Higher Engineering Mathematics

Fourier series

72

Fourier series over any range

72.1 Expansion of a periodic function

of periodL

(a) A periodic function f(x) of period L

repeats itself when x increases by L, i.e.

f(x+L)=f(x). The change from functions

dealt with previously having period 2πto func-

tions having periodLis not difficult since it may

be achieved by a change of variable.

(b) To find a Fourier series for a functionf(x)in

the range−

L

2

≤x≤

L

2

a new variableuis intro-

duced such thatf(x), as a function ofu, has

period 2π.Ifu=

2 πx

L

then, when x=−

L

2

,

u=−π and when x=

L

2

,u=+π. Also, let

f(x)=f

(

Lu

2 π

)

=F(u). The Fourier series for

F(u) is given by:

F(u)=a 0 +

∑∞

n= 1

(ancosnu+bnsinnu),

wherea 0 =

1

2 π

∫π

−π

F(u)du,

an=

1

π

∫π

−π

F(u) cosnudu

and bn=

1

π

∫π

−π

F(u) sinnudu

(c) It is however more usual to change the formula

of para. (b) to terms ofx. Sinceu=

2 πx

L

, then

du=

2 π

L

dx,

and the limits of integration are −

L

2

to+

L

2

instead of from−πto+π. Hence the Fourier

series expressed in terms ofxis given by:

f(x)=a 0 +

∑∞

n= 1

[

ancos

(

2 πnx

L

)

+bnsin

(

2 πnx

L

)]

where, in the range−

L

2

to+

L

2

:

and

a 0 =

1

L

∫ L

2

−L

2

f(x)dx,

an=

2

L

∫ L

2

−L

2

f(x) cos

(

2 πnx

L

)

dx

bn=

2

L

∫ L

2

−L

2

f(x) sin

(

2 πnx

L

)

dx

The limits of integration may be replaced by any

interval of lengthL, such as from 0 toL.

Problem 1. The voltage from a square wave

generator is of the form:

v(t)=

{

0, − 4 <t< 0

10, 0<t< 4

and has a period of 8 ms.

Find the Fourier series for this periodic function.

The square wave is shown in Fig. 72.1. From para.

(c), the Fourier series is of the form:

v(t)=a 0 +

∑∞

n= 1

[

ancos

(

2 πnt

L

)

+bnsin

(

2 πnt

L

)]