The properties of the Fourier series are important in applying it to signal
analysis and to interpreting it. The main properties are given here using the
notation that the Fourier series of a real valued function
x
(
t
)
x
(
t
)
over
{
0
≤
t
≤
T
}
{
0
≤
t
≤
T
}
is given by
ℱ
{
x
(
t
)
}
=
c
(
k
)
ℱ
{
x
(
t
)
}
=
c
(
k
)
and
x
˜
(
t
)
x
˜
(
t
)
denotes the periodic extensions of
x
(
t
)
x
(
t
)
.
-
Linear:
ℱ
{
x
+
y
}
=
ℱ
{
x
}
+
ℱ
{
y
}
ℱ
{
x
+
y
}
=
ℱ
{
x
}
+
ℱ
{
y
}
Idea
of superposition. Also scalability:
ℱ
{
a
x
}
=
a
ℱ
{
x
}
ℱ
{
a
x
}
=
a
ℱ
{
x
}
-
Extensions of
x
(
t
)
x
(
t
)
:
x
˜
(
t
)
=
x
˜
(
t
+
T
)
x
˜
(
t
)
=
x
˜
(
t
+
T
)
x
˜
(
t
)
x
˜
(
t
)
is periodic.
-
Even and Odd Parts:
x
(
t
)
=
u
(
t
)
+
j
v
(
t
)
x
(
t
)
=
u
(
t
)
+
j
v
(
t
)
and
C
(
k
)
=
A
(
k
)
+
j
B
(
k
)
=
|
C
(
k
)
|
e
j
θ
(
k
)
C
(
k
)
=
A
(
k
)
+
j
B
(
k
)
=
|
C
(
k
)
|
e
j
θ
(
k
)
u
u
v
v
A
A
B
B
|
C
|
|
C
|
θ
θ
even
0
even
0
even
0
odd
0
0
odd
even
0
0
even
0
even
even
π
/
2
π
/
2
0
odd
odd
0
even
π
/
2
π
/
2
-
Convolution: If continuous cyclic convolution is defined
by
y
(
t
)
=
h
(
t
)
∘
x
(
t
)
=
∫
0
T
h
˜
(
t
−
τ
)
x
˜
(
τ
)
ⅆ
τ
y
(
t
)
=
h
(
t
)
∘
x
(
t
)
=
∫
0
T
h
˜
(
t
−
τ
)
x
˜
(
τ
)
ⅆ
τ
then
ℱ
{
h
(
t
)
∘
x
(
t
)
}
=
ℱ
{
h
(
t
)
}
ℱ
{
x
(
t
)
}
ℱ
{
h
(
t
)
∘
x
(
t
)
}
=
ℱ
{
h
(
t
)
}
ℱ
{
x
(
t
)
}
-
Multiplication: If discrete convolution is defined
by
e
(
n
)
=
d
(
n
)
*
c
(
n
)
=
∑
m
=
−
∞
∞
d
(
m
)
c
(
n
−
m
)
e
(
n
)
=
d
(
n
)
*
c
(
n
)
=
∑
m
=
−
∞
∞
d
(
m
)
c
(
n
−
m
)
then
ℱ
{
h
(
t
)
x
(
t
)
}
=
ℱ
{
h
(
t
)
}
*
ℱ
{
x
(
t
)
}
ℱ
{
h
(
t
)
x
(
t
)
}
=
ℱ
{
h
(
t
)
}
*
ℱ
{
x
(
t
)
}
This
property is the inverse of property 4 and vice versa.
-
Parseval:
1
T
∫
0
T
|
x
(
t
)
|
2
ⅆ
t
=
∑
k
=
−
∞
∞
|
C
(
k
)
|
2
1
T
∫
0
T
|
x
(
t
)
|
2
ⅆ
t
=
∑
k
=
−
∞
∞
|
C
(
k
)
|
2
This
property says the energy calculated in the time domain is the same as that
calculated in the frequency (or Fourier) domain.
-
Shift:
ℱ
{
x
˜
(
t
−
t
0
)
}
=
C
(
k
)
e
−
j
2
π
t
0
k
/
T
ℱ
{
x
˜
(
t
−
t
0
)
}
=
C
(
k
)
e
−
j
2
π
t
0
k
/
T
A
shift in the time domain results in a linear phase shift in the frequency
domain.
-
Modulate:
ℱ
{
x
(
t
)
e
j
2
π
K
t
/
T
}
=
C
(
k
−
K
)
ℱ
{
x
(
t
)
e
j
2
π
K
t
/
T
}
=
C
(
k
−
K
)
Modulation
in the time domain results in a shift in the frequency domain. This property
is the inverse of property 7.
-
Orthogonality of basis functions:
∫
0
T
e
−
j
2
π
m
t
/
T
e
j
2
π
n
t
/
T
ⅆ
t
=
T
δ
(
n
−
m
)
=
{
T
if
n
=
m
0
if
n
≠
m
.
∫
0
T
e
−
j
2
π
m
t
/
T
e
j
2
π
n
t
/
T
ⅆ
t
=
T
δ
(
n
−
m
)
=
{
T
if
n
=
m
0
if
n
≠
m
.
(1)
