Review of Analog Signal Processing Basics - Digital Signal Processing: Fundamentals and Applications - page 792

TABLE B.5 Laplace Transform Table (continued)

Laplace Transform
XðsÞ [ LðxðtÞÞ

Line Time Function xðtÞ

sin ut cos ut þ B aA

u

e at u t

As þ B
ðs þ aÞ

10 A

2

þ u

2

n !
s nþ1

t n uðtÞ

11a

u t

1
ðn

1
s n

11b

Þ ! t n1

1

n !
ðs þ aÞ nþ1

e at t n uðtÞ

12a

u t

1
ðn

1
ðs þ aÞ n

12b

Þ ! e at t n1 1

A
s þ a þ ju

A
s þ a ju þ

ðutÞÞe at uðtÞ 13 ð 2Real ðAÞ cos ðutÞ 2Imag ðAÞ sin

dxðtÞ
dt

14

0 Þ

sXðsÞxð

Z t
0 xðtÞdt

XðsÞ
s

15

e as XðsÞ 16 xðt aÞuðt aÞ

e at xðtÞuðtÞ 17 Xðs þ aÞ

In Example B.9, we examine the Laplace transform in light of its definition.

EXAMPLE B.9
Derive the Laplace transform of the unit step function.
Solution:
By the definition in Equation (B.23) ,

Z N

u t e st dt X s ¼

0

Z N N

e st
s

e N
s

e 0

s ¼ s

e st dt ¼

¼ ¼

0

0

The answer is consistent with the result listed in Table B.5 . Now we use the results in Table B.5 to find the Laplace
transform of a function.