Multidimensional Systems and Signal Processing

, Volume 28, Issue 1, pp 75–103 | Cite as

The injectivity of the canonical signal module for multidimensional linear systems of difference equations with variable coefficients



We consider discrete behaviors with varying coefficients. Our results are new also for one-dimensional systems over the time-axis of natural numbers and for varying coefficients in a field, we derive the results, however, in much greater generality: Instead of the natural numbers we use an arbitrary submonoid N of an abelian group, for instance the standard multidimensional lattice of r-dimensional vectors of natural numbers or integers. We replace the base field by any commutative self-injective ring F, for instance a direct product of fields or a quasi-Frobenius ring or a finite factor ring of the integers. The F-module W of functions from N to F is the canonical discrete signal module and is a left module over the natural associated noncommutative ring A of difference operators with variable coefficients. Our main result states that this module is injective and therefore satisfies the fundamental principle: An inhomogeneous system of linear difference equations with variable coefficients has a solution if and only if the right side satisfies the canonical compatibility conditions. We also show that for the typical cases of partial difference equations and in contrast to the case of constant coefficients the A-module W is not a cogenerator. We also generalize the standard one-dimensional theory for periodic coefficients to the multidimensional situation by invoking Morita equivalence.


Difference equation Variable coefficient Injective signal module Fundamental principle Cogenerator Periodic 

Mathematics Subject Classfication

93C05 93C55 16D50 39A06 39A14 



We thank the two reviewers for their careful reading of and useful suggestions for the paper.


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.SATIEENS Cachan/CNAMCachanFrance
  2. 2.École Centrale de NantesNantes Cedex 3France
  3. 3.Institut für MathematikUniversität InnsbruckInnsbruckAustria

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