Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 26

The Double-Sided Laplace Transform

The double-sided Laplace transform (DLT) definition incorporates an integral
which is evaluated across all possible values of time (both negative and positive):

X d ð s Þ¼L d f x ð t Þg¼ Z

1

x ð t Þ e st dt ;

1

where s = r ? jx is the complex frequency variable. Note that:

L d f x ð t Þg¼F x ð t Þ e st

f g:

The inverse DLT, L d f X d ð s Þg , can be used to recover x(t) as follows:

r 1 þ1

Z

x ð t Þ¼L d f X d ð s Þg¼ 1 X d ð s Þ e st ds ;

2pj

r 1 1

where r 1 is any arbitrary value of r. Note that L 1 requires integration in the

d

complex plane.

The Single-Sided Laplace Transform

In real-world applications one normally deals with casual systems in which the
impulse response is only non-zero for positive values of time. In recognition of this
fact, the single-sided Laplace transform (SLT) is defined to allow for only causal
signals and systems. That is, the integral within its definition is only evaluated for
positive and zero values of time. The SLT is very useful in calculating the response
of a system to a causal input, especially when the system is described by a linear
constant coefficient differential equation with non-zero initial conditions.
Note that the properties of the DLT are not exactly the same as those of SLT.
This topic will concentrate only on the SLT, which will be referred to hereafter
simply as the Laplace Transform (LT). Its definition is:

X ð s Þ¼Lf x ð t Þg¼ Z

1

x ð t Þ e st dt ;

0

Note that 0 - is used in the above definition rather than 0 to allow for analyzing
delta functions x(t) = d(t). The inverse transform (ILT) is given by:

Z

r 1 þ1

x ð t Þ¼L 1 f X ð s Þg¼ 1
2pj

X d ð s Þ e st ds :

r 1 1