讲稿信号与系统chapter9-3the laplace transform

1、Chapter 9The Laplace Transform- The Inverse LT- Geometric Evaluation of the FT From the Pole-Zero PlotReview:Denotation:Note:This transform is often called the bilateral Laplace transform, to distinguish it from the unilateral Laplace transform.The relationship between FT and LTReReS-planeS-planeImI

2、m-a-aROC of Ex 9.1ROC of Ex 9.2 A convenient way to display the ROC is shown as the following figure:Rational Transforms Many (but by no means all) Laplace transforms of interest to us are rational functions of s (the preceding examples)N(s), D(s) polynomials in s Any x(t) consisting of a linear com

3、bination of complex exponentials for t0 and for t0 has a rational Laplace transform. Zeros and Poles Roots of D(s) = poles of X(s); Roots of N(s) = zeros of X(s). The representation of X(s) through its poles and zeros in the s-plane is referred to as the pole-zero plot of X(s) 9.3 The Inverse Laplac

4、e TransformFix s ROC and apply the inverse FTBut s=s+jw (s fixed) ds=jdwThis equation states that x(t) can be represented as a weighted integral of complex exponentials.will not useHow Do We Perform Inverse LT? In general, we will only deal with Laplace transform that are:1) Rational, i.e. X(s) = N(

5、s)/D(s) ;And/or:2) exponential, i.e. X(s) = esTFor case 2), use shift property (Similar to the FT property)x(t-T) X(s)e-sTFor case 1), use PFE (Partial Fraction Expansion). Appendix Partial Fraction ExpansionConsider a fraction polynomial:Discuss two cases of D(s)=0, for distinct root and same root.

6、(1) Distinct root:thusCalculate A1 : Multiply two sides by (s-1):Let s=1, so Generally(2) Same root:thusFor first order poles:Multiply two sides by (s-1)r : For r-order poles:So General form of X(s) x(t) ? ( Partial Fraction Expansion )ROCROC- right of the poleROC- left of the poleExampleSolution:Ho

7、w many possible ROCs ?ImReS-planeHomework: 9.5 9.9 9.13ROC II : Left-sided signal.ROC III : Two-sided signalROC I : Right-sided signal.9.4 Geometric Evaluation of the FT From the Pole-Zero PlotAs we have known, the Fourier transform of a signal is the Laplace transform evaluated on the jw-axis.In this section, we discuss a procedure for geometrically evaluating the Fourier transform1. A single zero 2. A single pole Magnitude:Phase:zero vectorMagnitude:Phase:pole vector3. General conditions We can getExample 1 First-order systemGraphical evaluation of H(