Digital Signal Processing Reference
In-Depth Information
Table 13.3. Discrete frequencies corresponding to a few selected points along the unit circle in the z-domain
√
√
−
√
√
−
√
√
√
√
1
+
j0
+
j
0
+
j1
+
j
−
1
+
j0
−
j
0
−
j1
−
j
z
-coordinates
2
2
2
2
2
2
2
2
Frequency,
Ω
0
π/
4
π/
2
3
π/
4
π
5
π
/4
3
π/
2
π
/4
In this example, we observe that the output response for the first input signal
x
1
[
k
]
=
u
[
k
] is bounded. On the other hand, the output produced by the second
input,
x
2
[
k
]
=
sin(
π
k
/
3)
u
[
k
], is unbounded. Note that the second input is a
sinusoidal sequence, which contains two complex exponentials:
π
k
3
u
[
k
]
=
1
2j
e
j
π
k
/
3
−
e
−
j
π
k
/
3
sin
,
with discrete frequencies
Ω
m
=π/
3. Since the frequencies of the c
o
mplex
exponentials are the same as the value of tan
√
−
1
(
b
m
/
a
m
)
=
tan
−
1
(
3
/
4)
=
√
π/
3, determined from the poles, at
z
3
/
2, on the unit circle, the
output response is unbounded. This is consistent with the marginal stability
condition mentioned above.
=
0
.
5
j
13.9 Frequency-res
ponse calculation in the z-domain
Based on Eq. (13.8), the DTFT transfer function is related to the z-transfer
function by the following relationship:
k
=−∞
h
[
k
]
z
∞
−
k
H
(
Ω
)
=
=
H
(
z
)
z
=
e
j
Ω
,
(13.47)
which may be used to derive the DTFT transfer function from the z-transfer
function. Equation (13.47) has wider implications, as we discuss in the follow-
ing.
(1) Taking the magnitude of both sides of the relationship
z
=
exp( j
Ω
)gives
z
=
1; therefore, Eq. (13.47) is only valid if the ROC of the z-transfer
function contains the unit circle. Otherwise, the substitution
z
=
exp( j
Ω
)
cannot be made and the DTFT transfer function does not exist.
(2) Equation (13.47) can also be used to compute the magnitude and phase
spectra of the LTID system by evaluating the z-transfer function at dif-
ferent frequencies (0
≤
≤
2
π
) along the unit circle. The correspon-
dence between the discrete frequency
Ω
and the
z
-coordinates is shown in
Fig. 13.9. A selected subset of the discrete frequencies along the unit circle
is shown in Table 13.3.
Ω
The computation of the magnitude and phase spectra from the z-transfer
function is illustrated in the following example.
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