Page 125 - Schaum's Outline of Theory and Problems of Signals and Systems

LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS [CHAP. 3

where a,,, equals the maximum real part of any of the poles of X(s). Thus, the ROC is
a half-plane to the right of the vertical line Reb) = a,,, in the s-plane and thus to the
right of all of the poles of Xb).
Property 4: If x(t) is a left-sided signal, that is, x(t) = O for t > t, > -=, then the ROC is of the
form

where a,,, equals the minimum real part of any of the poles of X(s). Thus, the ROC is
a half-plane to the left of the vertical line Re(s) =amin in the s-plane and thus to the left
of all of the poles of X(s).
Property 5: If x(t) is a two-sided signal, that is, x(t) is an infinite-duration signal that is neither
right-sided nor left-sided, then the ROC is of the form

where a, and a, are the real parts of the two poles of X(s). Thus, the ROC is a vertical
strip in the s-plane between the vertical lines Re(s) = a, and Re(s) = a,.
Note that Property 1 follows immediately from the definition of poles; that is, X(s) is
infinite at a pole. For verification of the other properties see Probs. 3.2 to 3.7.

33 LAPLACE TRANSFORMS OF SOME COMMON SIGNALS

A. Unit Impulse Function S( t ):
Using Eqs. (3.3) and (1.20), we obtain

J[s(t)] = /- s(t)e-" dt = 1 all s
- m

B. Unit Step Function u(t 1:

where O+ = lim, , + €1.
"(0

C. Laplace Transform Pairs for Common Signals:

The Laplace transforms of some common signals are tabulated in Table 3-1. Instead of
having to reevaluate the transform of a given signal, we can simply refer to such a table
and read out the desired transform.

3.4 PROPERTIES OF THE LAPLACE TRANSFORM
Basic properties of the Laplace transform are presented in the following. Verification
of these properties is given in Probs. 3.8 to 3.16.