Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University.
1
Course Notes on Digital Signal Processing
Discrete Fourier Series (DFS)
Consider a periodic sequence
[ ]
p
x
n
that is periodic with period ‘
N
’.,
so that
[
]
[ ]
p
p
x
n
rN
x
n
for any value of ‘
r
’.
Such a sequence
[ ]
p
x
n
can be represented by sum of sine and cosine sequences or
equivalently complex exponential sequences with frequencies that are integer multiples of
fundamental frequency
2
N
associated with periodic sequence.
The Discrete Fourier Series ‘Analysis’ and ‘Synthesis’ equations are expressed as
1
0
[
]
[ ]
,
0,1, 2,...,
1
N
Kn
p
p
N
n
X
K
x
n W
K
N
Analysis equation
1
0
1
[ ]
[
]
,
0,1, 2,...,
1
N
Kn
p
p
N
K
x
n
X
K W
n
N
N
Synthesis equation
where
2
j
N
N
W
e
, and both
[ ]
p
x
n
and
[
]
p
X
K
are periodic sequences with period
‘
N
’
.
Example 1
:
Determine the Fourier series coefficients of a periodic signal shown in
Figure.
Solution:
From the given figure, the periodic signal can be expressed as
[ ]
{0, 1, 2, 3}
x n
with period
N
= 4. Therefore
1
0
3
4
0
0
3
2
4
4
4
4
3
2
4
4
4
[
]
[ ]
,
0,1, 2, ...,
1
[ ]
,
0,1, 2, 3
[0]
[1]
[2]
[3]
,
0,1, 2, 3
0
2
3
,
0,1, 2, 3
N
Kn
p
p
N
n
Kn
p
n
K
K
K
p
p
p
p
K
K
K
X
K
x
n W
K
N
x
n W
K
x
W
x
W
x
W
x
W
K
W
W
W
K
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Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University.
2
,
2
0
0
0
0
4
4
4
4
2
1
2
3
1
4
2
4
4
4
4
2
4
6
2
1
1
4
4
4
4
4
4
3
6
9
3
4
4
4
4
[0]
0
2
3
0
1
2
3
6
1
[1]
2
3
2( 1)
3
2
2
cos
sin
2
2
[2]
2
3
1
2
3( 1)
2
1
[3]
2
3
2
2)
K
j
K
N
p
N
j
j
p
p
p
X
W
W
W
W
e
W
X
W
W
W
j
j
j
W
e
e
j
j
X
W
W
W
W
W
W
j
j
X
W
W
W
j
W
W
X
o
o
o
o
0
0
0
2
1
4
4
1
[
]
{6, ( 2
2),
2, ( 2
2)}
[
]
{6
0 , 2.8284
135 , 2
180 , 2.8284
135 }

[
]
{6, 2.8284, 2, 2.8284}
[
]
{0,135 ,180 ,
135 }
p
p
p
p
W
j
j
X
K
j
j
X
K
X
K
X
K
X
The Discrete Fourier Series can be written as
1
3
0
0
0
4
4
4
4
3
2
2
3
2
2
1
1
[ ]
[
]
,
0,1, 2, ...,
1
[
]
,
0,1, 2,3
4
1
6
( 2
2)
2
( 2
2)
4
1
6
( 2
2)
2
( 2
2)
4
3
1
1
2
2
2
2
N
Kn
Kn
p
p
p
N
N
K
K
n
n
n
j
n
j
n
j n
j
n
j
n
j n
x
n
X
K W
n
N
X
K W
n
N
W
j
W
W
j
W
j
e
e
j
e
j
j
e
e
e