Digital Signal Processing Reference
In-Depth Information
finite-length signal x
and it can be obtained by a type of Lagrangian in-
terpolation. As in the previous case, the values of DTFT at the roots of unity
are equal to the DFT coefficients; note, however, that the transform of a fi-
nite support sequence is very different from the DTFT of a periodized se-
quence. The latter, in accordance with the definition of the Dirac delta, is
defined only in the limit and for a finite set of frequencies; the former is just
a (smooth) interpolation of the DFT.
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]
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4.6 Fourier Transform Properties
4.6.1 DTFT Properties
The DTFT possesses the following properties.
Symmetries and Structure.
The DTFT of a time-reversed sequence is
DTFT
←→ X
(
e
−j
ω
)
x
[
−n
]
(4.45)
while, for the complex conjugate of a sequence we have
DTFT
←→
x
∗
[
X
∗
(
e
−j
ω
)
]
n
(4.46)
For the very important case of a
real
sequence
x
[
n
]
∈
,property4.46
implies that the DTFT is conjugate-symmetric:
e
j
ω
)=
X
∗
(
e
−j
ω
)
X
(
(4.47)
which leads to the following special symmetries for real signals:
•
The magnitude of the DTFT is symmetric:
X
=
X
e
j
ω
)
e
−j
ω
)
(
(
(4.48)
•
ThephaseoftheDTFTisantisymmetric:
X
(
e
j
ω
)=
−
X
(
e
−j
ω
)
(4.49)
•
The real part of the DTFT is symmetric:
Re
X
e
j
ω
)
=
Re
X
e
−j
ω
)
(
(
(4.50)
•
The imaginary part of the DTFT is antisymmetric:
Im
X
e
j
ω
)
=
−
Im
X
e
−j
ω
)
(
(
(4.51)
