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
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. ∞ |𝑦(𝑢)| 𝑒𝑢 = ∞ when −∞ Computer Engineering Department, Signal and Systems 2
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: 𝑓 𝑡𝑢 is an eigenfunction of any LTI system 𝑡 = 𝜏 + 𝑘𝜕 can be complex in general Computer Engineering Department, Signal and Systems 3
Book Chapter#: Section# The (Bilateral) Laplace Transform ∞ 𝑦(𝑢)𝑓 −𝑡𝑢 𝑒𝑢 = 𝑀{𝑦(𝑢) 𝑦(𝑢) ↔ 𝑌(𝑡) = −∞ s = σ+ j ω is a complex variable – Now we explore the full range of 𝑡 absolute integrability needed Basic ideas: ∞ 𝑦(𝑢)𝑓 −𝜏𝑢 ]𝑓 −𝑘𝜕𝑢 𝑒𝑢 = 𝐺{𝑦(𝑢)𝑓 −𝜏𝑢 𝑌(𝑡) = 𝑌(𝜏 + 𝑘𝜕) = 1. −∞ A critical issue in dealing with Laplace transform is 2. convergence: — X(s) generally exists only for some values of s, located in what is called the region of convergence(ROC): ∞ |𝑦 𝑢 𝑓 −𝜏𝑢 |𝑒𝑢 < ∞ 𝑆𝑃𝐷 = {𝑡 = 𝜏 + 𝑘𝜕 so that −∞ If 𝑡 = 𝑘𝜕 is in the ROC (i.e. σ= 0), then 3. absolute 𝑌(𝑡)| 𝑡=𝑘ω = 𝐺{𝑦(𝑢) integrability condition Computer Engineering Department, Signal and Systems 4
Book Chapter#: Section# Example #1: 𝑦 1 (𝑢) = 𝑓 −𝑏𝑢 𝑣(𝑢 (a – an arbitrary real or complex number) ) ∞ 𝑓 − 𝑡+𝑏 𝑢 = − ∞ 𝑓 −𝑏𝑢 𝑣 𝑢 𝑓 −𝑡𝑢 𝑒𝑢 = 𝑡+𝑏 𝑓 − 𝑡+𝑏 ∞ − 1 1 𝑌 1 𝑡 = −∞ 0 This converges only if Re ( s + a ) > 0, i.e. Re ( s ) > - Re ( a ) 1 𝑌 1 𝑡 = 𝑡+𝑏 , ℜ𝑓 𝑡 > −ℜ𝑓{𝑏 Computer Engineering Department, Signal and Systems 5
Book Chapter#: Section# Example #2: 𝑦 2 (𝑢) = −𝑓 −𝑏𝑢 𝑣(−𝑢 ) ∞ 𝑓 −𝑏𝑢 𝑣(−𝑢)𝑓 −𝑡𝑢 𝑒𝑢 𝑌 2 (𝑡) = − −∞ 0 𝑓 −(𝑡+𝑏)𝑢 𝑒𝑢 = − −∞ 1 1 𝑡+𝑏 𝑓 −(𝑡+𝑏)𝑢 | −∞ 0 𝑡+𝑏 [1 − 𝑓 (𝑡+𝑏)∞ = = This converges only if Re(s+a) < 0, i.e. Re(s) < -Re(a) 1 𝑡+𝑏 , ℜ𝑓{𝑡} < −ℜ𝑓{𝑏 Same as 𝑌 1 (s), but different ROC 𝑌 2 (𝑡) = 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 6
Book Chapter#: Section# Graphical Visualization of the ROC 1 𝑌 1 (𝑡) = 𝑡+𝑏 , ℜ𝑓{𝑡} > −ℜ𝑓{𝑏 Example1: 𝑦 1 (𝑢) = 𝑓 −𝑏𝑢 𝑣(𝑢) → 𝑠𝑗ℎ𝑢 − 𝑡𝑗𝑒𝑓𝑒 1 𝑌 2 (𝑡) = 𝑡+𝑏 , ℜ𝑓{𝑡} < −ℜ𝑓{𝑏 Example2: 𝑦 2 (𝑢) = −𝑓 −𝑏𝑢 𝑣(−𝑢) → 𝑚𝑓𝑔𝑢 − 𝑡𝑗𝑒𝑓𝑒 Computer Engineering Department, Signal and Systems 7
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 8
Book Chapter#: Section# Example #3 𝑦(𝑢) = 3𝑓 2𝑢 𝑣(𝑢) − 2𝑓 −𝑢 𝑣(𝑢 ) ∞ 3𝑓 2𝑢 − 2𝑓 −𝑢 ]𝑓 −𝑡𝑢 𝑒𝑢 𝑌(𝑡) = 0 ∞ 𝑓 −(𝑡−2)𝑢 𝑒𝑢 − 2 ∞ 𝑓 −(𝑡+1)𝑢 𝑒𝑢 𝑌(𝑡) = 3 0 0 3 2 𝑡+7 𝑌(𝑡) = 𝑡−2 − 𝑡+1 = 𝑡 2 −𝑡−2 , ℜ𝑓{𝑡} > 2 Computer Engineering Department, Signal and Systems 9
Book Chapter#: Section# Laplace Transforms and ROCs Some signals do not have Laplace Transforms (have no ROC) ∞ |𝑦(𝑢)𝑓 −𝜏𝑢 |𝑒𝑢 = ∞ for all 𝜏 𝑏)𝑦(𝑢) = 𝐷𝑓 −𝑢 for all t since −∞ 𝑐)𝑦(𝑢) = 𝑓 𝑘𝜕 0 𝑢 for all t ) 𝐺𝑈: 𝑌(𝑘ω) = 2𝜌𝜀(𝜕 − 𝜕 0 ∞ |𝑦(𝑢)𝑓 −𝜏𝑢 |𝑒𝑢 = ∞ 𝑓 −𝜏𝑢 𝑒𝑢 = ∞ for all 𝜏 −∞ −∞ X(s) is defined only in ROC; we don ’ t allow impulses in LTs Computer Engineering Department, Signal and Systems 10
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 1. to the j ω -axis in the s -plane (i.e. the ROC only depends on σ ).Why? ∞ |𝑦(𝑢)𝑓 −𝑡𝑢 |𝑒𝑢 = ∞ |𝑦(𝑢)𝑓 −𝜏𝑢 | 𝑒𝑢 < ∞ depends −∞ −∞ only on } 𝜏 = ℜ𝑓{𝑡 If X ( s ) is rational, then the ROC does not contain any 2. poles. Why? Poles are places where D ( s ) = 0 ⇒ X(s) = N(s)/ D(s) = ∞ Not convergent. Computer Engineering Department, Signal and Systems 11
Book Chapter#: Section# More Properties If x(t) is of finite duration and is absolutely integrable, then the ROC is the entire s-plane. ∞ 2 𝑦(𝑢)𝑓 −𝑡𝑢 𝑒𝑢 < ∞ 𝑗𝑔 2 |𝑦(𝑢)| 𝑒𝑢 < ∞ 𝑈 𝑈 𝑦(𝑢)𝑓 −𝑡𝑢 𝑒𝑢 = 𝑌(𝑡) = 𝑈 𝑈 1 1 −∞ Computer Engineering Department, Signal and Systems 12
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) = 𝜏 0 is in the ROC, then all values of s for which Re(s) > 𝜏 0 are also in the ROC. ROC is a right half plane (RHP) Computer Engineering Department, Signal and Systems 13
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) = 𝜏 0 is in the ROC, then all values of s for which Re(s) < 𝜏 0 are also in the ROC. ROC is a left half plane (LHP) Computer Engineering Department, Signal and Systems 14
Book Chapter#: Section# Still More ROC Properties If x(t) is two-sided and if the line Re(s) = 𝜏 0 is in the ROC, then the ROC consists of a strip in the s-plane Computer Engineering Department, Signal and Systems 15
Book Chapter#: Section# Example: 𝑦(𝑢) = 𝑓 −𝑐|𝑢| Intuition? Okay: multiply by constant ( 𝑓 𝜏𝑢 ) and will be integrable Looks bad: no 𝑓 𝜏𝑢 will dampen both sides Computer Engineering Department, Signals and Systems 16
Book Chapter#: Section# Example (continued): 1 1 𝑦(𝑢) = 𝑓 𝑐𝑢 𝑣(−𝑢) + 𝑓 −𝑐𝑢 𝑣(𝑢) ⇒ − 𝑡−𝑐 , ℜ𝑓{𝑡} < 𝑐 + 𝑡+𝑐 , ℜ𝑓{𝑡} > −𝑐 −2𝑐 Overlap if 𝑐 > 0 ⇒ 𝑌 𝑡 = 𝑡 2 −𝑐 2 , with ROC: What if b < 0? ⇒ No overlap ⇒ No Laplace Transform Computer Engineering Department, Signal and Systems 17
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. a) If x(t) is left-sided, the ROC is to the left of the leftmost pole. b) If ROC of X(s) includes the jω -axis, then FT of x(t) exists. Computer Engineering Department, Signal and Systems 18
Book Chapter#: Section# Example: Three possible ROCs Fourier Transform exists? 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 Computer Engineering Department, Signal and Systems 19
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