Digital Signal Processing Reference
In-Depth Information
The equation for the inverse
z
-transform, (11.2), is the same for both the bi-
lateral and the unilateral
z
-transforms. Hence, (11.2) also gives the inverse unilater-
al
z
-transform, provided that is replaced with In addition, the inverse
z
-transform of the unilateral
z
-transform gives the function
F
b
(z)
F(z).
f[n]
for all time and, in
particular, gives the value
If is
z
-transformable [if the summation in (11.4) exists], evaluating (11.4)
will yield a function
f[n] = 0, n 6 0 [1].
f[n]
F(z).
The evaluation of the inverse transform of
F(z)
by the
complex inversion integral, (11.2), will then yield
f[n].
We denote this relationship
with
f[n]
Î
z
"
F(z).
(11.5)
Two important properties of the
z
-transform will now be demonstrated. The
(unilateral)
z
-transform is used in this derivation; however, it is seen that the de-
rivation applies equally well to the bilateral
z
-transform.
Consider the sum
f[n] = (f
1
[n] + f
2
[n]).
From (11.4), the
z
-transform of
f[n]
is given by
q
[f
1
[n] + f
2
[n]]z
-n
z
[f[n]]
=
z
[f
1
[n] + f
2
[n]] =
a
n= 0
(11.6)
q
q
f
1
[n]z
-n
f
2
[n]z
-n
=
a
+
a
= F
1
(z) + F
2
(z).
n= 0
n= 0
Hence, the
z
-transform of the sum of two functions is equal to the sum of the
z
-transforms of the two functions. (It is assumed that the involved
z
-transforms
exist.) We extend this property to the sum of any number of functions by replacing
in the foregoing derivation with the sum and so on.
To derive a second property of the
z
-transform, we consider the
z
-transform
of
af
[
n
], where
a
is a constant:
f
2
[n]
(f
3
[n] + f
4
[n]),
q
q
af[n]z
-n
f[n]z
-n
z
[af[n]] =
a
= a
a
= aF(z).
(11.7)
n= 0
n= 0
Thus, the
z
-transform of a function multiplied by a constant is equal to the constant
multiplied by the
z
-transform of the function. A transform with the properties
(11.6) and (11.7) is said to be a
linear transform;
the
z
-transform is then a linear
transform. These two properties are often stated as a single equation:
z
[a
1
f
1
[n] + a
2
f
2
[n]] = a
1
F
1
(z) + a
2
F
2
(z).
(11.8)
Suppose, in (11.7), that the constant
a
is replaced with the function
g
[
n
]. Then,
q
q
q
f[n]g[n]z
-n
f[n]z
-n
a
g[n]z
-n
.
z
[f[n]g[n]] =
a
Z
a
n= 0
n= 0
n= 0
