Digital Signal Processing Reference
In-Depth Information
Fig. 4.12. Waveform
reconstructed from the first 1000
CTFS coefficients in Example
4.11.
1
0.5
0
−0.5
t
−1
−1
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
4.5 Exponential Fourier series
In Section 4.4, we considered the trigonometric CTFS expansion using a set
of sinusoidal terms as the basis functions. An alternative expression for the
CTFS is obtained if complex exponentials
{
exp( j
n
ω
0
t
)
, for
n
∈
Z
, are used
as the basis functions to expand a CT periodic signal. The resulting CTFS
representation is referred to as the exponential CTFS, which is defined below.
Definition 4.7
An arbitrary periodic function x
(
t
)
with a fundamental period
T
0
can be expressed as follows:
∞
D
n
e
j
n
ω
0
t
,
x
(
t
)
=
(4.43)
m
=
0
where the exponential CTFS coefficients D
n
are calculated as
=
1
T
0
−
j
n
ω
0
t
d
t
,
D
n
x
(
t
)e
(4.44)
T
0
ω
0
being the fundamental frequency given by
ω
0
=
2
π/
T
0
.
Equation (4.43) is known as the exponential CTFS representation of
x
(
t
). Since
the basis functions corresponding to the trigonometric and exponential CTFS
are related by Euler's identity,
−
j
n
ω
0
t
e
=
cos(
n
ω
0
t
)
−
j sin(
n
ω
0
t
)
,
it is intuitively pleasing to believe that the exponential and trigonometric CTFS
coefficients are also related to each other. The exact relationship is derived by
expanding the trigonometric CTFS series as follows:
∞
x
(
t
)
=
a
0
+
(
a
n
cos(
n
ω
0
t
)
+
b
n
sin(
n
ω
0
t
))
n
=
1
∞
∞
a
n
2
b
n
(e
j
n
ω
0
t
−
j
n
ω
0
t
)
+
2j
(e
j
n
ω
0
t
−
j
n
ω
0
t
)
.
=
a
0
+
+
e
−
e
n
=
1
n
=
1
Combining terms with the same exponential functions, we obtain
∞
∞
+
1
2
+
1
2
−
j
b
n
)e
j
n
ω
0
t
−
j
n
ω
0
t
.
x
(
t
)
=
a
0
(
a
n
(
a
n
+
j
b
n
)e
n
=
1
n
=
1
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