Digital Signal Processing Reference
In-Depth Information
100
100
50
50
0
0
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
(a) Norm Freq, x
1
[n]
(e) Norm Freq, x
2
[n]
2
2
0
0
−2
−2
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
(b) Norm Freq, x
1
[n]
(f) Norm Freq, x
2
[n]
100
100
0
0
−100
−100
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
(c) Norm Freq, x
1
[n]
(g) Norm Freq, x
2
[n]
100
100
0
0
−100
−100
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
(d) Norm Freq, x
1
[n]
(h) Norm Freq, x
2
[n]
Figure 1.4:
DTFT of first sequence, with its magnitude and phase and real and imaginary parts being
shown, respectively, in plots (a)-(d); DTFT of second sequence, which is the first sequence offset in
frequency by
π/
4 radian, with its magnitude, phase, and real and imaginary parts being shown, respectively,
in plots (e)-(h). All frequencies are normalized, i.e., in units of
π
radians.
1.5 INVERSE DTF T
The Inverse DTFT, i.e., the original time domain sequence
x
[
n
]
from which a given DTFT was produced,
can be reconstructed by evaluating the following integral:
π
1
2
π
X(e
jω
)e
jωn
dω
[
]=
x
n
(1.7)
−
π
We will illustrate this with several examples, one analytic and the other numerical.
Example 1.4.
Using Eq. (1.7), compute
x
[
0
]
,
x
[
1
]
, and
x
[
2
]
from the DTFT obtained in Eq. (1.6).
We get
