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Next: Bidiagonal Representation of TIB Up: Triangular Input Balanced Realizations Previous: Meixner Functions and TIB

Gauss-Meixner Summation

The impulse response of a TIB realization with a single eigenvalue tex2html_wrap_inline1150 is given by
equation716
where tex2html_wrap_inline1348 and, as usual, c is the vector of coefficients. In order to compute c we exploit Gaussian quadrature (in this case summation) properties of the orthogonal polynomials. The normalized three-term recurrence for the tex2html_wrap_inline1330 Meixner polynomials results in the Jacobi matrix
equation724
and the associated Gauss-Meixner summation formula
equation737
where p is any polynomial of degree tex2html_wrap_inline1358. The tex2html_wrap_inline1360 are the eigenvalues of J and the tex2html_wrap_inline1364 are proportional to the squares of the first components of the corresponding orthonormal eigenvectors. We fix the tex2html_wrap_inline1364 by choosing the proportionality constant so that
equation749
for consistency, (i.e. consider p=1).

We put this in terms of a summation formula for the Meixner functions:
eqnarray756
This allows us to compute Meixner coefficients, that is for tex2html_wrap_inline1370 of the form
equation776
by the formula
eqnarray783
Now we can compute tex2html_wrap_inline1372 recursively without explicitly computing tex2html_wrap_inline1374 as follows.
eqnarray802
then using the recursion for tex2html_wrap_inline1376:
equation818
and
equation831
where we put tex2html_wrap_inline1378. A check that this has worked is that tex2html_wrap_inline1380 for k=1,...,d.

Gauss-Meixner summation works very well when we can bound the order d of the model in advance, and it is one attractive way of computing the TIB filter realization of the maximally flat impulse response.




Fri Jun 27 03:10:38 EDT 1997

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