Digital Signal Processing Reference
In-Depth Information
•
For sequences that are composites of the above criteria, the sequence should be considered as
the sum of a number of subsequences, and the ROC is generally the intersection of the ROCs
of each subsequence.
2.3.5 TRIVIAL POLES AND ZEROS
The generalized
z
-transform for an finite-length sequence may be written as
b
N
−
1
z
−
(N
−
1
)
where the sequence length is
N
. If the right-hand side of this equation is multiplied by
z
N
−
1
/z
N
−
1
(i.e.,
1), the result is
b
1
z
−
1
b
2
z
−
2
B(z)
=
b
0
+
+
+
...
+
(b
0
z
N
−
1
b
1
z
N
−
2
b
2
z
N
−
3
b
N
−
1
)/z
N
−
1
B(z)
=
+
+
+
...
+
which has
(N
1
)
poles at
z
= 0. These are referred to as
Trivial Poles
.
Likewise, the generalized expression for an IIR
−
a
N
−
1
z
−
(N
−
1
)
)
if multiplied on the right by
z
N
−
1
/z
N
−
1
will yield a transfer function with
N
-1
Trivial Zeros
at
z
=0.
The trivial poles of a finite sequence (or FIR) cause the
z
-transform to diverge at
z
= 0 as mentioned
a
1
z
−
1
a
2
z
−
2
A(z)
=
1
/(a
0
−
−
−
...
−
above.
2.3.6 BASIC PROPERTIES OF THE Z-TRANSFORM
Linearity
If the
z
-transforms of two functions
x
1
[
n
]
and
x
2
[
n
]
are
X
1
(z)
and
X
2
(z)
, respectively, then the
z
-
transform of
Z(c
1
x
1
[
n
]+
c
2
x
2
[
n
]
)
=
c
1
X
1
(z)
+
c
2
X
2
(z)
;
ROC: ROC(
x
1
[
n
]
)
∩
ROC(
x
2
[
n
]
)
for all values of
c
1
and
c
2
.
Shifting or Delay
If
X
is
z
−
d
X
[
z
]
is the
z
-transform of
x
[
n
]
, then the
z
-transform of the delayed sequence
x
[
n
−
d
]
[
z
]
with the ROC the same as that for
x
. This is a very useful property since it allows one to write the
z
-transform of difference equations by inspection.
[
n
]
Example 2.9.
Using the shifting property, write the
z
-transform of the following causal system:
y
[
n
]=
x
[
n
]+
ay
[
n
−
1
]+
bx
[
n
−
1
]
The
z
-transform may be written as
az
−
1
Y(z)
bz
−
1
X(z)
=
+
+
Y(z)
X(z)
which simplifies to
bz
−
1
Y(z)
X(z)
=
1
+
az
−
1
;
ROC:
|
z
|
>
|
a
|
1
−
