Digital Signal Processing Reference
In-Depth Information
6.1
Filter Analysis
To see the usefulness of the
z
-transform in the context of the analysis and
the design of realizable filters, it is sufficient to consider the following two
formal properties of the
z
-transform operator:
l
g
r
,
y
i
d
.
,
©
,
L
s
•
Linearity:
given two sequences
x
[
n
]
and
y
[
n
]
and their respective
z
-transforms
X
(
z
)
and
Y
(
z
)
,wehave
α
]
=
α
x
[
n
]+
β
y
[
n
X
(
z
)+
β
Y
(
z
)
[
]
(
)
•
Time-shift:
given a sequence
x
n
and its
z
-transform
X
z
,wehave
x
]
=
z
−N
X
[
n
−
N
(
z
)
Intheabove,wehaveconvenientlyignored all convergence issues for the
z
-transform; these will be addressed shortly but, for the time being, let us
just make use of the formalism as it stands.
6.1.1
Solving CCDEs
Consider the generic filter CCDE (Constant-Coefficient Difference Equation)
in (5.46):
M
−
1
N
−
1
y
[
n
]=
b
k
x
[
n
−
k
]
−
a
k
y
[
n
−
k
]
k
=
0
k
=
1
If we apply the
z
-transform operator to both sides and exploit the linearity
and time-shifting properties, we have
M
−
1
N
−
1
b
k
z
−k
X
a
k
z
−k
Y
Y
(
z
)=
(
z
)
−
(
z
)
(6.2)
k
=
0
k
=
1
M
−
1
b
k
z
−k
k
=
0
=
X
(
z
)
(6.3)
N
−
1
a
k
z
−k
1
+
=
k
1
=
(
)
(
)
H
z
X
z
(6.4)
H
is called the
transfer function
of the LTI filter described by the CCDE.
The following properties hold:
(
z
)
