Digital Signal Processing Reference
In-Depth Information
Thus, the time-shift property is given by
x[n - n
0
]
Î
"
e
-jÆn
0
X(Æ).
(12.8)
Note also that this property can be written directly from Table 5.1. We illustrate this
property with a numerical example.
Illustration of the time-shift property of the DTFT
EXAMPLE 12.5
We find the discrete-time Fourier transform of the sequence shown in Figure 12.3. The
sequence is described mathematically by
=
q
k= 3
Á
(0.5)
k- 3
d[n - k]
x[n] = d[n - 3] + 0.5d[n - 4] + 0.25d[n - 5] +
= (0.5)
n- 3
u[n - 3].
From Table 12.1, for
ƒ a ƒ 6 1,
1
1 - ae
-jÆ
.
[a
n
u[n]] =
Then, from this transform and (12.8),
e
-j3Æ
1 - 0.5e
-jÆ
.
[(0.5)
n- 3
u[n - 3]] =
x
[
n
]
1
•••
•••
3
2
1
0
1
2
3
4
5
n
Figure 12.3
A discrete-time sequence.
■
The time-shift property gives the effects in the frequency domain of a shift in the
time domain. We now give the time-domain manipulation that results in a shift in
the frequency domain:
[e
jÆ
0
n
x[n]] =
q
n=-
q
e
jnÆ
0
x[n]e
-jnÆ
=
q
n=-
q
x[n]e
-jn(Æ-Æ
0
)
= X(Æ-Æ
0
).
This property is then
e
jnÆ
0
x[n]
Î
"
X(Æ-Æ
0
).
(12.9)
