Page 158 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 31 LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS

Hence, the system function H(s) is

Assuming the system is causal and taking the inverse Laplace transform of H(s), the impulse
response h(t ) is

3.30. Consider a continuous-time LTI system for which the input x(t) and output y(t) are
related by

~"(1) +yl(t) - 2y(t) =x(t) (3.86)

(a) Find the system function H(s).
(b) Determine the impulse response h(t) for each of the following three cases: (i)
the system is causal, (ii) the system is stable, (iii) the system is neither causal nor
stable.
(a) Taking the Laplace transform of Eq. (3.86), we have
s2~(s) + sY(s) - 2Y(s) = X(s)

or (s2 + s - ~)Y(s) =X(s)
Hence, the system function H(s) is

(b) Using partial-fraction expansions, we get

(i) If the system is causal, then h(t) is causal (that is, a right-sided signal) and the
ROC of H(s) is Re(s) > 1. Then from Table 3-1 we get

(ii) If the system is stable, then the ROC of H(s) must contain the jo-axis. Conse-
quently the ROC of H(s) is - 2 < Re(s) < 1. Thus, h(t) is two-sided and from
Table 3-1 we get

(iii) If the system is neither causal nor stable, then the ROC of H(s) is Re(s) < -2.
Then h(r) is noncausal (that is, a left-sided signal) and from Table 3-1 we get

331. The feedback interconnection of two causal subsystems with system functions F(s)
and G(s) is depicted in Fig. 3-13. Find the overall system function H(s) for this
feedback system.