Laplace transform - Continuous and Discrete Time Signals and Systems - page 314

6.3 Using the partial fraction expansion approach, calculate the inverse Laplace
transform for the following rational functions of s :

s 2 + 2 s + 1
( s + 1)( s 2 + 5 s + 6) ;

(a) X ( s ) = ROC:Re s > − 1;

s 2 + 2 s + 1
( s + 1)( s 2 + 5 s + 6) ;

(b) X ( s ) = ROC:Re s < − 3;

s 2 + 3 s − 4
( s + 1)( s 2 + 5 s + 6) ;

(c) X ( s ) = ROC:Re s > − 1;

s 2 + 3 s − 4
( s + 1)( s 2 + 5 s + 6) ;

(d) X ( s ) = ROC:Re s < − 3;

s 2 + 1
s ( s + 1)( s 2 + 2 s + 17) ;

(e) X ( s ) = ROC:Re s > 0;

s + 1
( s + 2) 2 ( s 2 + 7 s + 12) ;

(f) X ( s ) = ROC:Re s > − 2;

s 2 − 2 s + 1
( s + 1) 3 ( s 2 + 16) ;

(g) X ( s ) = ROC:Re s < − 1.

6.4 The Laplace transforms of two CT signals x 1 ( t ) and x 2 ( t ) are given by the
following expressions:

s
s 2 + 5 s + 6

←→

with ROC( R 1 ):Re s > − 2 x 1 ( t )

and

1
s 2 + 5 s + 6

←→

x 2 ( t ) with ROC( R 2 ):Re s > − 2 .

Determine the Laplace transform and the associated ROC R of the com-
bined signal x 1 ( t ) + 2 x 2 ( t ). Explain how the ROC R of the combined
signal exceeds the intersection ( R 1 ∩ R 2 ) of the individual ROCs R 1
and R 2 .

6.5 Calculate the time-domain representation of the bilateral Laplace transform

s 2
( s 2 − 1)( s 2 − 4 s + 5)( s 2 + 4 s + 5)

X ( s ) =

if the ROC R is specified as follows:
(a) R :Re s < − 2;
(b) R : − 2 < Re s < − 1;
(c) R : − 1 < Re s < 1;

(d) R :1 < Re s < 2;
(e) R :Re s > 2.

6.6 Prove the frequency-shifting property, Eq. (6.20), as stated in Section 6.4.4.

6.7 Prove the time-integration property for the unilateral and bilateral Laplace
transform as stated in Section 6.4.6.

6.8 Prove the initial-value theorem, Eq. (6.27), as stated in Section 6.4.8.