Fundamentals of System Analysis and Synthesis - Digital Signal Processing in Power System Protection and Control - page 34

Fig. 4.2 Signal a and its
Fourier transform ˇ

(a)

x ( t )

1

t

T 0

1

1

(b)

X ( j )

T

0.5

0

-0.5

-4 -2 0 2 4

2 T

π

Z

1

j x ð t Þ j dt\ 1:

1

Whatever the way to get Fourier transform a pair of Fourier integrals allows to
go from time to frequency domain and the other way round, [ 5 , 8 ]. The integrals
describing simple and inverse transformations are as follows:

X ð jx Þ¼ Z

1

x ð t Þ exp ð jxt Þ dt ; ð 4 : 3 Þ

1

Z

1

x ð t Þ¼ 1
2p

X ð jx Þ exp ð jxt Þ dx : ð 4 : 4 Þ

1

Example 4.2 Applying Fourier transform determine spectrum of the signal x(t)
that is equal 1.0 for the time range from zero to T and equal 0.0 otherwise
(Fig. 4.2 a). Draw the magnitude of this spectrum as a function of frequency.

Solution Application of the formula ( 4.3 ) yields:

X ð jx Þ¼ Z

T

exp ð jxt Þ dt ¼ T sin ð p Þ
p

exp ð jp Þ;

0

where

p ¼ xT
2