Digital Signal Processing Reference
In-Depth Information
Finally, if
x
[
n
]
is real and symmetric, then the DTFT is real:
e
j
ω
)
∈
[
]
∈
[
−
]=
[
]
⇐⇒
(
x
n
,
x
n
x
n
X
(4.52)
l
g
r
,
y
i
d
.
,
©
,
L
s
while, for real antisymmetric signals we have that the DTFT is purely imagi-
nary:
Re
X
e
j
ω
)
=
x
[
n
]
∈
,
x
[
−
n
]=
−
x
[
n
]
⇐⇒
(
0
(4.53)
Linearity and Shifts.
The DTFT is a linear operator:
DTFT
←→
α
e
j
ω
)+
β
e
j
ω
)
α
x
[
n
]+
β
y
[
n
]
X
(
Y
(
(4.54)
A shift in the discrete-time domain leads to multiplication by a phase term
in the frequency domain:
DTFT
←→ e
−j
ω
n
0
X
(
e
j
ω
)
x
[
n−n
0
]
(4.55)
while multiplication of the signal by a complex exponential (i.e. signal
mod-
ulation
by a complex “carrier” at frequency
ω
0
)leadsto
X
e
j
(
ω
−
ω
0
)
DTFT
←→
e
j
ω
0
n
x
[
n
]
(4.56)
which means that the spectrum is shifted by
ω
0
.Thislastresultisknownas
the
modulation theorem
.
Energy Conservation.
TheDTFT satisfies the
Plancherel-Parseval
equal-
ity:
x
]
=
X
e
j
ω
)
1
2
e
j
ω
)
[
n
]
,
y
[
n
(
,
Y
(
(4.57)
π
and
L
2
[
−
π
π
]
:
or, using the respective definitions of inner product for
2
(
)
,
π
∞
1
2
x
∗
[
X
∗
(
e
j
ω
)
e
j
ω
)
]
[
]=
(
ω
n
y
n
Y
d
(4.58)
π
n
=
−∞
−
π
(note the explicit normalization factor 1
/
2
π
). The above equality specializes
into
Parseval's theorem
as
π
∞
x
]
X
e
j
ω
)
1
2
2
2
[
n
=
(
d
ω
(4.59)
π
n
=
−∞
−
π
which establishes the conservation of energy property between the time and
the frequency domains.
