Oscillating systems 5

0

0

0

0

1

()d,

2

()cos2 d

2

and ( ) sin 2 d.

T

T

k

T

kk

axtt

T

axt ft

T

bxt ft

T

π

π

=

=

=

∫

∫

∫

kt

t

k)

(1.4)

a 0 is the mean value of the function. For a periodic oscillation this value is by
definition equal to zero but we shall for completeness retain it in the following
derivations. In standard textbooks on mathematics one will find the coefficients readily
calculated for a large number of functions. Before we give some examples, two
alternative expressions for such Fourier series are worth looking into, expressions that
are more common in signal analysis. The first alternative form of Equation (1.1) is

(^0) (
1
() kkcos 2
k
xt c c πftθ
∞

=+∑ +, (1.5)

where we may easily show that the Fourier coefficients (Fourier amplitudes) ck and the
Fourier phase angles θk are given by the following expressions:

00

22

,

,

and arctg.

kkk

k

k

k

ca

cab

b

a

θ

=

=+

=

Introducing complex numbers we may derive the second alternative form of Equation
(1.1). Moivre’s formula gives

cos (nnθθ)+⋅j sin( )=ejnθ,

where j is the complex unit, and we may then write

() j2 k

k

ft

k

k

xt X eπ

=+∞

=−∞

=∑ , (1.6)

()

00

jj 2

0

where

1

and kk( ) d with 1, 2, 3,...

T

ft

kk k

Xa

Xf Xe xte t k

T

−−θπ

=

==∫ =±±±

(1.7)

The quantity |Xk| is the modulus of the complex Fourier amplitude. Even if x(t) is a real
function we may mathematically represent it in complex form using both positive and
negative frequencies. There is, however, no reason to ascribe any physical significance to
these negative frequencies. They show up at an intermediate stage in the calculations and