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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-06-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/12/1945 |
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| Summary: | 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. |
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| ISSN: | 2227-7390 |