Digital Signal Processing Reference
In-Depth Information
We now take the limit of both sides of this equation as
z
approaches unity; as a
result, the terms
(z
-i + 1
- z
i
)
approach zero. Thus,
lim
z:1
z
[f[n + 1] - f[n]] =
lim
n:
q
[f[n + 1] - f[0]].
(11.28)
We have replaced
k
with
n
on the right side, for clarity, in the remainder of this
derivation.
From the shifting property (11.25),
z
[f[n + 1] - f[n]] = z[F(z) - f[0]] - F(z) = (z - 1)F(z) - zf[0],
and thus,
z:1
z
[f[n + 1] - f[n]] =
lim
lim
z:1
[(z - 1)F(z) - zf[0]].
(11.29)
Equating the right sides of (11.28) and (11.29) yields
lim
n:
q
[f[n + 1] - f[0]] =
z:1
[(z - 1)F(z) - zf[0]].
lim
Because
f [0]
is a constant, this term cancels and the final-value property is given by
f[n] = f[
q
] =
lim
n:
q
lim
z:1
(z - 1)F(z),
(11.30)
provided that the limit on the left side exists—that is, provided that has a final
value. [It is shown later that has a final value, provided that all poles of
are
inside
the unit circle, except for possibly a single pole at
f [n]
f [n]
F(z)
z = 1.
In addition, from
(11.30), if
f [n]
has a final value, that value is nonzero only for the case that
F(z)
has
a pole at
z = 1.]
Illustrations of initial- and final-value properties
EXAMPLE 11.5
We illustrate the initial- and final-value properties with an example. Consider the unit step
function
u
[
n
]:
z
z - 1
.
z
[u[n]] =
From the initial-value property (11.27),
z
z - 1
=
1
1 - 1/z
= 1.
f[0] =
lim
z:
q
lim
z:
q
We know that the final value of
u
[
n
] exists; hence, from the final-value property (11.30),
z
z - 1
=
f[q] =
lim
z:1
(z - 1)
lim
z:1
z = 1.
Both of these values are seen to be correct.
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