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| Accelerated plane-wave destruction | |
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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
|
(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)
|
(25) |
which is equivalent to the following linear maximally flat conditions
(Thiran, 1971):
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(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,
|
(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,
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(28) |
therefore the coefficients
|
(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:
|
(30) |
where
,
and is the Lagrange polynomial related to the basis ,
|
(31) |
Substituting the above equation, and
into equation A-7,
we can prove equation A-7.
It follows that
|
(32) |
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(33) |
with
|
(34) |
Thus hence
and
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(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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| Accelerated plane-wave destruction | |
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Next: Bibliography
Up: Chen, Fomel & Lu:
Previous: Acknowledgments
2013-07-26