Digital Signal Processing Reference
In-Depth Information
circle, and is called a
bracelet
of zeros generated by
z
k
. If each zero of
C
(
z
) is replaced like this, then each factor (1
z
−
1
z
0
) is replaced with
−
P−
1
z
0
W
k
z
−
1
)
,
(1
−
k
=0
and the new FIR filter is precisely the righthand side of Eq. (D.20). Equa-
tion (D.20) shows that the determinant of the blocked version
C
(
z
)(or
rather
C
(
z
P
)) is precisely this bracelet polynomial.
4.
Eigenvalues and eigenvectors.
From Eq. (D.19) we have
C
(
z
P
)
T
z
=
T
z
Λ
c
(
z
)
,
where
T
z
=
Λ
(
z
−
1
)
W
−
1
.
This shows that the diagonal elements of
Λ
c
(
z
)
are the eigenvalues of
C
(
z
P
)
.
The corresponding eigenvectors are given by
the columns of
T
z
.
Summarizing,
C
(
z
P
)has
eigenvalues
C
(
z
)
,C
(
zW
−
1
)
,...,C
(
zW
−
(
P−
1)
)
,
and the
eigenvector
corresponding to
C
(
zW
−k
)is
]
T
,
α
k
α
P−
1
k
[1
α
k
...
where
α
k
=
zW
−k
.
The
eigenvectors are universal
, in the sense that they
do not depend on
C
(
z
). Only the
eigenvalues
depend on
C
(
z
)
.
z
-plane
P
= 8
z
1
z
0
0
Figure D.1
. The bracelet of zeros created from a real zero
z
0
,
and from a
complex zero
z
1
.
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