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If all the coefficients of
are polynomials of
,
equation 4 is also a polynomial of
,
and the plane-wave destruction equation becomes
in turn a polynomial equation of
.
The problem is to design a
points filter
with polynomial coefficients
such that the allpass system
can approximate
the phase-shift operator
.
Denoting the phase response of the system as
,
that is
,
the group delay of the system is
![\begin{displaymath}
\tau(\omega)=\frac{\partial \theta(\omega)}{\partial \omega}.
\end{displaymath}](img107.png) |
(24) |
The maximally flat criteria designs a filter
with a smoothest phase response.
There are
unknown coefficients in
,
so we can add
flat constraints for the first
th order deviratives
of the phase response.
It becomes
(Zhang, 2009, equation 7)
![\begin{displaymath}
\left\{\begin{array}{l}
\tau(\omega)=p \\
\displaystyle{\fr...
...partial \omega^n}}=0
\qquad n=1,2,\dots,2N
\end{array}\right.,
\end{displaymath}](img110.png) |
(25) |
which is equivalent to the following linear maximally flat conditions
(Thiran, 1971):
![\begin{displaymath}
\sum_{k=-N}^N (d-k)^{2n+1}b_k =0,
\end{displaymath}](img111.png) |
(26) |
where
and
is the fractional delay of
or
.
In order to solve
from the above equations,
Thiran (1971) used an additional condition
,
which leads
to be a fractional function of
.
Differently from that, we use the following condition,
![\begin{displaymath}
\sum_{k=-N}^N b_k =1,
\end{displaymath}](img119.png) |
(27) |
where
can be proved to be polynomials of
.
Let vector
.
Combining equations A-3 and A-4,
we rewrite them into the following matrix form:
The matrix on the left side, denoted as
,
can be split into four blocks
as shown above.
Following the lemma of matrix inversion,
![\begin{displaymath}
\tensor V^{-1}=\left[\begin{array}{cc}
(\tensor A-\tensor B\...
...\tensor C)^{-1}\tensor B\tensor D^{-1} \\
\end{array}\right],
\end{displaymath}](img124.png) |
(28) |
therefore the coefficients
![\begin{displaymath}
\mathbf b=\tensor V^{-1}[1,0,\dots,0]^\textrm T=
\left[\begi...
... A-\tensor B\tensor D^{-1}\tensor C)^{-1}
\end{array}\right].
\end{displaymath}](img125.png) |
(29) |
Let subindex
and
.
Submatrix
can be expressed as
so
.
Denoting
with elements
,
as
is a Vandermonde matrix,
and Lagrange intepolating polynomials have the following relationship:
![\begin{displaymath}
\sum_{j=1}^{2N}u_{ij}x^{2j-2}=\ell_i(x),
\end{displaymath}](img135.png) |
(30) |
where
,
and
is the Lagrange polynomial related to the basis
,
![\begin{displaymath}
\ell_i(x)=
\prod_{-N\leq m\leq N}^{m\neq i,m\neq 0}
\frac{x^2-(d+m)^2}{(d+i)^2-(d+m)^2}.
\end{displaymath}](img139.png) |
(31) |
Substituting the above equation,
and
into equation A-7,
we can prove equation A-7.
It follows that
![\begin{displaymath}[\tensor E^{-1}\tensor C]_i=d\ell_i(d),
\end{displaymath}](img141.png) |
(32) |
![\begin{displaymath}[\tensor D^{-1}\tensor C]_i=
[\tensor X^{-1}\tensor E^{-1}\tensor C]_i=
\frac{d}{d+i}\ell_i(d),
\end{displaymath}](img142.png) |
(33) |
with
![\begin{displaymath}
\ell_i(d)=\frac{(-1)^{i+1}N!N!}{(N+i)!(N-i)!}
\frac{d+i}{d}
\prod_{m=-N}^N \frac{2d+m}{2d+m+i}.
\end{displaymath}](img143.png) |
(34) |
Thus hence
and
![\begin{displaymath}[\tensor A-\tensor B\tensor D^{-1}\tensor C]^{-1} =
\frac{(2N)!(2N)!}{(4N)!N!N!}
\prod_{m=N+1}^{2N}(m^2-p^2).
\end{displaymath}](img147.png) |
(36) |
It is the coefficient
, a
-th degree polynomial of
.
Substituting it into equation A-6,
the coefficients at
are expressed as
With the additional condition A-4 in
points approximation,
all the coefficients are polynomials of
of
-th degree.
Thus the plane-wave destruction equation 6
therefore is proved to be a polynomial equation of
-th degree.
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Next: Bibliography
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Previous: Acknowledgments
2013-07-26