On General Factorizations for n-D Polynomial Matrices

Multivariate (n-D) polynomial matrix factorizations are basic research subjects in multidimensional (n-D) systems and signal processing. In this paper, several results on general matrix factorizations are provided for extracting a matrix factor from a given n-D polynomial matrix whose lower order minors satisfy certain conditions. These results are further generalizations of previous results in (Lin et al. in Circuits Syst. Signal Process. 20(6):601–618, 2001). As a consequence, the application range of the constructive algorithm in (Lin et al. in Circuits Syst. Signal Process. 20(6):601–618, 2001) has been greatly extended. Three examples are worked out in detail to show the practical value of the proposed method for obtaining general factorizations for a class of n-D polynomial matrices.

[1]  Zhiping Lin,et al.  Factorizations fornD polynomial matrices , 2001 .

[2]  G. Greuel,et al.  A Singular Introduction to Commutative Algebra , 2002 .

[3]  Georg Regensburger,et al.  Gröbner bases in control theory and signal processing , 2007 .

[4]  Ettore Fornasini,et al.  nD Polynomial Matrices with Applications to Multidimensional Signal Analysis , 1997, Multidimens. Syst. Signal Process..

[5]  N. K. Bose,et al.  Multidimensional FIR filter bank design using Grobner bases , 1999 .

[6]  Dengguo Feng,et al.  On Lin-Bose problem , 2004 .

[7]  Zhiping Lin,et al.  Notes on n-D Polynomial Matrix Factorizations , 1999, Multidimens. Syst. Signal Process..

[8]  N. Bose,et al.  A Generalization of Serre's Conjecture and Some Related Issues , 2001 .

[9]  C. P. Kwong,et al.  On multivariate polynomial matrix factorization problems , 2005, Math. Control. Signals Syst..

[10]  Zhiping Lin,et al.  On primitive factorizations for n-D polynomial matrices , 1993, 1993 IEEE International Symposium on Circuits and Systems.

[11]  Mingsheng Wang,et al.  Notes on factor prime factorizations for n-D polynomial matrices , 2010, Multidimens. Syst. Signal Process..

[12]  U. Oberst,et al.  Transfer Operators and State Spaces for Discrete Multidimensional Linear Systems , 1999 .

[13]  J. Guiver,et al.  Polynomial matrix primitive factorization over arbitrary coefficient field and related results , 1982 .

[14]  J. Pommaret.  Solving bose conjecture on linear multidimensional systems , 2001, 2001 European Control Conference (ECC).

[15]  Zhiping Lin.  On primitive factorizations for 3-D polynomial matrices , 1992 .

[16]  Vasudevan Srinivas.  A generalized Serre problem , 2004 .

[17]  Zhiping Lin,et al.  On minor prime factorizations for n-D polynomial matrices , 2005, IEEE Transactions on Circuits and Systems II: Express Briefs.

[18]  Mingsheng Wang,et al.  On Factor Prime Factorizations for $n$-D Polynomial Matrices , 2007, IEEE Transactions on Circuits and Systems I: Regular Papers.

[19]  Dante C. Youla,et al.  Notes on n-Dimensional System Theory , 1979 .

[20]  T. Lam.  Serre's Problem on Projective Modules , 2006 .

[21]  Dante C. Youla,et al.  The Quillen - Suslin theorem and the structure of n-dimensional elementary polynomial matrices , 1984 .

[22]  Zhiping Lin,et al.  A Tutorial on GrÖbner Bases With Applications in Signals and Systems , 2008, IEEE Transactions on Circuits and Systems I: Regular Papers.

[23]  Mingsheng Wang,et al.  Remarks on $n$ -D Polynomial Matrix Factorization Problems , 2008, IEEE Transactions on Circuits and Systems II: Express Briefs.

[24]  N. Bose.  Multidimensional systems theory and applications , 1995 .

[25]  N. Bose.  Applied multidimensional systems theory , 1982 .

[26]  Nikos E. Mastorakis,et al.  A general factorization method for multivariable polynomials , 1994, Multidimens. Syst. Signal Process..