TY - JOUR
T1 - Matrix splitting with symmetry and dyadic framelet filter banks over algebraic number fields
AU - Mo, Qun
AU - Zhuang, Xiaosheng
PY - 2012/11/15
Y1 - 2012/11/15
N2 - Algebraic number fields are of particular interest and play an important role in both mathematics and engineering since an algebraic number field can be viewed as a finite dimensional linear space over the rational number field Q. Algorithms using algebraic number fields can be efficiently implemented involving only integer arithmetics. In this paper, we properly formulate the matrix splitting problem over any general subfield of it C, including an algebraic number field as a special case, and provide a simple necessary and sufficient condition for a 2×2 matrix of Laurent polynomials with symmetry to be able to be factorized by a 2×2 matrix of Laurent polynomials with certain symmetry structure. We propose an effective algorithm on how to obtain the factorization matrix step by step. As an application, we obtain a satisfactory algorithm for constructing dyadic framelet filter banks with the perfect reconstruction property and with symmetry over algebraic number fields. Several examples are provided to illustrate the algorithms proposed in this paper. © 2012 Elsevier Inc. All rights reserved.
AB - Algebraic number fields are of particular interest and play an important role in both mathematics and engineering since an algebraic number field can be viewed as a finite dimensional linear space over the rational number field Q. Algorithms using algebraic number fields can be efficiently implemented involving only integer arithmetics. In this paper, we properly formulate the matrix splitting problem over any general subfield of it C, including an algebraic number field as a special case, and provide a simple necessary and sufficient condition for a 2×2 matrix of Laurent polynomials with symmetry to be able to be factorized by a 2×2 matrix of Laurent polynomials with certain symmetry structure. We propose an effective algorithm on how to obtain the factorization matrix step by step. As an application, we obtain a satisfactory algorithm for constructing dyadic framelet filter banks with the perfect reconstruction property and with symmetry over algebraic number fields. Several examples are provided to illustrate the algorithms proposed in this paper. © 2012 Elsevier Inc. All rights reserved.
KW - Algebraic framelet filters
KW - Algebraic number fields
KW - Extended Euclidean algorithm
KW - Framelet filter banks
KW - Matrix splitting
KW - Symmetry
KW - Wavelets
UR - http://www.scopus.com/inward/record.url?scp=84865696892&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-84865696892&origin=recordpage
U2 - 10.1016/j.laa.2012.06.039
DO - 10.1016/j.laa.2012.06.039
M3 - 21_Publication in refereed journal
VL - 437
SP - 2650
EP - 2679
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
SN - 0024-3795
IS - 10
ER -