Page 136 - Schaum's Outline of Theory and Problems of Signals and Systems
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CHAP. 31 LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
I. Differentiation in the Time Domain:
provided that lim , ,, x(t )e-"' = 0. Repeated application of this property yields
where
2. Integration in the Time Domain:
C. System Function:
Note that with the unilateral Laplace transform, the system function H(s) = Y(s)/X(s)
is defined under the condition that the LTI system is relaxed, that is, all initial conditions
are zero.
D. Transform Circuits:
The solution for signals in an electric circuit can be found without writing integrodif-
ferential equations if the circuit operations and signals are represented with their Laplace
transform equivalents. [In this subsection the Laplace transform means the unilateral
Laplace transform and we drop the subscript I in X,(s).] We refer to a circuit produced
from these equivalents as a transform circuit. In order to use this technique, we require the
Laplace transform models for individual circuit elements. These models are developed in
the following discussion and are shown in Fig. 3-10. Applications of this transform model
technique to electric circuits problems are illustrated in Probs. 3.40 to 3.42.
I. Signal Sources:
where u(t) and i(t) are the voltage and current source signals, respectively.
2. Resistance R:
