2. Signals and Systems

3. Chapter 4 the Laplace Transform

4. 4.4 The Inverse Laplace Transform

5. Review Multiplying both sides by s =σ+ jω ， ds =jdω The Inverse Laplace Transform:

6.  This equation states that x(t) can be represented as a weighted integral of complex exponentials.  The contour of integration is the straight line in the s-plane corresponding to all points s satisfying Re(s)=  The formal evaluation of the integral for a general X(s) requires the use of contour integration ( 围线积分 ) in the complex plane,a topic that we will not consider here.  The inverse Laplace transform can be determined by partial-fraction expansion.

7. For example:determine x(t). Firstly, perform a partial-fraction expansion to obtain: Secondly, evaluate the coefficients: thus,the partial-fraction expansion for X(s) is : So,

8. Partial-fraction Expansion  The procedure consists of expanding the rational algebraic expression into a linear combination of lower order terms.

9. Assuming no multiple-order poles and that the order of the denominator polynomial is greater than the order of the numerator polynomial,we can expand X(s) in the form: ROC σ> -a i (right sided signal) σ< -a i (left sided signal) Adding the inverse transforms of the individual terms,then yields the inverse transform of X(s).

10. Discussing 1 ： no multiple-order poles,the first order different poles,real numbers or complex numbers

11. For example:

13. Homework  Page 250 #4-4(3)(6)(9)(15)