Laplace transform - Continuous and Discrete Time Signals and Systems - page 298

given by

9
4 s

5
s 2

3
8 ( s + 2 )
ROC:Re s <− 2

+ 21

8( s − 2) .
ROC:Re s < 2
Taking the inverse Laplace transform, the impulse response for a non-causal
LTIC system is given by

H 1 ( s ) =−

− −

ROC:Re s < 0 ROC:Re s < 0

9
4

+ 5 t + 3 − 21

8

− 2 t

e 2 t

h 1 ( t ) =

8 e

u ( − t ) .

(ii) Using the partial fraction expansion, H 2 ( s ) may be expressed as follows:

3
10( s + 1)

3 s − 1
10( s 2 + 4 s + 13) .

H 2 ( s ) = −

If H 2 ( s ) represents a causal system, then its ROC is given by R c :Re { s > − 1.
The ROCs associated with the individual terms in H 2 ( s ) are given by

3
10( s + 1 )
ROC:Re s >− 1

3 s − 1
1 0( s 2 + 4 s + 13 )

H 2 ( s ) = − .

ROC:Re s >− 2

Taking the inverse Laplace transform, the impulse response for a causal LTIC
system is given by

3
10 e

− 3

− 2 t cos(3 t ) + 7 − t − 2 t sin(3 t )

h 2 ( t ) =

10 e 30 e

u ( t ) .

If H 2 ( s ) represents a non-causal system, then several different choices of ROC
are possible. One possible choice is given by R nc :Re { s < − 2. The ROCs
associated with the individual terms in H 2 ( s ) are given by

3
10( s + 1 )
ROC:Re s <− 1

3 s − 1
1 0( s 2 + 4 s + 13 )

H 2 ( s ) = − .

ROC:Re s <− 2

Taking the inverse Laplace transform, the impulse response for a causal LTIC
system is given by

− 3

− t + 3 − 2 t cos(3 t ) − 7 − 2 t sin(3 t )

h 2 ( t ) =

10 e 10 e 30 e

u ( − t ) .

6.8 Stable and cau sal LTIC systems

In Section 3.7.3, we showed that the impulse response h ( t ) of a BIBO stable
system satisfies the condition

∞

h ( t ) d t < ∞. (6.39)

−∞