Digital Signal Processing Reference
In-Depth Information
The concept of symmetry can be reinterpreted as follows: a symmetric sig-
nal is equal to its time reversed version; therefore, for a length-
N
signal to be
symmetric, the first member of (4.71) must equal the first member of (4.72),
that is
l
g
r
,
y
i
d
.
,
©
,
L
s
1,2,...,
(
2
x
[
k
]=
x
[
N
−
k
]
,
k
=
N
−
1
)
/
(4.74)
Note that, in the above definition, the index
k
runs from 1 of
(
2
;
this means that symmetry does not place any constraint on the value of
x
[
0
]
and, similarly, the value of
x
N
−
1
)
/
is also unconstrained for even-length sig-
nals. Figure 4.14 shows some examples of symmetric length-
N
signals for
different values of
N
. Of course the same definition can be used for anti-
symmetric signals with just a change of sign.
[
N
/
2
]
Symmetries and Structure.
The symmetries and structure derived for
the DFS can be rewritten specifically for the DFT as
DFT
←→
x
[
−
n
mod
N
]
X
[
−
k
mod
N
]
(4.75)
DFT
←→
x
∗
[
X
∗
[
−
n
]
k
mod
N
]
(4.76)
The following symmetries hold only for
real
signals:
X
∗
[
−
X
[
k
]=
k
mod
N
]
(4.77)
X
=
X
[
k
]
[
−
k
mod
N
]
(4.78)
X
[
k
]=
−
X
[
−
k
mod
N
]
(4.79)
Re
X
]
=
Re
X
]
[
k
[
−
k
mod
N
(4.80)
Im
X
]
=
−
Im
X
]
[
k
[
−
k
mod
N
(4.81)
Finally, if
x
[
n
]
is real and symmetric (using the symmetry definition in (4.74),
then the DFT is real:
[
]=
[
]
=
(
)
/
[
]
∈
x
k
x
N
−
k
,
k
1,2,...,
N
−
1
2
⇐⇒
X
k
(4.82)
while, for real antisymmetric signals we have that the DFT is purely imagi-
nary.
Linearity and Shifts.
The DFT is obviously a linear operator. A circular
shift in the discrete-time domain leads to multiplication by a phase term in
the frequency domain:
x
(
mod
N
DFT
←→
W
kn
N
X
n
−
n
0
)
[
k
]
(4.83)
while the finite-length equivalent of the modulation theorem states
X
(
mod
N
DFT
←→
W
−nk
0
N
x
[
n
]
k
−
k
0
)
(4.84)
