Digital Signal Processing Reference
In-Depth Information
where Hence, we have the time-scaling property (as defined earli-
er), for
k
a positive integer:
z
[f[n]] = F(z).
f[n/k]
Î
z
"
F(z
k
).
(11.34)
The time scaling creates additional sample values, all of which we
choose to set to zero. The derivations presented here apply for only this choice;
other rules can be used to assign values for the samples created, and then (11.34)
does not apply.
We do not consider the time scaling where
k
is a positive integer. Recall
from Chapter 9 that this transform results in the loss of sample values. We now
illustrate time scaling with an example.
m = n/k
f [nk],
Illustration of time-scaling property
EXAMPLE 11.7
f[n] = a
n
.
Consider the exponential function
From Table 11.2,
z
z - a
= F(z).
f[n] = a
n
Î
z
"
We wish to find the
z
-transform of
f
t
[n] = f[n/2].
We first find
F
t
(z)
from its definition; then
we use property (11.33) for verification.
From definition (11.32),
z
[f[n/2]] = 1 + az
-2
+ a
2
z
-4
+
Á
=
q
n= 0
a
n
z
-2n
=
q
n= 0
(az
-2
)
n
.
From Appendix C,
q
n= 0
1
1 - b
;
b
n
=
ƒ b ƒ 6 1.
b = az
-2
,
Thus, with
z
2
1
1 - az
-2
=
z
[f[n/2]] =
- a
.
(11.35)
z
2
Direct substitution into the scaling property, (11.33), with
k = 2
verifies this result.
■
We now derive the transform for the convolution summation. From (10.13) and the
definition of convolution,
q
k=-
q
x[k]y[n - k] =
q
k= 0
x[n]*y[n] =
x[k]y[n - k],
(11.36)
