Digital Signal Processing Reference
In-Depth Information
Solution
Using the time-differencing property, the z-transform of
u
[
k
]
−
u
[
k
−
1] is
given by
←→
−
1
)
Z
u
[
k
]
,
u
[
k
]
−
u
[
k
−
1]
(1
−
z
ROC:
z
>
1
.
−
1
) and noting that
u
[
k
]
−
u
[
k
−
Substituting the value of
Z
u
[
k
]
=
1
/
(1
−
z
1]
= δ
[
k
], we obtain
←→
δ
[
k
]
1
.
Since the z-transform of the unit impulse function is finite for all values of
z
,
the ROC of the aforementioned z-transform pair is the entire z-plane.
13.4.5 z-domain differentiation
←→
If
x
[
k
]
with ROC
R
x
, then
X
(
z
)
←→ −
z
d
X
(
z
)
d
z
kx
[
k
]
,
ROC:
R
x
.
(13.23)
The z-domain differentiation property is satisfied by both unilateral and bilateral
z-transforms.
Proof
By definition,
∞
−
k
.
X
(
z
)
=
x
[
k
]
z
k
=
0
Differentiating both sides with respect to
z
yields
∞
∞
−
k
d
X
(
z
)
d
z
d
z
−
k
−
1
.
=
x
[
k
]
=
x
[
k
](
−
k
)
z
d
z
k
=
0
k
=
0
Multiplying both sides by
−
z
, we obtain
∞
d
X
(
z
)
d
z
−
k
,
−
z
=
kx
[
k
]
z
k
=
0
which proves Eq. (13.23).
Example 13.8
Given the z-transform pair
1
1
− α
z
−
1
,
←→
α
k
u
[
k
]
ROC:
z
> α,
calculate the z-transform of the function
k
α
k
u
[
k
].
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