Digital Signal Processing Reference
In-Depth Information
phase spectra of the DTFT
X
(
Ω
), the Hermitian symmetry property can be
expressed as follows:
X
(
−
Ω
)
=
X
(
Ω
)
and
<
X
(
−
Ω
)
=−<
X
(
Ω
)
,
(11.39c)
implying that the magnitude spectrum is even and that the phase spectrum is
odd.
As extensions of the Hermitian symmetry properties, we consider the special
cases when: (a)
x
[
k
] is real-valued and even and (b)
x
[
k
] is imaginary-valued
and odd.
Case 1
If
x
[
k
] is both real-valued and even, then its DTFT
X
(
Ω
) is also real-
valued and even, with the imaginary component Im
{
X
(
Ω
)
=
0. In other words,
Re
X
(
−
Ω
)
=
Re
X
(
Ω
)
and
Im
X
(
−
Ω
)
=
0
.
(11.39d)
Case 2
If
x
[
k
] is both imaginary-valued and odd, then its DTFT
X
(
Ω
) is also
imaginary-valued and odd, with the real component Re
{
X
(
Ω
)
=
0. In other
words,
Re
X
(
−
Ω
)
=
0
and
Im
X
(
−
Ω
)
=−
Im
X
(
Ω
)
.
(11.39e)
11.5.3 Linearity
Like the CTFT, both the DTFT and DTFS satisfy the linearity property.
DTFT
If
x
1
[
k
] and
x
2
[
k
] are two DT sequences with the following DTFT pairs:
DTFT
←−−→
DTFT
←−−→
x
1
[
k
]
X
1
(
Ω
)
and
x
2
[
k
]
X
2
(
Ω
)
,
then the linearity property states that
DTFT
←−−→
a
1
x
1
[
k
]
+
a
2
x
2
[
k
]
a
1
X
1
(
Ω
)
+
a
2
X
2
(
Ω
)
,
(11.40a)
for any arbitrary constants
a
1
and
a
2
, which may be complex-valued.
DTFS
If
x
1
[
k
] and
x
2
[
k
] are two periodic DT sequences with the same funda-
mental period
K
0
and the following DTFS pairs:
DTFS
←−−→
DTFS
←−−→
D
x
1
n
D
x
n
,
x
1
[
k
]
and
x
2
[
k
]
then the DTFS coefficients of the periodic DT sequence
x
3
[
k
]
=
a
1
x
1
[
k
]
+
a
2
x
2
[
k
], which also has a period of
K
0
, are given by
DTFS
←−−→
a
1
D
x
1
+
a
2
D
x
2
a
1
x
1
[
k
]
+
a
2
x
2
[
k
]
,
(11.40b)
n
n
x
3
[
k
]
D
n
3
n
for any arbitrary constants
a
1
and
a
2
, which may be complex-valued.
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