Decrease in Computational Load and Increase in Accuracy for Filtering of Random Signals
This paper describes methods for optimal filtering of random signals that involve large matrices. We developed a procedure that allows us to significantly decrease the computational load associated with numerically implementing the associated filter and increase its accuracy. The procedure is based...
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MDPI AG
2025-06-01
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| author | Phil Howlett Anatoli Torokhti Pablo Soto-Quiros |
| author_facet | Phil Howlett Anatoli Torokhti Pablo Soto-Quiros |
| author_sort | Phil Howlett |
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| description | This paper describes methods for optimal filtering of random signals that involve large matrices. We developed a procedure that allows us to significantly decrease the computational load associated with numerically implementing the associated filter and increase its accuracy. The procedure is based on the reduction of a large covariance matrix to a collection of smaller matrices. This is done in such a way that the filter equation with large matrices is equivalently represented by a set of equations with smaller matrices. The filter we developed is represented by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">x</mi><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></msubsup><msub><mi mathvariant="bold">M</mi><mi>j</mi></msub><msub><mi mathvariant="bold">y</mi><mi>j</mi></msub></mrow></semantics></math></inline-formula> and minimizes the associated error over all matrices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">M</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi mathvariant="bold">M</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula>. As a result, the proposed optimal filter has two degrees of freedom that increase its accuracy. They are associated, first, with the optimal determination of matrices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">M</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi mathvariant="bold">M</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula> and second, with an increase in the number <i>p</i> of components in the filter. The error analysis and results of numerical simulations are provided. |
| format | Article |
| id | doaj-art-ad0bb3f6d5864feb96e06d7a43d7ed7a |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-ad0bb3f6d5864feb96e06d7a43d7ed7a2025-08-20T03:16:34ZengMDPI AGMathematics2227-73902025-06-011312194510.3390/math13121945Decrease in Computational Load and Increase in Accuracy for Filtering of Random SignalsPhil Howlett0Anatoli Torokhti1Pablo Soto-Quiros2STEM Discipline, University of South Australia, Mawson Lakes, Adelaide, SA 5001, AustraliaSTEM Discipline, University of South Australia, Mawson Lakes, Adelaide, SA 5001, AustraliaEscuela de Matemática, Instituto Tecnológico de Costa Rica, Cartago 30101, Costa RicaThis paper describes methods for optimal filtering of random signals that involve large matrices. We developed a procedure that allows us to significantly decrease the computational load associated with numerically implementing the associated filter and increase its accuracy. The procedure is based on the reduction of a large covariance matrix to a collection of smaller matrices. This is done in such a way that the filter equation with large matrices is equivalently represented by a set of equations with smaller matrices. The filter we developed is represented by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">x</mi><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></msubsup><msub><mi mathvariant="bold">M</mi><mi>j</mi></msub><msub><mi mathvariant="bold">y</mi><mi>j</mi></msub></mrow></semantics></math></inline-formula> and minimizes the associated error over all matrices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">M</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi mathvariant="bold">M</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula>. As a result, the proposed optimal filter has two degrees of freedom that increase its accuracy. They are associated, first, with the optimal determination of matrices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">M</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi mathvariant="bold">M</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula> and second, with an increase in the number <i>p</i> of components in the filter. The error analysis and results of numerical simulations are provided.https://www.mdpi.com/2227-7390/13/12/1945large covariance matricesleast squares linear estimatesingular value decompositionerror minimization |
| spellingShingle | Phil Howlett Anatoli Torokhti Pablo Soto-Quiros Decrease in Computational Load and Increase in Accuracy for Filtering of Random Signals Mathematics large covariance matrices least squares linear estimate singular value decomposition error minimization |
| title | Decrease in Computational Load and Increase in Accuracy for Filtering of Random Signals |
| title_full | Decrease in Computational Load and Increase in Accuracy for Filtering of Random Signals |
| title_fullStr | Decrease in Computational Load and Increase in Accuracy for Filtering of Random Signals |
| title_full_unstemmed | Decrease in Computational Load and Increase in Accuracy for Filtering of Random Signals |
| title_short | Decrease in Computational Load and Increase in Accuracy for Filtering of Random Signals |
| title_sort | decrease in computational load and increase in accuracy for filtering of random signals |
| topic | large covariance matrices least squares linear estimate singular value decomposition error minimization |
| url | https://www.mdpi.com/2227-7390/13/12/1945 |
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