Digital Signal Processing Reference
In-Depth Information
Continuation of Example 11.5
EXAMPLE 11.6
The sinusoidal function
sin(pn/2)
is now considered. From Table 11.2,
z sin(p/2)
z
z
[sin(pn/2)] =
- 2z cos(p/2) + 1
=
+ 1
.
z
2
z
2
From the initial-value property (11.27),
z
f[0] =
lim
z:
q
+ 1
= 0,
z
2
which is the correct value. From the final-value property (11.30),
z
f[
q
] =
lim
z:1
(z - 1)
+ 1
= 0,
z
2
which is incorrect, because
sin(pn/2)
oscillates continually and therefore does not have a
final value.
■
We have now derived several properties of the
z
-transform. Additional prop-
erties are derived in the next section.
11.5
ADDITIONAL PROPERTIES
Two additional properties of the
z
-transform will now be derived; then a table of
properties will be given.
Independent-variable transformations were introduced in Chapter 9. We now con-
sider the effects of these discrete-time transformations on the
z
-transform of a
function.
Consider first the
z
-transform of
f [m];
for convenience, we now denote
discrete time by the variable
m
:
F(z) =
z
[f[m]] =
q
m= 0
+
Á
.
f[m]z
-m
= f[0] + f[1]z
-1
+ f[2]z
-2
An example of
f [m]
is plotted in Figure 11.4(a). We now consider the time-scal-
ing transformation
m = n/k (n = mk),
where
k
is a positive integer, and use the
notation
`
f
t
[n] = f[m]
m=n/k
= f[n/k]
(11.31)
