Signals and Systems - Electrical Engineering

3.4 Inverse Laplace Transform 213

convergenceRe(s) <2. That this is so is confirmed by the intersection of these two regions of

convergence that gives

[Re(s) >− 2 ]∩[Re(s) < 2 ]=− 2 <Re(s) < 2

As such, we have

x(t)=−0.25e−^2 tu(t)−0.25e^2 tu(−t) n

nExample 3.21

Consider the transfer function

H(s)=

s

(s+ 2 )(s− 1 )

=

2 / 3

s+ 2

+

1 / 3

s− 1

with a zero ats=0, and poles ats=−2 ands=1. Find out how many impulse responses can be

connected withH(s)by considering different possible regions of convergence and by determining

in which cases the system withH(s)as its transfer function is BIBO stable.

Solution

The following are the different possible impulse responses:

n If ROC:Re(s) >1, the impulse response

h 1 (t)=( 2 / 3 )e−^2 tu(t)+( 1 / 3 )etu(t)

corresponding toH(s)with this region of convergence is causal. The corresponding system is

unstable—due to the pole in the right-hands-plane, which will make the impulse response

grow astincreases.

n If ROC:− 2 <Re(s) <1, the impulse response corresponding toH(s)with this region of

convergence is noncausal, but the system is stable. The impulse response would be

h 2 (t)=( 2 / 3 )e−^2 tu(t)−( 1 / 3 )etu(−t)

Notice that the region of convergence includes thejaxis, and this guarantees the stability

(verify thath 2 (t)is absolutely integrable), and as we will see later, also the existence of the

Fourier transform ofh 2 (t).

n If ROC:Re(s) <−2, the impulse response in this case would be anti-causal, and the system is

unstable (please verify it), as the impulse response is

h 3 (t)=−( 2 / 3 )e−^2 tu(−t)−( 1 / 3 )etu(−t) n

Two very important generalizations of the results in this example are:

n An LTI with a transfer functionH(s)and region of convergenceRis BIBO stable if thejaxis is contained

in the region of convergence.