Page 134 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 31 LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS

or

Thus,

Hence, H(s) is always rational. Note that the ROC of H(s) is not specified by Eq. (3.40)
but must be inferred with additional requirements on the system such as the causality or
the stability.

D. Systems Interconnection:

For two LTI systems [with hl(t) and h2(t), respectively] in cascade [Fig. 3-Nu)], the
overall impulse response h(t) is given by [Eq. (2.811, Prob. 2.141

Thus, the corresponding system functions are related by the product

This relationship is illustrated in Fig. 3-8(b).
Similarly, the impulse response of a parallel combination of two LTI systems
[Fig. 3-9(a)] is given by (Prob. 2.53)

Thus,

This relationship is illustrated in Fig. 3-9(b).

Fig. 3-8 Two systems in cascade. (a) Time-domain representation; (b) s-domain representation.