Digital Signal Processing Reference
In-Depth Information
for any pair of functions taken from the set
{
p
n
(
t
)
}
. Section 4.3 proves that
the complex exponentials
{
exp( j
n
ω
0
t
)
}
, for
−∞
<
n
<
∞
, and sinusoidal
functions
{
sin(
n
ω
0
t
), 1, cos(
m
ω
0
t
)
}
, for 0
<
n
,
m
<
∞
, form two complete
orthogonal sets over any interval [
t
1
,
t
1
+
2
π/ω
0
] of duration
T
0
=
2
π/ω
0
.We
refer to
ω
0
as the angular frequency and to its inverse
T
0
2
π/ω
0
as the funda-
mental period. Expressing a periodic signal
x
(
t
) as a linear combination of the
sinusoidal set of functions
{
sin(
n
ω
0
t
), 1, cos(
m
ω
0
t
)
}
leads to the trigonometric
representation of the CTFS. The trigonometric CTFS is defined as follows:
=
∞
x
(
t
)
=
a
0
+
(
a
n
cos(
n
ω
0
t
)
+
b
n
sin(
n
ω
0
t
))
,
n
=
1
where
ω
0
=
2
π/
T
0
is the fundamental frequency of
x
(
t
) and coefficients
a
0
,
a
n
,
and
b
n
are referred to as the trigonometric CTFS coefficients. The coefficients
are calculated using the following formulas:
1
T
0
a
0
=
x
(
t
)d
t
,
T
0
2
T
0
a
n
=
x
(
t
) cos(
n
ω
0
t
)d
t
,
T
0
and
=
2
T
0
b
n
x
(
t
) sin(
n
ω
0
t
)d
t
.
T
0
The trigonometric CTFS is presented in Section 4.4, while its counterpart, the
exponential CTFS, is covered in Section 4.5. The exponential CTFS is obtained
by expressing the periodic signal
x
(
t
) as a linear combination of complex expo-
nentials
{
exp( j
n
ω
0
t
)
}
and is given by
m
=−∞
D
n
e
j
n
ω
0
t
,
where the exponential CTFS coefficients
D
n
are calculated using the following
expression:
∞
x
(
t
)
=
=
1
T
0
−
j
n
ω
0
t
d
t
.
D
n
x
(
t
)e
T
0
The exponential CTFS has several interesting properties that are useful in the
analysis of CT signals.
(1) The
linearity
property states that the exponential CTFS coefficients of a
linear combination of periodic signals are given by the same linear combi-
nation of the exponential CTFS coefficients of each of the periodic signals.
(2) A time shift of
t
0
in the periodic signal does not affect the magnitude of the
exponential CTFS coefficients. However, the phase changes by an additive
Search WWH ::
Custom Search