Signal and Systems Prof. H. Sameti Chapter 9: Laplace Transform  Motivatio n 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# Computer Engineering Department, Signal and Systems 2

Motivation for the Laplace Transform (continued) Book Chapter#: Section# Computer Engineering Department, Signal and Systems 3

The (Bilateral) Laplace Transform Book Chapter#: Section# Computer Engineering Department, Signal and Systems 4 absolute integrability needed absolute integrability condition

Example #1: Book Chapter#: Section# Computer Engineering Department, Signal and Systems 5

Example #2: Book Chapter#: Section# Computer Engineering Department, Signal and Systems 6

Graphical Visualization of the ROC Book Chapter#: Section# Computer Engineering Department, Signal and Systems 7

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. Book Chapter#: Section# Computer Engineering Department, Signal and Systems 8

Example #3 Book Chapter#: Section# Computer Engineering Department, Signal and Systems 9

Laplace Transforms and ROCs Book Chapter#: Section# Computer Engineering Department, Signal and Systems 10

Properties of the ROC Book Chapter#: Section# Computer Engineering Department, Signal and Systems 11

More Properties Book Chapter#: Section# Computer Engineering Department, Signal and Systems 12

ROC Properties that Depend on Which Side You Are On - I Book Chapter#: Section# Computer Engineering Department, Signal and Systems 13 ROC is a right half plane (RHP)

ROC Properties that Depend on Which Side You Are On -II Book Chapter#: Section# Computer Engineering Department, Signal and Systems 14 ROC is a left half plane (LHP)

Still More ROC Properties Book Chapter#: Section# Computer Engineering Department, Signal and Systems 15

Example: Book Chapter#: Section# Computer Engineering Department, Signal and Systems 16

Example (continued): Book Chapter#: Section# Computer Engineering Department, Signal and Systems 17

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 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.  If ROC of X(s) includes the jω-axis, then FT of x(t) exists. Book Chapter#: Section# Computer Engineering Department, Signal and Systems 18

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 Book Chapter#: Section# Computer Engineering Department, Signal and Systems 19 Fourier Transform exists?

The z-Transform Last time: Unit circle (r = 1) in the ROC ⇒ DTFT exists Rational transforms correspond to signals that are linear combinations of DT exponentials Computer Engineering Department, Signal and Systems 20 -depends only on r = |z|, just like the ROC in s-plane only depends on Re(s) Book Chapter10: Section 1

Some Intuition on the Relation between ZT and LT Computer Engineering Department, Signal and Systems 21 The (Bilateral) z-Transform Can think of z-transform as DT version of Laplace transform with Let t=nT Book Chapter10: Section 1

More intuition on ZT-LT, s-plane - z-plane relationship  LHP in s-plane, Re(s) < 0 ⇒ |z| = | e sT | < 1, inside the |z| = 1 circle. Special case, Re(s) = -∞ ⇔ |z| = 0.  RHP in s-plane, Re(s) > 0 ⇒ |z| = | e sT | > 1, outside the |z| = 1 circle. Special case, Re(s) = +∞ ⇔ |z| = ∞.  A vertical line in s-plane, Re(s) = constant ⇔ | e sT | = constant, a circle in z-plane. Computer Engineering Department, Signal and Systems 22 Book Chapter10: Section 1

Computer Engineering Department, Signal and Systems 23 Computer Engineering Department, Signal and Systems 23 Inverse Laplace Transform Fix σ ∈ ROC and apply the inverse Fourier But s = σ + jω (σ fixed) ⇒ ds =jdω Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 24 Inverse Laplace Transforms Via Partial Fraction Expansion and Properties Example: Three possible ROC’s — corresponding to three different signals Recall Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 25 ROC I:—Left-sided signal. ROC II:—Two-sided signal, has Fourier Transform. ROC III: —Right-sided signal. Book Chapter 9 : Section 2

 Many parallel properties of the CTFT, but for Laplace transforms we need to determine implications for the ROC  For example: Book Chapter 9 : Section 2 Computer Engineering Department, Signal and Systems 26 Properties of Laplace Transforms 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 27 Time Shift Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 28 Time-Domain Differentiation ROC could be bigger than the ROC of X(s), if there is pole-zero cancellation. E.g., s-Domain Differentiation Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 29 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)=e t u(t),and h(t)=-e -t u(-t) ROC could be larger than the overlap of the two. Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 30 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: Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 31 Geometric Evaluation of Rational Laplace Transforms Example #1:A first-order zero Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 32 Example #2:A first-order pole Still reason with vector, but remember to "invert" for poles Example #3:A higher-order rational Laplace transform Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 33 First-Order System Graphical evaluation of H(jω) Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 34 Bode Plot of the First-Order System Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 35 Second-Order System complex poles — Underdamped double pole at s = −ω n — Critically damped 2 poles on negative real axis — Overdamped Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 36 DemoPole-zero diagrams, frequency response, and step response of first-order and second-order CT causal systems Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 37 Bode Plot of a Second-Order System Top is flat when ζ= 1/√2 = ⇒ a LPF for ω < ω n Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 38 Unit-Impulse and Unit-Step Response of a Second- Order System No oscillations when ζ ≥ 1 ⇒ Critically (=) and over (>) damped. Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 39 First-Order All-Pass System 1.Two vectors have the same lengths 2. Book Chapter 9 : Section 2

CT System Function Properties x(t) 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 40

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 41

Checking if All Poles Are In the Left-Half Plane Poles are the root of D(s)=s n +a n-1 s n-1 +…+a1s+a 0 Method #1:Calculate all the roots and see! Method #2:Routh-Hurwitz – Without having to solve for roots. 42

Initial- and Final-Value Theorems Ifx(t) = 0 fort < 0 and there are no impulses or higher order discontinuities at the origin, then Initial value Final value Ifx(t) = 0 fort < 0 and x(t) has a finite limit as t → ∞, then 43

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 ： 44

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. 45

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+H 1 (s)H 2 (s), instead of H 1 (s) 46

Block Diagram for Causal LTI Systems with Rational System Functions Example ： — Can be viewed as cascade of two systems. 47

Example (continued) Instead of We can construct H(s) using: Notation: 1/s—an integrator 48

Note also that Lesson to be learned ： There are many different ways to construct a system that performs a certain function. 49

The Unilateral Laplace Transform (The preferred tool to analyze causal CT systems described by LCCDEs with initial conditions) Note: 1)If x(t) = 0 for t < 0, 2)Unilateral LT of x(t) = Bilateral LT of x(t)u(t-) 3)For example, if h(t) is the impulse response of a causal LTI system then, 4)Convolution property: If x 1 (t) = x 2 (t) = 0 for t < 0, Same as Bilateral Laplace transform 50

Differentiation Property for Unilateral Laplace Transform Initial condition! Derivation ： Note ： integration by parts ∫f ． dg=fg － ∫g ． df 51

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 52

Example (continued) Response for LTI system initially at rest (β = γ = 0 ) Response to initial conditions alone (α = 0). For example: 53