Fundamentals of System Analysis and Synthesis - Digital Signal Processing in Power System Protection and Control - page 33

Z

s

a 0 ¼ 2
T 0

s
T 0 :

A dt ¼ A

0

Graphical presentation of the calculated coefficients as a function of relative
angular frequency for the signal magnitude A equal unity and s = T 0 ¼ 0 : 25 is
shown in Fig. 4.1 b.
Further, according to Eq. 4.2 one obtains:

Z A

np sin

:

T 0

c n ¼ 1
T 0

s
T 0

s
T 0

A exp ð jnx 0 t Þ dt ¼ exp jnp np

0

It is seen that the coefficients are complex numbers defining in an integrated
way both magnitudes and arguments of particular signal components. The absolute
values of c n represent magnitudes of given frequency components:

c jj¼ A
np

s
T 0

sin np ;

with

s
T 0

lim

n ! 0 c jj¼ A

that are shown graphically as a function of frequency in Fig. 4.1 c. The graph was
prepared for A = 1 and s = T 0 ¼ 0 : 25 ; similarly as in previous example. Intro-
ducing calculated values c n to ( 4.2 ) one gets the complex Fourier series of the
considered signal:

exp ½ jnx 0 ð t s = 2 Þ:

X

1

x ð t Þ¼ A
p

1
n sin

s
T 0

np

n ¼1

4.3 Fourier Transform

Non-periodic signals can be represented in frequency domain as well. One of
many transitions between Fourier series and Fourier transform (integral) rely on
representing non-periodic functions as periodic ones with their period approaching
infinity. This transition causes that instead of samples in frequency domain (as in
Fourier series) one obtains continuous spectra. Without discussing any particular
assumptions concerning the transition it can be stated that Fourier transformation
concerns energy signals, i.e. satisfying the condition: