A novel general formulation of up/downsampling commutativity

The paper establishes the commutativity conditions of upsampling and downsampling for multidimensional signals defined on discrete Abelian groups (lattices). The general condition includes results available in the literature as particular cases. Complete generality is achieved by giving an abstract definition of up/downsampling and by working in the signal domain instead of the traditional approach based on z and Fourier transforms. Examples of applications are outlined 1) for the one-dimensional (1-D) case verifying existing results, 2) for television scanning, 3) for degenerate (reduced-dimensionality) lattices, and 4) for multiplicative groups.

[1]  P. P. Vaidyanathan,et al.  The role of integer matrices in multidimensional multirate systems , 1993, IEEE Trans. Signal Process..

[2]  Geoffrey Fox,et al.  Intégration dans les groupes topologiques , 1971 .

[3]  Shin-ichi Matsushita.  Topological Groups , 1952 .

[4]  P. J. Higgins.  Introduction to Topological Groups: Topological groups , 1975 .

[5]  Stéphane Coulombe,et al.  Nonuniform perfect reconstruction filter banks over lattices with application to transmultiplexers , 1999, IEEE Trans. Signal Process..

[6]  Ton Kalker,et al.  A group theoretic approach to multidimensional filter banks: theory and applications , 1996, IEEE Trans. Signal Process..

[7]  A. Kalker.  Commutativity of up/down sampling , 1992 .

[8]  E. Hansen.  Theory of circular harmonic image reconstruction , 1981 .

[9]  P. J. Higgins,et al.  An introduction to topological groups , 1975 .

[10]  Roberto Manduchi,et al.  Multistage sampling structure conversion of video signals , 1993, IEEE Trans. Circuits Syst. Video Technol..

[11]  P. P. Vaidyanathan,et al.  Commutativity of D-dimensional decimation and expansion matrices, and application to rational decimation systems , 1992, [Proceedings] ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[12]  Jelena Kovacevic,et al.  The commutativity of up/downsampling in two dimensions , 1991, IEEE Trans. Inf. Theory.

[13]  C. A. Rogers,et al.  An Introduction to the Geometry of Numbers , 1959 .

[14]  Steven Roman.  Advanced Linear Algebra , 1992 .

[15]  P. Vaidyanathan.  Multirate Systems And Filter Banks , 1992 .

[16]  W. Rudin,et al.  Fourier Analysis on Groups. , 1965 .

[17]  E. Dubois,et al.  The sampling and reconstruction of time-varying imagery with application in video systems , 1985, Proceedings of the IEEE.

[18]  Ramesh A. Gopinath,et al.  On upsampling, downsampling, and rational sampling rate filter banks , 1994, IEEE Trans. Signal Process..

[19]  P. P. Vaidyanathan,et al.  Multirate digital filters, filter banks, polyphase networks, and applications: a tutorial , 1990, Proc. IEEE.