《信号与系统教学资料》Laplace Transform.pptVIP

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Signals and Systems Spring 2014 Lecture #22 15 May, 2014 Chen Zubin (陈祖斌) Room 304, Geology Palace, 0431 czb@jlu.edu.cn Signals and Systems 1. Motivation and Definition of the (Bilateral) Laplace Transform2. Examples of Laplace Transforms and Their Regions of Convergence (ROCs)3. Properties of ROCs 1. 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 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: 1. Motivation for the Laplace Transform The (Bilateral) Laplace Transform The (Bilateral) Laplace Transform Example #1: The (Bilateral) Laplace Transform Graphical Visualization of the ROC Example #1 Example #2 Rational Transforms The (Bilateral) Laplace Transform Laplace Transforms and ROCs Some signals do not have Laplace Transforms (have no ROC) X(s) is defined only in ROC; we don’t allow impulses in LTs Properties of the ROC More Properties ROC Properties that Depend on Which Side You Are On -I ROC Properties that Depend on Which Side You Are On -II Still More ROC Properties Example Example (continued) 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. 8) Suppose X(s) is