Gauss-Meixner Summation

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Gauss-Meixner Summation

The impulse response of a TIB realization with a single eigenvalue is given by
equation716
where 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 Meixner polynomials results in the Jacobi matrix
equation724
and the associated Gauss-Meixner summation formula
equation737
where p is any polynomial of degree . The are the eigenvalues of J and the are proportional to the squares of the first components of the corresponding orthonormal eigenvectors. We fix the 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 of the form
equation776
by the formula
eqnarray783
Now we can compute recursively without explicitly computing as follows.
eqnarray802
then using the recursion for :

and
equation831
where we put . A check that this has worked is that 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