Page 131 - Schaum's Outline of Theory and Problems of Signals and Systems

120 LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS [CHAP. 3

B. Use of Tables of Laplace Transform Pairs:

In the second method for the inversion of X(s), we attempt to express X(s) as a sum
X(s) = X,(s) + . . . +Xn(s) (3.26)
where X,(s), . . . , Xn(s) are functions with known inverse transforms xl(t), . . . , xn(t). From
the linearity property (3.15) it follows that

x(t) =xl(t) + - - +xn(t) (3.27)

C. Partial-Fraction Expansion:
If X(s) is a rational function, that is, of the form

a simple technique based on partial-fraction expansion can be used for the inversion of
Xb).
(a) When X(s) is a proper rational function, that is, when m < n:

1. Simple Pole Case:
If all poles of X(s), that is, all zeros of D(s), are simple (or distinct), then X(s) can be
written as

where coefficients ck are given by

If D(s) has multiple roots, that is, if it contains factors of the form (s -pi)', we say that
pi is the multiple pole of X(s) with multiplicity r. Then the expansion of X(s) will consist of
terms of the form

where

(b) When X(s) is an improper rational function, that is, when m 2 n:
If m 2 n, by long division we can write X(s) in the form

where N(s) and D(s) are the numerator and denominator polynomials in s, respectively,
of X(s), the quotient Q(s) is a polynomial in s with degree rn - n, and the remainder R(s)