Signals and Systems

Chapter 4 the Laplace Transform

4.4 The Inverse Laplace Transform

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

 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.

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,

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

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).

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

For example:

Homework  Page 250 #4-4(3)(6)(9)(15)