Chapter 71

The complex or

exponential form

of a Fourier series

71.1 Introduction

The form used for the Fourier series in Chapters 66 to
70 consisted of cosine and sine terms. However, there is
another form that is commonly used—one that directly
gives the amplitudeterms in thefrequency spectrum and
relates to phasor notation.This form involves the use of
complex numbers (see Chapters 20 and 21). It is called
theexponentialorcomplex formof a Fourier series.

71.2 Exponential or complex

notation

It was shown on page 226, equations (4) and (5) that:

ejθ=cosθ+jsinθ (1)

and e−jθ=cosθ−jsinθ (2)

Adding equations (1) and (2) gives:

ejθ+e−jθ=2cosθ

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 630, the Fourier series f(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 πLnx

−e−j

2 πLnx

2

)]