Fundamentals of System Analysis and Synthesis - Digital Signal Processing in Power System Protection and Control - page 31

4.2 Fourier Series

Let there be given a periodic time function (signal) having a period of T 0 , i.e. such
that x ð t þ kT 0 Þ¼ x ð t Þ; k 2 C : The function can be represented by infinite series
called Fourier series according to the following equation [ 2 , 9 ]:

x ð t Þ¼ X

1

½ a n cos ð nx 0 t Þþ b n sin ð nx 0 t Þ ; ð 4 : 1 Þ

n ¼ 0

where

Z

T 0

a n ¼ 2
T 0

x ð t Þ cos ð nx 0 t Þ dt ;

0

Z

T 0

b n ¼ 2
T 0

x ð t Þ sin ð nx 0 t Þ dt ;

0

Z

T 0

a 0 ¼ 1
T 0

x ð t Þ dt ;

0

x 0 ¼ 2p
T 0

:

It can be noted here that above signal equivalent is obtained with minimization
of mean square error and that the set of functions sin, cos is a complete set of
orthogonal functions. There are also other sets of orthogonal functions, for
instance Walsh, Haar and others which can be used as well for analysis of periodic
signals.
The formula of Fourier series ( 4.1 ) can be expressed multifold. If one uses the
Euler substitution:

cos ð a Þ¼ 1

2 ½ exp ð ja Þþ exp ð ja Þ;

sin ð a Þ¼ 1

2j ½ exp ð ja Þ exp ð ja Þ;

where j ¼

p

is an imaginary operator, then after simple rearrangements the
complex Fourier series is yielded, in the form:

1

x ð t Þ¼ X

1

c n exp ð jnx 0 t Þ; ð 4 : 2 Þ

n ¼ 0