Digital Signal Processing Reference
In-Depth Information
Fig. 13.9. Determination of the
magnitude and phase spectra
from the z-transfer function.
Im{
z
}
p
2
⇒
W
=
(0, 1)
)
⇒
W
=
p
1
1
2
)
⇒
W
=
3
p
,
1
1
2
,
−
2
4
2
4
(− 1, 0) ⇒
W
=
p
Re{
z
}
(1, 0) ⇒
W
= 0
)
⇒
W
=
−−
1
2
1
5
p
)
⇒
W
=
7
p
1
1
2
,
−
2
4
(0, −1)
2
4
3
p
2
⇒
W
=
H
(
W
)
16
<
H
(
W
)
3
0.245
p
.
W
W
−
p
−
p
/2
p
/2
p
−
p
−
p
/2
p
/2
p
−0.245
p
(a)
(b)
Fig. 13.10. (a) Magnitude
spectrum and (b) phase
spectrum of the LTID system
considered in Example 13.18.
The responses are shown in the
frequency range
Ω
Example 13.18
Consider the system with z-transfer function given by
2
z
2
z
2
−
(3
/
4)
z
+
(1
/
8)
2
H
(
z
)
=
=
−
2
.
= [−π, π].
1
−
(3
/
4)
z
−
1
+
(1
/
8)
z
Calculate and plot the amplitude and phase spectra of the system.
Solution
The DTFT transfer function is given by
=
2
1
−
(3
/
4)e
−
j
Ω
+
(1
/
8)e
−
j2
Ω
.
The magnitude spectrum
H
(
Ω
)
and the phase spectrum
<
H
(
Ω
) are plotted in
Fig. 13.10, which are identical to the spectra shown in Fig. 11.18.
H
(
Ω
)
=
H
(
z
)
z
=
e
j
Ω
13.10 DTFT and the z-transform
In Chapter 11 and in this chapter, we presented two different frequency-domain
approaches to analyze DT signals and systems. The DTFT-based approach,
introduced in Chapter 11, uses the real frequency
Ω
, whereas the z-transform-
based approach uses the complex frequency
σ
+
j
Ω
. The output response of
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