One-dimensional and multidimensional spectral factorization using Gröbner basis approach

Spectral factorization is an important step in the process of designing quadrature-mirror filter bank (QMF Bank) and other types of filter banks. In the one-dimensional case, the spectral factorization can be done effectively by finding all roots of the polynomial to be factorized and pairs up the appropriate roots to form the required factors. On the other hand, this simple approach is not general- izable to the multidimensional case since the roots of the multivariate polynomial are generally not isolated. Here, the new approach based on the usage of Grobner basis is proposed. This new approach is not only applicable to the one-dimensional case, but it is also generalizable to the multidimensional case. Furthermore, for lower order polynomial, the proposed algorithm can provide a symbolic solution. The factorization algorithm is described along with the numerical example to show the effectiveness of the proposed method.