Fundamentals of System Analysis and Synthesis - Digital Signal Processing in Power System Protection and Control - page 36

Fig. 4.3 Periodic signal
a and its Fourier transform b

(a)

1
) x ( t

0

-1

t

-2 -1 0 1 2

T

(b)

1.5

X ( j ω)

1

π

0.5

0

ω

-2 -1 0 1 2

ω

0

X ð r þ jx Þ¼ Z

1

x ð t Þ exp ð r þ jx Þ t ½ dt : ð 4 : 6 Þ

1

The inverse transformation can be done according to:

Z

1

x ð t Þ¼ 1
2pj

X ð r þ jx Þ exp r þ jx ½ ð Þ t dx : ð 4 : 7 Þ

1

Substituting s ¼ r þ jx (i.e. ds ¼ j dx Þ one obtains, [ 2 , 3 ]:

X ð s Þ¼ Z

1

x ð t Þ exp ð st Þ dt ð 4 : 8 Þ

1

and

r þ j 1

Z

x ð s Þ¼ 1
2pj

X ð s Þ exp ð st Þ ds ; ð 4 : 9 Þ

r j 1

which represent the Laplace transform and its inverse.
The main features of the Laplace transform that facilitate calculation for more
complex signals are collected in Table 4.1 , whereas the Laplace transforms of
selected most commonly analyzed signals are given in Table 4.2 .

Example 4.4 Applying the definition ( 4.8 ) and/or the theorems from Table 4.1
calculate the Laplace transforms of the signals: (a) x(t) = 1(t), (b) x ð t Þ¼

cos ð xt Þ 1 ð t Þ: