Accelerated plane-wave destruction

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# Appendix A: Polynomial form of PWD

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):
 (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 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)

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 I-7, we can prove equation I-7. It follows that

 (32)

 (33)

with
 (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

Next: Bibliography Up: Chen, Fomel & Lu: Previous: Acknowledgments

2013-03-02