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Multidimensional system

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In mathematical <a href="Systems_theory" title="Systems theory">systems theory</a>, a multidimensional system or m-D system is a system in which not only one <a href="Independent_variable" class="mw-redirect" title="Independent variable">independent variable</a> exists (like time), but there are several independent variables.

Important problems such as <a href="Factorization" title="Factorization">factorization</a> and <a href="Stability_theory" title="Stability theory">stability</a> 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 <a href="Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a> does not exist in the <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> of m-D (m > 1) <a href="Polynomials" class="mw-redirect" title="Polynomials">polynomials</a>.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

Multidimensional systems or m-D systems are the necessary mathematical background for modern <a href="Digital_image_processing" title="Digital image processing">digital image processing</a> with many applications in <a href="Biomedicine" title="Biomedicine">biomedicine</a>, <a href="X-ray_technology" class="mw-redirect" title="X-ray technology">X-ray technology</a> and <a href="Satellite_communications" class="mw-redirect" title="Satellite communications">satellite communications</a>.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a>

There are also some studies combining m-D systems with <a href="Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a> (PDEs).

<div class="mw-heading mw-heading2"><h2 id="Linear_multidimensional_state-space_model">Linear multidimensional state-space model</h2></div>

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:<a href="#cite_note-Tzafestas-3">[3]</a><a href="#cite_note-Kaczorek-4">[4]</a>

Represent the input vector at each point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,j)}">

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<annotation encoding="application/x-tex">{\displaystyle (i,j)}</annotation>

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</math><img src="./8ef21910f980c6fca2b15bee102a7a0d861ed712.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle (i,j)}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(i,j)}">

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<annotation encoding="application/x-tex">{\displaystyle u(i,j)}</annotation>

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</math><img src="./f051e2e25bf6926b55409017dd95d680e18e4d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.933ex; height:2.843ex;" alt="{\displaystyle u(i,j)}" loading="lazy">, the output vector by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(i,j)}">

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<annotation encoding="application/x-tex">{\displaystyle y(i,j)}</annotation>

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</math><img src="./31b7111654b899d4d9ecc51749fb0ef416a9348c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.759ex; height:2.843ex;" alt="{\displaystyle y(i,j)}" loading="lazy"> the horizontal state vector by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(i,j)}">

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<annotation encoding="application/x-tex">{\displaystyle R(i,j)}</annotation>

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</math><img src="./063a40a7060a7792304bd9beb4e3c1a4d53f2c43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.368ex; height:2.843ex;" alt="{\displaystyle R(i,j)}" loading="lazy"> and the vertical state vector by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(i,j)}">

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<annotation encoding="application/x-tex">{\displaystyle S(i,j)}</annotation>

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</math><img src="./21e815291e13c789b81dad82a5df7458c9e315d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.103ex; height:2.843ex;" alt="{\displaystyle S(i,j)}" loading="lazy">. Then the operation at each point is defined by:

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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</math><img src="./d05f6e77b180caef5ccdfa8eed3e7e091e3896d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:46.718ex; height:9.176ex;" alt="{\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}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1},A_{2},A_{3},A_{4},B_{1},B_{2},C_{1},C_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{1},A_{2},A_{3},A_{4},B_{1},B_{2},C_{1},C_{2}}</annotation>

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</math><img src="./24a83f170e8ca9ca735a54e5d9e0e50f00593695.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.496ex; height:2.509ex;" alt="{\displaystyle A_{1},A_{2},A_{3},A_{4},B_{1},B_{2},C_{1},C_{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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These equations can be written more compactly by combining the matrices:

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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</math><img src="./4e092b2703ef9a74efabe5aa224c879b5d904b2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:44.451ex; height:9.843ex;" alt="{\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}}}" loading="lazy"></dd></dl>

Given input vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(i,j)}">

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</math><img src="./f051e2e25bf6926b55409017dd95d680e18e4d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.933ex; height:2.843ex;" alt="{\displaystyle u(i,j)}" loading="lazy"> at each point and initial state values, the value of each output vector can be computed by recursively performing the operation above.

<div class="mw-heading mw-heading2"><h2 id="Multidimensional_transfer_function">Multidimensional transfer function</h2></div>

A discrete linear two-dimensional system is often described by a partial difference equation in the form:

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<annotation encoding="application/x-tex">{\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)}</annotation>

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</math><img src="./544fdc50450e2e706d1fc9f1ec5b1f892bf8c932.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.063ex; width:49.082ex; height:7.343ex;" alt="{\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)}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(i,j)}">

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</math><img src="./fa564861b679d04c45d07bfcae975b0bbc17b576.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.933ex; height:2.843ex;" alt="{\displaystyle x(i,j)}" loading="lazy"> is the input and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(i,j)}">

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<annotation encoding="application/x-tex">{\displaystyle y(i,j)}</annotation>

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</math><img src="./31b7111654b899d4d9ecc51749fb0ef416a9348c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.759ex; height:2.843ex;" alt="{\displaystyle y(i,j)}" loading="lazy"> is the output at point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,j)}">

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</math><img src="./8ef21910f980c6fca2b15bee102a7a0d861ed712.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle (i,j)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{p,q}}">

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</math><img src="./587eb795764575ecac8e7f8d844de714f0fcf661.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.503ex; height:2.343ex;" alt="{\displaystyle a_{p,q}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{p,q}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{p,q}}</annotation>

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</math><img src="./e7abb7c15250705ecb3de4268fd21f51c6696454.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.27ex; height:2.843ex;" alt="{\displaystyle b_{p,q}}" loading="lazy"> are constant coefficients.

To derive a transfer function for the system the 2d Z-transform is applied to both sides of the equation above.

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<annotation encoding="application/x-tex">{\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})}</annotation>

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</math><img src="./b0fae9778c279996175ecb81402fca63d7181bee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-left: -0.063ex; width:53.102ex; height:7.343ex;" alt="{\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})}" loading="lazy"></dd></dl>

Transposing yields the transfer function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(z_{1},z_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle T(z_{1},z_{2})}</annotation>

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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</math><img src="./ab2c22968b199f2b790fc57dff7259401af4cbcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:44.495ex; height:7.843ex;" alt="{\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}}}" loading="lazy"></dd></dl>

So given any pattern of input values, the 2d Z-transform of the pattern is computed and then multiplied by the transfer function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(z_{1},z_{2})}">

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<div class="mw-heading mw-heading2"><h2 id="Realization_of_a_2d_transfer_function">Realization of a 2d transfer function</h2></div>

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:

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<annotation encoding="application/x-tex">{\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})}</annotation>

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</math><img src="./d85c786921caafeea0edee5ec6f7ccf13db72802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:40.698ex; height:7.843ex;" alt="{\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})}" loading="lazy"></dd></dl>

Two cases are individually considered 1) the bottom summation is simply the constant 1 2) the top summation is simply a constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">. 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.

<div class="mw-heading mw-heading3"><h3 id="Example:_all_zero_or_finite_impulse_response">Example: all zero or finite impulse response</h3></div>

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<annotation encoding="application/x-tex">{\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})}</annotation>

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</math><img src="./2a1cb599545f835ec8e1b76ce8dcc0e67794c694.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.943ex; height:7.343ex;" alt="{\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})}" loading="lazy"></dd></dl>

The state-space vectors will have the following dimensions:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(1\times m),\quad S(1\times n),\quad x(1\times 1)}</annotation>

</semantics>

</math><img src="./4f6e9887eedf0499355a7a9cd48f6a15c0fd30f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.34ex; height:2.843ex;" alt="{\displaystyle R(1\times m),\quad S(1\times n),\quad x(1\times 1)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(1\times 1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y(1\times 1)}</annotation>

</semantics>

</math><img src="./82fc519b6a29e3aeb15883e59d1906efb7f30cc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.13ex; height:2.843ex;" alt="{\displaystyle y(1\times 1)}" loading="lazy"></dd></dl>

Each term in the summation involves a negative (or zero) power of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{1}}</annotation>

</semantics>

</math><img src="./c3621e468231ab352b7caa30bcf0ce9b452241a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{1}}" loading="lazy"> and of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{2}}</annotation>

</semantics>

</math><img src="./5abf655fa14f7ea44ad0ca781b59ff59c5f49117.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{2}}" loading="lazy"> which correspond to a delay (or shift) along the respective dimension of the input <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(i,j)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x(i,j)}</annotation>

</semantics>

</math><img src="./fa564861b679d04c45d07bfcae975b0bbc17b576.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.933ex; height:2.843ex;" alt="{\displaystyle x(i,j)}" loading="lazy">. This delay can be effected by placing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">’s along the super diagonal in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{1}}</annotation>

</semantics>

</math><img src="./6bc2435b217c1a0f46f8a517ffa225c6f9440e81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{1}}" loading="lazy">. and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{4}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{4}}</annotation>

</semantics>

</math><img src="./8e3eaac426165e199a4a745aa46bb8028fed100d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{4}}" loading="lazy"> matrices and the multiplying coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{i,j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{i,j}}</annotation>

</semantics>

</math><img src="./c01bc815e19e0446968a07c762eb225c3738a65b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.932ex; height:2.843ex;" alt="{\displaystyle b_{i,j}}" loading="lazy"> in the proper positions in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{2}}</annotation>

</semantics>

</math><img src="./3ec73b8bc9abc3efb934f5a6ec2803713771f4bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{2}}" loading="lazy">. The value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0,0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0,0}}</annotation>

</semantics>

</math><img src="./4dac2aba3397b8dd5cdffd10715b630c6869886f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.331ex; height:2.843ex;" alt="{\displaystyle b_{0,0}}" loading="lazy"> is placed in the upper position of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{1}}</annotation>

</semantics>

</math><img src="./1fa091eb428443c9c5c5fcf32a69d3665c89e00c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle B_{1}}" loading="lazy"> matrix, which will multiply the input <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(i,j)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x(i,j)}</annotation>

</semantics>

</math><img src="./fa564861b679d04c45d07bfcae975b0bbc17b576.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.933ex; height:2.843ex;" alt="{\displaystyle x(i,j)}" loading="lazy"> and add it to the first component of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{i,j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{i,j}}</annotation>

</semantics>

</math><img src="./7bb6f9df8716f0eb2d1e8bc277932500b81eb62b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.699ex; height:2.843ex;" alt="{\displaystyle R_{i,j}}" loading="lazy"> vector. Also, a value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0,0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0,0}}</annotation>

</semantics>

</math><img src="./4dac2aba3397b8dd5cdffd10715b630c6869886f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.331ex; height:2.843ex;" alt="{\displaystyle b_{0,0}}" loading="lazy"> is placed in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> matrix which will multiply the input <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(i,j)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x(i,j)}</annotation>

</semantics>

</math><img src="./fa564861b679d04c45d07bfcae975b0bbc17b576.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.933ex; height:2.843ex;" alt="{\displaystyle x(i,j)}" loading="lazy"> and add it to the output <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">.

The matrices then appear as follows:

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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}}}</annotation>

</semantics>

</math><img src="./1e547aa96103c038b551f3c300333fe94eadd093.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.838ex; width:30.152ex; height:20.843ex;" alt="{\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}}}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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}}}</annotation>

</semantics>

</math><img src="./988068cc880d4b25c4d9e2d0d5f8d6c373d7c785.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.838ex; width:30.152ex; height:20.843ex;" alt="{\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}}}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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}}}</annotation>

</semantics>

</math><img src="./27a243d71d066bd701d127d0457caa3a14136a60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:55.661ex; height:21.843ex;" alt="{\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}}}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

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</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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}}}</annotation>

</semantics>

</math><img src="./b7acaaf187ac7ecd73217253f8e79e5b51f2ea75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.838ex; width:30.152ex; height:20.843ex;" alt="{\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}}}" loading="lazy">

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{1}={\begin{bmatrix}1\\0\\0\\0\\\vdots \\0\\0\end{bmatrix}}}</annotation>

</semantics>

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle C_{1}={\begin{bmatrix}b_{1,0}&b_{2,0}&b_{3,0}&\cdots &b_{m-1,0}&b_{m,0}\\\end{bmatrix}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle C_{2}={\begin{bmatrix}0&0&0&\cdots &0&1\\\end{bmatrix}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle D={\begin{bmatrix}b_{0,0}\end{bmatrix}}}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBose1985" class="citation book cs1">Bose, N.K., ed. (1985). Multidimensional Systems Theory, Progress, Directions and Open Problems in Multidimensional Systems. Dordre http, Holland: D. Reidel Publishing Company.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFBose1979" class="citation book cs1">Bose, N.K., ed. (1979). Multidimensional Systems: Theory and Applications. IEEE Press.</cite>

</li>

<li id="cite_note-Tzafestas-3">^ <a href="#cite_ref-Tzafestas_3-0">a</a> <a href="#cite_ref-Tzafestas_3-1">b</a> <cite id="CITEREFTzafestas1986" class="citation book cs1">Tzafestas, S.G., ed. (1986). Multidimensional Systems: Techniques and Applications. New York: Marcel-Dekker.</cite>

</li>

<li id="cite_note-Kaczorek-4">^ <a href="#cite_ref-Kaczorek_4-0">a</a> <a href="#cite_ref-Kaczorek_4-1">b</a> <cite id="CITEREFKaczorek1985" class="citation book cs1">Kaczorek, T. (1985). Two-Dimensional Linear Systems. Lecture Notes Contr. and Inform. Sciences. Vol. 68. Springer-Verlag.</cite>

</li>

</ol></div></div></div><div><div class="zim-footer">

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