Digital Signal Processing Reference
In-Depth Information
e
jnÆ
0
Hence, the multiplication of a time-domain signal by the complex exponential
results in a shift of
Æ
0
in the frequency domain.
EXAMPLE 12.6
Application of the frequency-shift property of the DTFT
From Example 12.1,
1
1 - ae
-jÆ
, |a| 6 1
a
n
u[n]
Î
"
.
Then, from (12.9),
1
1 - ae
-j(Æ-Æ
0
)
, |a| 6 1
e
jnÆ
0
a
n
u[n]
Î
"
.
■
We state the symmetry properties of the discrete-time Fourier transform without
proof. Expressing
X(Æ)
in its real and imaginary parts yields
X(Æ) = Re[X(Æ)] + j Im[X(Æ)].
For the case that
x
[
n
] is a real-valued sequence, we list the symmetry properties:
Re[X(Æ)] is even;
Im[X(Æ)] is odd;
(12.10)
ƒX(Æ) ƒ is even;
arg X(Æ) is odd.
Consider the time-reversed signal
x[-n].
Then
q
x[-n]e
-jnÆ
.
[x[-n]] =
a
n=-
q
Now let
-n = k.
It follows that
q
q
x[k]e
jkÆ
=
a
x[k]e
-jk(-Æ)
[x[-n]] =
a
= X(-Æ).
k=-
q
k=-
q
The time-reversal property is then
x[-n] · X(-Æ).
(12.11)
The effect of time reversal is frequency reversal.
