Higher Engineering Mathematics

Fourier series

74

The complex or exponential form of

a Fourier series

74.1 Introduction

The form used for the Fourier series in Chapters 69 to

73 consisted of cosine and sine terms. However, there

is another form that is commonly used—one that

directly gives the amplitude terms in the frequency

spectrum and relates to phasor notation. This form

involves the use of complex numbers (see Chapters

23 and 24). It is called theexponentialorcomplex

formof a Fourier series.

74.2 Exponential or complex notation

It was shown on page 264, equations (4) and (5) that:

ejθ=cosθ+jsinθ (1)

and e−jθ=cosθ−jsinθ (2)

Adding equations (1) and (2) gives:

ejθ+e−jθ=2 cosθ

from which, cosθ=

ejθ+e−jθ

2

(3)

Similarly, equation (1) – equation (2) gives:

ejθ−e−jθ= 2 jsinθ

from which, sinθ=

ejθ−e−jθ

2 j

(4)

Thus, from page 676, the Fourier seriesf(x) over

any rangeL,

f(x)=a 0 +

∑∞

n= 1

[

ancos

(

2 πnx

L

)

+bnsin

(

2 πnx

L

)]

may be written as:

f(x)=a 0 +

∑∞

n= 1

[

an

(

ej

2 πnx

L +e−j

2 πnx

L

2

)

+bn

(

ej

2 πnx

L −e−j

2 πnx

L

2 j

)]

Multiplying top and bottom of thebnterm by−j

(and remembering thatj^2 =−1) gives:

f(x)=a 0 +

∑∞

n= 1

[

an

(

ej

2 πnx

L +e−j

2 πnx

L

2

)

−jbn

(

ej

2 πnx

L −e−j

2 πnx

L

2

)]

Rearranging gives:

f(x)=a 0 +

∑∞

n= 1

[(

an−jbn

2

)

ej

2 πnx

L

+

(

an+jbn

2

)

e−j

2 πnx

L

]

(5)

The Fourier coefficients a 0 ,an and bn may be

replaced by complex coefficientsc 0 ,cnandc−nsuch

that

c 0 =a 0 (6)

cn=

an−jbn

2

(7)

and c−n=

an+jbn

2

(8)

wherec−nrepresents the complex conjugate ofcn

(see page 251).

Thus, equation (5) may be rewritten as:

f(x)=c 0 +

∑∞

n= 1

cnej

2 πnx

L +

∑∞

n= 1

c−ne−j

2 πnx

L (9)