Multidimensional systems

In mathematical systems theory, a multidimensional system or m-D system is a system in which not only one independent variable exists (like time), but there are several independent variables.

Important problems such as factorization and stability of m-D systems (m > 1) have recently attracted the interest of many researchers and practitioners. The reason is that the factorization and stability is not a straightforward extension of the factorization and stability of 1-D systems because, for example, the fundamental theorem of algebra does not exist in the ring of m-D (m > 1) polynomials.

Multidimensional systems or m-D systems are the necessary mathematical background for modern digital image processing with many applications in biomedicine, X-ray technology and satellite communications. There are also some studies combining m-D systems with partial differential equations (PDEs).

A state-space model is a representation of a system in which the effect of all "prior" input values is contained by a state vector. In the case of an m-d system, each dimension has a state vector that contains the effect of prior inputs relative to that dimension. The collection of all such dimensional state vectors at a point constitutes the total state vector at the point.

Consider a uniform discrete space linear two-dimensional (2d) system that is space invariant and causal. It can be represented in matrix-vector form as follows:

Represent the input vector at each point ${\displaystyle (i,j)}$ by ${\displaystyle u(i,j)}$, the output vector by ${\displaystyle y(i,j)}$ the horizontal state vector by ${\displaystyle R(i,j)}$ and the vertical state vector by ${\displaystyle S(i,j)}$. Then the operation at each point is defined by:

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