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EXPONENTIAL FOURIER SERIES
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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
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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
$
%
' (
)
$
)
%
' (
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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
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"
3
1
!
"
3
1
!
"
$
)
%
' (
This expression holds for all values of
, positive, negative, and
zero.
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Example: Half-rectified sine wave.
A
T t0 T/20
"
!
$
)
%
' (
#
1
$
%
' (
3
$
)
%
' (
$
)
%
' (
#
1
$
%
'
)
(
3
$
)
%
'
(
3
$
%
)
3
"
"
3
$
%
3
"
"
"or 3
"
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Now notice
$
%
"
1
!
"
3
3
"
Thus we get
for
odd
"
and
"
"
3
for
even
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The special cases of
"
have to be handled separately. For
",
#
1
$
%
' (
3
$
)
%
' (
$
)
%
' (
#
1
"
3
$
)
%
' (
1
Similarly, for
3
",
)
3
1
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Therefore, the exponential fourier series is
3
"
$
)
%
' (
3
$
)
%
' (
1
$
)
%
' (
3
1
$
%
' (
3
$
%
' (
3
"
$
%
' (
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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
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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
.
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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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