Accelerated plane-wave destruction |

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) |

(25) |

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,

Let vector
.
Combining equations I-3 and I-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*,

(28) |

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:

where , and is the Lagrange polynomial related to the basis ,

Substituting the above equation, and
into equation I-7,
we can prove equation I-7.
It follows that

(32) |

(33) |

(34) |

Thus hence

(35) |

and

(36) |

It is the coefficient , a -th degree polynomial of .
Substituting it into equation I-6,
the coefficients at
are expressed as

(37) |

With the additional condition I-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.

Accelerated plane-wave destruction |

2013-03-02