Laplace transform - Continuous and Discrete Time Signals and Systems - page 273

6.3 Inverse Laplac e transform

Evaluation of the inverse Laplace transform is an important step in the analysis
of LTIC systems. The inverse Laplace transform can be calculated directly by
solving the complex integral in the synthesis equation, Eq. (6.7). This approach
involves contour integration, which is beyond the scope of this text. In cases
where the Laplace transform takes the following rational form:

= b m s m + b m − 1 s m − 1 + b m − 2 s m − 2 ++ b 1 s + b 0
s n + a n − 1 s n − 1 + a n − 2 s n − 2 ++ a 1 s + a 0

X ( s ) = N ( s )
D ( s )

, (6.11)

an alternative approach based on the partial fraction expansion is commonly
used. The approach eliminates the need for computing Eq. (6.7) and consists
of the following steps.

(1) Calculate the roots of the characteristic equation of the rational fraction, Eq.
(6.11). The characteristic equation is obtained by equating the denominator
D ( s ) in Eq. (6.11) to zero, i.e.

D ( s ) = s n + a n − 1 s n − 1 + a n − 2 s n − 2 ++ a 1 s + a 0

= 0 . (6.12)

For an n th-order characteristic equation, there will be n first-order roots.
Depending on the value of the coefficients { b l } ,0 ≤ l ≤ ( n − 1), roots
{ p r } ,1 ≤ r ≤ n , of the characteristic equation may be real-valued and/or
complex-valued. Assuming that roots are real-valued and do not repeat, the
Laplace transform X ( s ) is represented as

N ( s )

X ( s ) =

p n ) .

(6.13)

( s − p 1 )( s − p 2 ) ( s − p n − 1 )( s −

(2) Using Heaviside's partial fraction expansion formula, explained in
Appendix D, decompose X ( s ) into a summation of the first- or second-
order fractions. If no roots are repeated, X ( s ) is decomposed as follows:

k 1
( s −

k 2
( s −

k n − 1
( s −

k n
( s −

X ( s ) = + ++ +

p n ) ,

(6.14)

p 1 ) p 2 ) p n − 1 )

where the coefficients { k r } ,1 ≤ r ≤ n , are obtained from

p r ) N ( s )
D ( s )

k r = ( s − . (6.15)

s = p r

If there are repeated or complex roots, X ( s ) takes a slightly different form.
See Appendix D for more details.
(3) From Table 6.1,

1
( s −

←→

e p r t u ( t ) with ROC: Re s > p r .

p r )