Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 27

Properties of the LT

The properties of the LT can be found in Tables. Some of the most useful prop-
erties are:

ð 1 Þ dx ð t Þ
dt

! sX ð s Þ x 0 ðÞ

L

ð 2 Þ Z

t

X ð s Þ
s

x ð k Þ dk L !

0

These properties effectively transform differentiation and integration into
algebraic quantities. Because of these properties the LT can be used to transform
differential equations into algebraic equations. This has application in many areas,
most notably the analysis of electric circuits.

Region of Convergence of the LT

The region of convergence (ROC) is the region of the s-plane in which the LT is
convergent (i.e., has finite values).
Example Find the Laplace transform and its ROC for the signal x(t) = e a tu(t),
where a is a constant.

Solution:

X ð s Þ¼ Z x ð t Þ e st dt ¼ Z

1 1

e at e st dt

0 0

¼ Z

1

1

e ð a þ s Þ t dt ¼ e ð a þ s Þ t
a þ s

0

0

h n o e 0 i

¼ 1
a þ s

t !1 e ð a þ r Þ t e jxt

lim

The term e -jxt is always bounded, the ROC therefore depends on only the term
e -(a+r)t . Now,

t !1 e ð a þ r Þ t ¼ 1;

when ð a þ r Þ [ 0orr\ a

lim ð 1 : 23 Þ

0 ; when ð a þ r Þ \0orr [ a

Hence, the ROC is the region defined by r = Re{s} [ -a (see Fig. 1.17 ). For
values of a which do lead to a convergent LT, the LT is given by:

X ð s Þ¼ 1

s þ a ð note that a can be positive or negative Þ: