systems

Geometric Evaluation of Rational Laplace Transforms Book Chapter 9 : Section 2 Geometric Evaluation of Rational Laplace Transforms Example #1: A first-order zero Computer Engineering Department, Signal and Systems

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 Computer Engineering Department, Signal and Systems

First-Order System Graphical evaluation of H(jω) Book Chapter 9 : Section 2 First-Order System Graphical evaluation of H(jω) Computer Engineering Department, Signal and Systems

Bode Plot of the First-Order System Book Chapter 9 : Section 2 Bode Plot of the First-Order System Computer Engineering Department, Signal and Systems

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 Computer Engineering Department, Signal and Systems

Book Chapter 9 : Section 2 Demo Pole-zero diagrams, frequency response, and step response of first-order and second-order CT causal systems Computer Engineering Department, Signal and Systems

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 Computer Engineering Department, Signal and Systems

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. Computer Engineering Department, Signal and Systems

First-Order All-Pass System Book Chapter 9 : Section 2 First-Order All-Pass System Two vectors have the same lengths Computer Engineering Department, Signal and Systems

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

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

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.

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

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：

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.

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)

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

Example (continued) Instead of We can construct H(s) using: Notation: 1/s—an integrator

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

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

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

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

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