Page 121 - Schaum's Outline of Theory and Problems of Signals and Systems
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Chapter 3
Laplace Transform and Continuous-Time
LTI Systems
3.1 INTRODUCTION
A basic result from Chapter 2 is that the response of an LTI system is given by
convolution of the input and the impulse response of the system. In this chapter and the
following one we present an alternative representation for signals and LTI systems. In this
chapter, the Laplace transform is introduced to represent continuous-time signals in the
s-domain (s is a complex variable), and the concept of the system function for a
continuous-time LTI system is described. Many useful insights into the properties of
continuous-time LTI systems, as well as the study of many problems involving LTI systems,
can be provided by application of the Laplace transform technique.
3.2 THE LAPLACE TRANSFORM
In Sec. 2.4 we saw that for a continuous-time LTI system with impulse response h(t),
the output y(0 of the system to the complex exponential input of the form e" is
where
A. Definition:
The function H(s) in Eq. (3.2) is referred to as the Laplace transform of h(t). For a
general continuous-time signal x(t), the Laplace transform X(s) is defined as
The variable s is generally complex-valued and is expressed as
The Laplace transform defined in Eq. (3.3) is often called the bilateral (or two-sided)
Laplace transform in contrast to the unilateral (or one-sided) Laplace transform, which is
defined as
