Digital Signal Processing Reference
In-Depth Information
11.1
DEFINITIONS OF
z
-TRANSFORMS
We begin by defining the direct
z
-transform and the inverse
z
-transform. We usually
omit the term
direct
and call the direct
z
-transform simply the
z
-transform. By defi-
nition, the (direct)
z-transform F
(
z
) of a discrete-time function
f[n]
is given by the
summation
q
n=-
q
f[n]z
-n
,
z
b
[f[n]] = F
b
(z) =
(11.1)
z
b
[
#
]
where
indicates the
z
-transform. Expanding the
z
-transform yields
Á
Á .
+ f[-2]z
2
+ f[-1]z + f[0] + f[1]z
-1
+ f[2]z
-2
F
b
(z) =
+
In general,
z
is complex, with
z =©+jÆ.
(Recall that the Laplace-transform vari-
able
s
is also complex, with )
Definition (11.1) is called the bilateral, or two-sided,
z
transform—hence, the
subscript
b
. The inverse
z
-transform is given by
s = s + jv.
1
2pj
C
≠
z
-1
[F
b
(z)] = f[n] =
F
b
(z)z
n- 1
dz,
j =
2
-1,
(11.2)
z
-1
[
#
]
where indicates the inverse
z
-transform and is a particular counterclock-
wise closed path in the
z
-plane. Equation (11.2) is called the
complex inversion inte-
gral
. Because of the difficulty of evaluating this integral, we seldom, if ever, use it to
find inverse transforms. Instead, we use tables, as we do with other transforms.
Equations (11.1) and (11.2) are called the
bilateral z-transform pair
. We now
modify definition (11.1) to obtain a form of the
z
-transform that is useful in many
applications. First, we express (11.1) as
≠
q
-1
n=-
q
f[n]z
-n
f[n]z
-n
.
z
b
[f[n]] = F
b
(z) =
+
a
(11.3)
n= 0
Next, we
define f
[
n
] to be zero for such that the first summation in (11.3) is
zero. The resulting transform is called the
unilateral,
or
single-sided, z
-transform,
and is given by the power series
n 6 0,
z
[f[n]] = F(z) =
q
n= 0
f[n]z
-n
,
(11.4)
z
[
#
]
where denotes the unilateral
z
-transform. This transform is usually called, simply,
the
z
-transform, and we follow this custom. When any confusion can result, we refer
to the transform of (11.1) as the bilateral
z
-transform. We take the approach of
making the unilateral transform a special case of the bilateral transform. This
approach is not necessary; we could start with (11.1), with
f[n] = 0
for
n 6 0,
as a
definition.
