exponential fourier series

8/8/2019 Exponential Fourier Series

1/12

EXPONENTIAL FOURIER SERIES

EXPONENTIAL FOURIER SERIES 1


2/12

We have already learnt that any periodic function can be

experessed in the form

!

The sines and cosines can be expressed as

"

#

$

%

' (

$

)

%

' (

!

"

#

1

$

%

' (

3

$

)

%

' (

Substituting, we get

$

%

' (

$

)

%

' (

#

$

%

' (

3

$

)

%

' (

#

1



3/12

In order to simplify, we group together the similar terms:

3

1

#

$

%

' (

1

#

$

)

%

' (

Let us now introduce new coefficients:

3

1

#

)

1

#

giving the form

$

%

' (

)

$

)

%

' (



4/12

If we let

range from3

to

including zero, we get thecompact representation

)

$

%

' (

This is the exponential form of the Fourier series.

Now we have to obtain the complex coefficients

.

Substitute

#

for

#

!

for



5/12

"

3

1

!

"

3

1

!

"

$

)

%

' (

This expression holds for all values of

, positive, negative, and

zero.



6/12

Example: Half-rectified sine wave.

A

T t0 T/20

"

!

$

)

%

' (

#

1

$

%

' (

3

$

)

%

' (

$

)

%

' (

#

1

$

%

'

)

(

3

$

)

%

'

(

3

$

%

)

3

"

"

3

$

%

3

"

"

"or 3

"



7/12

Now notice

$

%

"

1

!

"

3

3

"

Thus we get

for

odd

"

and

"

"

3

for

even



8/12

The special cases of

"

have to be handled separately. For

",

#

1

$

%

' (

3

$

)

%

' (

$

)

%

' (

#

1

"

3

$

)

%

' (

1

Similarly, for

3

",

)

3

1



9/12

Therefore, the exponential fourier series is

3

"

$

)

%

' (

3

$

)

%

' (

1

$

)

%

' (

3

1

$

%

' (

3

$

%

' (

3

"

$

%

' (



10/12

How do we interpret the exponential Fourier series?A term

$

%

' (

represents a rotating phasor (except

which is stationary). Themagnitude of the phasor is

"

#

Terms of the form

$

%

' ( rotate in a counter-clockwise direction,

and those of the form

$

)

%

' ( rotate in clockwise direction. The

angle of the phasor at

is

!

)

3



11/12

cn

c

c

c

a /2

b /2

n

n

n

n

n

n

n

The imaginary parts cancel, giving the real value. Thus complex

conjugate components

and

)

must occur.

The function is obtained as an infinite summation of spinning

phasors rotating at speeds that are integral multiples of the

fundamental frequency

.



12/12

Because of the even symmetry of the magnitude plot, it is

sometimes the practice to consider only the positive side (

is

divided into two halves

#each, with a half assigned to the

positive side).

This way the summation of the components for only positive

yields half the magnitude of the function.

cn| |

0 1 3 4 5 626 5 4 3 1 2

n


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