Digital Signal Processing Reference
In-Depth Information
M
ATLAB
file
filter
used to compute the output response
y
[
k
] from spec-
ified sample values of the input sequence
x
[
k
] and the ancillary conditions. In
this section, we focus on the z-transfer function representation,
−
1
++
b
0
z
−
m
H
(
z
)
=
Y
(
z
)
X
(
z
)
=
b
m
+
b
m
−
1
z
,
(13.48)
a
n
+
a
n
−
1
z
−
1
++
a
0
z
−
n
which can also be factorized as follows:
−
1
)(1
−
z
1
z
−
1
)
(1
−
z
M
z
−
1
)
H
(
z
)
=
Y
(
z
)
X
(
z
)
(1
−
z
0
z
=
K
−
1
)
.
(13.49)
(1
−
p
0
z
−
1
)(1
−
p
1
z
−
1
)
(1
−
p
N
z
Since M
ATLAB
assumes that the numerator and denominator of the z-transfer
function are expressed in increasing powers of
z
−
1
, we prefer the aforemen-
tioned format for the z-transfer function.
13.11.1 Partial fraction expansion
To calculate the partial fraction expansion of a rational z-transfer function,
M
ATLAB
provides the
residuez
function, which has the following syntax:
>> [R,P,K] = residuez(B,A);
In terms of the transfer function in Eq. (13.48), the input variables
B
and
A
are
defined as follows:
A=[
a
n
a
n
−
1
...
a
0
]
and
B=[
b
m
b
m
−
1
...
b
0
]
.
The output parameter
R
returns the values of the partial fraction coefficients,
P
returns the location of the poles, while
K
contains the direct term in the row
vector.
Example 13.20
To illustrate the usage of the built-in function
residuez
, let us calculate the
partial fraction expansion of the z-transfer function,
2
z
(3
z
+
17)
(
z
−
1)(
z
2
−
6
z
+
25)
,
H
(
z
)
=
considered in Example 13.4(iii).
Solution
Expressing the z-transfer function in increasing powers of
z
−
1
yields
−
1
+
34
z
−
2
6
z
H
(
z
)
=
−
3
.
−
1
+
31
z
−
2
−
25
z
1
−
7
z
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