Application of the Algebraic Extension Method to the Construction of Orthogonal Bases for Partial Digital Convolutions
Mathematical tools have been developed that are analogous to the tool that allows one to reduce the description of linear systems in terms of convolution operations to a description in terms of amplitude-frequency characteristics. These tools are intended for use in cases where the system under cons...
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| author | Aruzhan Kadyrzhan Akhat Bakirov Dina Shaltykova Ibragim Suleimenov |
| author_facet | Aruzhan Kadyrzhan Akhat Bakirov Dina Shaltykova Ibragim Suleimenov |
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| description | Mathematical tools have been developed that are analogous to the tool that allows one to reduce the description of linear systems in terms of convolution operations to a description in terms of amplitude-frequency characteristics. These tools are intended for use in cases where the system under consideration is described by partial digital convolutions. The basis of the proposed approach is the Fourier–Galois transform using orthogonal bases in corresponding fields. As applied to partial convolutions, the Fourier–Galois transform is decomposed into a set of such transforms, each of which corresponds to operations in a certain Galois field. It is shown that for adequate application of the Fourier–Galois transform to systems described by partial convolutions, it is necessary to ensure the same number of cycles in each of the transforms from the set specified above. To solve this problem, the method of algebraic extensions was used, a special case of which is the transition from real numbers to complex numbers. In this case, the number of cycles varies from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow><mo>/</mo><mrow><mi>k</mi></mrow></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow></semantics></math></inline-formula> is a prime number, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi></mrow></semantics></math></inline-formula> are integers, and an arbitrary number divisor of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></semantics></math></inline-formula> can be chosen as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi></mrow></semantics></math></inline-formula>. This allows us to produce partial Fourier–Galois transforms corresponding to different Galois fields, for the same number of cycles. A specific example is presented demonstrating the constructiveness of the proposed approach. |
| format | Article |
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| spelling | doaj-art-ba8a618d5ffd477090c78ffff34d08782025-08-20T01:53:42ZengMDPI AGAlgorithms1999-48932024-11-01171149610.3390/a17110496Application of the Algebraic Extension Method to the Construction of Orthogonal Bases for Partial Digital ConvolutionsAruzhan Kadyrzhan0Akhat Bakirov1Dina Shaltykova2Ibragim Suleimenov3Institute of Communications and Space Engineering, Gumarbek Daukeev Almaty University of Power Engineering and Communications, Almaty 050040, KazakhstanInstitute of Communications and Space Engineering, Gumarbek Daukeev Almaty University of Power Engineering and Communications, Almaty 050040, KazakhstanNational Engineering Academy of the Republic of Kazakhstan, Almaty 050010, KazakhstanNational Engineering Academy of the Republic of Kazakhstan, Almaty 050010, KazakhstanMathematical tools have been developed that are analogous to the tool that allows one to reduce the description of linear systems in terms of convolution operations to a description in terms of amplitude-frequency characteristics. These tools are intended for use in cases where the system under consideration is described by partial digital convolutions. The basis of the proposed approach is the Fourier–Galois transform using orthogonal bases in corresponding fields. As applied to partial convolutions, the Fourier–Galois transform is decomposed into a set of such transforms, each of which corresponds to operations in a certain Galois field. It is shown that for adequate application of the Fourier–Galois transform to systems described by partial convolutions, it is necessary to ensure the same number of cycles in each of the transforms from the set specified above. To solve this problem, the method of algebraic extensions was used, a special case of which is the transition from real numbers to complex numbers. In this case, the number of cycles varies from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow><mo>/</mo><mrow><mi>k</mi></mrow></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow></semantics></math></inline-formula> is a prime number, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi></mrow></semantics></math></inline-formula> are integers, and an arbitrary number divisor of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></semantics></math></inline-formula> can be chosen as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi></mrow></semantics></math></inline-formula>. This allows us to produce partial Fourier–Galois transforms corresponding to different Galois fields, for the same number of cycles. A specific example is presented demonstrating the constructiveness of the proposed approach.https://www.mdpi.com/1999-4893/17/11/496orthogonal basesalgebraic extensionsdigital convolutionsFourier–Galois transformstransfer functionlinear systems |
| spellingShingle | Aruzhan Kadyrzhan Akhat Bakirov Dina Shaltykova Ibragim Suleimenov Application of the Algebraic Extension Method to the Construction of Orthogonal Bases for Partial Digital Convolutions Algorithms orthogonal bases algebraic extensions digital convolutions Fourier–Galois transforms transfer function linear systems |
| title | Application of the Algebraic Extension Method to the Construction of Orthogonal Bases for Partial Digital Convolutions |
| title_full | Application of the Algebraic Extension Method to the Construction of Orthogonal Bases for Partial Digital Convolutions |
| title_fullStr | Application of the Algebraic Extension Method to the Construction of Orthogonal Bases for Partial Digital Convolutions |
| title_full_unstemmed | Application of the Algebraic Extension Method to the Construction of Orthogonal Bases for Partial Digital Convolutions |
| title_short | Application of the Algebraic Extension Method to the Construction of Orthogonal Bases for Partial Digital Convolutions |
| title_sort | application of the algebraic extension method to the construction of orthogonal bases for partial digital convolutions |
| topic | orthogonal bases algebraic extensions digital convolutions Fourier–Galois transforms transfer function linear systems |
| url | https://www.mdpi.com/1999-4893/17/11/496 |
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