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multidimensional system wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 applications 2 linear multidimensional state space model 3 multidimensional transfer function 4 realization of a 2d transfer function toggle realization of a 2d transfer function subsection 4 1 example all zero or finite impulse response 5 references toggle the table of contents multidimensional system add languages add links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia redirected from multidimensional systems system in which not only one independent variable exists 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 applications edit 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 1 2 there are also some studies combining m d systems with partial differential equations pdes linear multidimensional state space model edit 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 3 4 represent the input vector at each point i j displaystyle i j by u i j displaystyle u i j the output vector by y i j displaystyle y i j the horizontal state vector by r i j displaystyle r i j and the vertical state vector by s i j displaystyle s i j then the operation at each point is defined by r i 1 j a 1 r i j a 2 s i j b 1 u i j s i j 1 a 3 r i j a 4 s i j b 2 u i j y i j c 1 r i j c 2 s i j d u i j displaystyle begin aligned r i 1 j a_ 1 r i j a_ 2 s i j b_ 1 u i j s i j 1 a_ 3 r i j a_ 4 s i j b_ 2 u i j y i j c_ 1 r i j c_ 2 s i j du i j end aligned where a 1 a 2 a 3 a 4 b 1 b 2 c 1 c 2 displaystyle a_ 1 a_ 2 a_ 3 a_ 4 b_ 1 b_ 2 c_ 1 c_ 2 and d displaystyle d are matrices of appropriate dimensions these equations can be written more compactly by combining the matrices r i 1 j s i j 1 y i j a 1 a 2 b 1 a 3 a 4 b 2 c 1 c 2 d r i j s i j u i j displaystyle begin bmatrix r i 1 j s i j 1 y i j end bmatrix begin bmatrix a_ 1 a_ 2 b_ 1 a_ 3 a_ 4 b_ 2 c_ 1 c_ 2 d end bmatrix begin bmatrix r i j s i j u i j end bmatrix given input vectors u i j displaystyle u i j at each point and initial state values the value of each output vector can be computed by recursively performing the operation above multidimensional transfer function edit a discrete linear two dimensional system is often described by a partial difference equation in the form p q 0 0 m n a p q y i p j q p q 0 0 m n b p q x i p j q displaystyle sum _ p q 0 0 m n a_ p q y i p j q sum _ p q 0 0 m n b_ p q x i p j q where x i j displaystyle x i j is the input and y i j displaystyle y i j is the output at point i j displaystyle i j and a p q displaystyle a_ p q and b p q displaystyle b_ p q are constant coefficients to derive a transfer function for the system the 2d z transform is applied to both sides of the equation above p q 0 0 m n a p q z 1 p z 2 q y z 1 z 2 p q 0 0 m n b p q z 1 p z 2 q x z 1 z 2 displaystyle sum _ p q 0 0 m n a_ p q z_ 1 p z_ 2 q y z_ 1 z_ 2 sum _ p q 0 0 m n b_ p q z_ 1 p z_ 2 q x z_ 1 z_ 2 transposing yields the transfer function t z 1 z 2 displaystyle t z_ 1 z_ 2 t z 1 z 2 y z 1 z 2 x z 1 z 2 p q 0 0 m n b p q z 1 p z 2 q p q 0 0 m n a p q z 1 p z 2 q displaystyle t z_ 1 z_ 2 y z_ 1 z_ 2 over x z_ 1 z_ 2 sum _ p q 0 0 m n b_ p q z_ 1 p z_ 2 q over sum _ p q 0 0 m n a_ p q z_ 1 p z_ 2 q so given any pattern of input values the 2d z transform of the pattern is computed and then multiplied by the transfer function t z 1 z 2 displaystyle t z_ 1 z_ 2 to produce the z transform of the system output realization of a 2d transfer function edit often an image processing or other md computational task is described by a transfer function that has certain filtering properties but it is desired to convert it to state space form for more direct computation such conversion is referred to as realization of the transfer function consider a 2d linear spatially invariant causal system having an input output relationship described by y z 1 z 2 p q 0 0 m n b p q z 1 p z 2 q p q 0 0 m n a p q z 1 p z 2 q x z 1 z 2 displaystyle y z_ 1 z_ 2 sum _ p q 0 0 m n b_ p q z_ 1 p z_ 2 q over sum _ p q 0 0 m n a_ p q z_ 1 p z_ 2 q x z_ 1 z_ 2 two cases are individually considered 1 the bottom summation is simply the constant 1 2 the top summation is simply a constant k displaystyle k case 1 is often called the all zero or finite impulse response case whereas case 2 is called the all pole or infinite impulse response case the general situation can be implemented as a cascade of the two individual cases the solution for case 1 is considerably simpler than case 2 and is shown below example all zero or finite impulse response edit y z 1 z 2 p q 0 0 m n b p q z 1 p z 2 q x z 1 z 2 displaystyle y z_ 1 z_ 2 sum _ p q 0 0 m n b_ p q z_ 1 p z_ 2 q x z_ 1 z_ 2 the state space vectors will have the following dimensions r 1 m s 1 n x 1 1 displaystyle r 1 times m quad s 1 times n quad x 1 times 1 and y 1 1 displaystyle y 1 times 1 each term in the summation involves a negative or zero power of z 1 displaystyle z_ 1 and of z 2 displaystyle z_ 2 which correspond to a delay or shift along the respective dimension of the input x i j displaystyle x i j this delay can be effected by placing 1 displaystyle 1 s along the super diagonal in the a 1 displaystyle a_ 1 and a 4 displaystyle a_ 4 matrices and the multiplying coefficients b i j displaystyle b_ i j in the proper positions in the a 2 displaystyle a_ 2 the value b 0 0 displaystyle b_ 0 0 is placed in the upper position of the b 1 displaystyle b_ 1 matrix which will multiply the input x i j displaystyle x i j and add it to the first component of the r i j displaystyle r_ i j vector also a value of b 0 0 displaystyle b_ 0 0 is placed in the d displaystyle d matrix which will multiply the input x i j displaystyle x i j and add it to the output y displaystyle y the matrices then appear as follows a 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 displaystyle a_ 1 begin bmatrix 0 0 0 cdots 0 0 1 0 0 cdots 0 0 0 1 0 cdots 0 0 vdots vdots vdots ddots vdots vdots 0 0 0 cdots 0 0 0 0 0 cdots 1 0 end bmatrix a 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 displaystyle a_ 2 begin bmatrix 0 0 0 cdots 0 0 0 0 0 cdots 0 0 0 0 0 cdots 0 0 vdots vdots vdots ddots vdots vdots 0 0 0 cdots 0 0 0 0 0 cdots 0 0 end bmatrix a 3 b 1 n b 2 n b 3 n b m 1 n b m n b 1 n 1 b 2 n 1 b 3 n 1 b m 1 n 1 b m n 1 b 1 n 2 b 2 n 2 b 3 n 2 b m 1 n 2 b m n 2 b 1 2 b 2 2 b 3 2 b m 1 2 b m 2 b 1 1 b 2 1 b 3 1 b m 1 1 b m 1 displaystyle a_ 3 begin bmatrix b_ 1 n b_ 2 n b_ 3 n cdots b_ m 1 n b_ m n b_ 1 n 1 b_ 2 n 1 b_ 3 n 1 cdots b_ m 1 n 1 b_ m n 1 b_ 1 n 2 b_ 2 n 2 b_ 3 n 2 cdots b_ m 1 n 2 b_ m n 2 vdots vdots vdots ddots vdots vdots b_ 1 2 b_ 2 2 b_ 3 2 cdots b_ m 1 2 b_ m 2 b_ 1 1 b_ 2 1 b_ 3 1 cdots b_ m 1 1 b_ m 1 end bmatrix a 4 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 displaystyle a_ 4 begin bmatrix 0 0 0 cdots 0 0 1 0 0 cdots 0 0 0 1 0 cdots 0 0 vdots vdots vdots ddots vdots vdots 0 0 0 cdots 0 0 0 0 0 cdots 1 0 end bmatrix b 1 1 0 0 0 0 0 displaystyle b_ 1 begin bmatrix 1 0 0 0 vdots 0 0 end bmatrix b 2 b 0 n b 0 n 1 b 0 n 2 b 0 2 b 0 1 displaystyle b_ 2 begin bmatrix b_ 0 n b_ 0 n 1 b_ 0 n 2 vdots b_ 0 2 b_ 0 1 end bmatrix c 1 b 1 0 b 2 0 b 3 0 b m 1 0 b m 0 displaystyle c_ 1 begin bmatrix b_ 1 0 b_ 2 0 b_ 3 0 cdots b_ m 1 0 b_ m 0 end bmatrix c 2 0 0 0 0 1 displaystyle c_ 2 begin bmatrix 0 0 0 cdots 0 1 end bmatrix d b 0 0 displaystyle d begin bmatrix b_ 0 0 end bmatrix 3 4 references edit bose n k ed 1985 multidimensional systems theory progress directions and open problems in multidimensional systems dordre http holland d reidel publishing company bose n k ed 1979 multidimensional systems theory and applications ieee press 1 2 tzafestas s g ed 1986 multidimensional systems techniques and applications new york marcel dekker 1 2 kaczorek t 1985 two dimensional linear systems lecture notes contr and inform sciences vol 68 springer verlag retrieved from https en wikipedia org w index php title multidimensional_system oldid 1310209881 categories digital imaging partial differential equations stability theory multidimensional signal processing hidden categories articles with short description short description matches wikidata this page was last edited on 8 september 2025 at 08 07 utc page 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