Laplace Transform
INTRODUCTION:
A basic result 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.
THE LAPLACE TRANSFORM:
we know that for a continuous-time LTI system with impulse response h(t), the output y(t) of the system to the complex exponential input of the form est is
where
Definition:
The function H(s) in the above Eq 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 the above Eq. is often called the bilateral (or two-sided) Laplace transform in contrast to the unilateral (or one-sided) Laplace transform, which is defined as
where 0-= limζ-0(O - ζ ) . Clearly the bilateral and unilateral transforms are equivalent only if x(t) = 0 for t < 0. The unilateral Laplace transform is discussed in. We will omit the word "bilateral" except where it is needed to avoid ambiguity. The above Equation is sometimes considered an operator that transforms a signal x(t) into a function X(s) symbolically represented by
and the signal x(t) and its Laplace transform X(s) are said to form a Laplace transform pair denoted as
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