Digital Signal Processing Reference
In-Depth Information
Let us now have a closer look at the last term ln
fA
(
z
,
n
)
g
in (7.9)
(
)
N
pre
i¼
1
a
i
(
n
)
z
i
ln{
A
(
z
,
n
)}
¼
ln 1
X
(
)
¼
ln
X
N
pre
a
i
(
n
)
z
i
:
(7
:
10)
i¼
0
By representing this expression as a product of zeros with modified coefficients
b
i
(
n
)
we can write
(
)
(
)
ln
X
N
pre
¼
ln
Y
N
pre
a
i
(
n
)
z
i
[1
b
i
(
n
)
z
1
]
i¼
0
i¼
0
¼
X
N
pre
ln{1
b
i
(
n
)
z
1
}
:
(7
:
11)
i¼
0
By exploiting the following series expansion [3]
ln{1
bz
1
}
¼
1
k¼
1
b
k
k
z
k
,
for
jzj
.
jbj
,
(7
:
12)
that holds for factors that converge within the unit circle, which is the case here since
A
(
z
,
n
) is analytic inside the unit circle [29], we can further rewrite (7.11) [and thus
also (7.10)]
N
pre
ln{(
A
(
z
,
n
)}
¼
X
1
b
i
(
n
)
k
z
k
i¼
0
k¼
1
¼
1
k¼
1
X
N
pre
b
i
(
n
)
k
z
k
:
(7
:
13)
i¼
0
If we now compare (7.9) with (7.13) we observe that the two sums do not have equal
limits. This means that the
c
i
(
n
) are equal to zero for
i
,
0. For
i
.
0 we can set the
c
i
(
n
) equal to the inner sum in (7.13). For
i ¼
0wehave
c
0
(
n
)
¼
ln
fs
(
n
)
g
from
(7.9). In conclusion, we can state
<
N
pre
X
b
i
m
(
n
)
i
,
for
i
.
0,
c
i
(
n
)
¼
(7
:
14)
m¼
0
:
ln{
s
(
n
)},
for
i ¼
0,
0,
for
i <
0
:
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