Digital Signal Processing Reference
In-Depth Information
2
2
1
1
0
0
0
0.5
1
1.5
2
0
0.5
1
1.5
2
(a) Norm Freq, x
1
[n]
(e) Norm Freq, x
2
[n]
5
5
0
0
−5
−5
0
0.5
1
1.5
2
0
0.5
1
1.5
2
(b) Norm Freq, x
1
[n]
(f) Norm Freq, x
2
[n]
2
2
0
0
−2
−2
0
0.5
1
1.5
2
0
0.5
1
1.5
2
(c) Norm Freq, x
1
[n]
(g) Norm Freq, x
2
[n]
2
2
0
0
−2
−2
0
0.5
1
1.5
2
0
0.5
1
1.5
2
(d) Norm Freq, x
1
[n]
(h) Norm Freq, x
2
[n]
Figure 1.6:
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/
8 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.
The DTFT of the first sequence is
X
1
(e
jω
)
e
−
jω
2
=
1
+
and for the second sequence
X
2
(e
jω
)
e
−
jω
3
=
1
−
The product is
e
−
jω
2
e
−
jω
3
e
−
jω
5
+
−
−
1
or stated more completely
