Tensor z-Transform
The multi-input multioutput (MIMO) systems involving multirelational signals generated from distributed sources have been emerging as the most generalized model in practice. The existing work for characterizing such a MIMO system is to build a corresponding transform tensor, each of whose entries tu...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2024-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2024/6614609 |
| _version_ | 1797206060119883776 |
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| author | Shih Yu Chang Hsiao-Chun Wu |
| author_facet | Shih Yu Chang Hsiao-Chun Wu |
| author_sort | Shih Yu Chang |
| collection | DOAJ |
| description | The multi-input multioutput (MIMO) systems involving multirelational signals generated from distributed sources have been emerging as the most generalized model in practice. The existing work for characterizing such a MIMO system is to build a corresponding transform tensor, each of whose entries turns out to be the individual z-transform of a discrete-time impulse response sequence. However, when a MIMO system has a global feedback mechanism, which also involves multirelational signals, the aforementioned individual z-transforms of the overall transfer tensor are quite difficult to formulate. Therefore, a new mathematical framework to govern both feedforward and feedback MIMO systems is in crucial demand. In this work, we define the tensor z-transform to characterize a MIMO system involving multirelational signals as a whole rather than the individual entries separately, which is a novel concept for system analysis. To do so, we extend Cauchy’s integral formula and Cauchy’s residue theorem from scalars to arbitrary-dimensional tensors, and then, to apply these new mathematical tools, we establish to undertake the inverse tensor z-transform and approximate the corresponding discrete-time tensor sequences. Our proposed new tensor z-transform in this work can be applied to design digital tensor filters including infinite-impulse-response (IIR) tensor filters (involving global feedback mechanisms) and finite-impulse-response (FIR) tensor filters. Finally, numerical evaluations are presented to demonstrate certain interesting phenomena of the new tensor z-transform. |
| first_indexed | 2024-04-24T09:01:00Z |
| format | Article |
| id | doaj.art-a0d07e166f25402eb66bee22cf857053 |
| institution | Directory Open Access Journal |
| issn | 1687-0042 |
| language | English |
| last_indexed | 2024-04-24T09:01:00Z |
| publishDate | 2024-01-01 |
| publisher | Hindawi Limited |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj.art-a0d07e166f25402eb66bee22cf8570532024-04-16T00:00:04ZengHindawi LimitedJournal of Applied Mathematics1687-00422024-01-01202410.1155/2024/6614609Tensor z-TransformShih Yu Chang0Hsiao-Chun Wu1Department of Applied Data ScienceSchool of Electrical Engineering and Computer ScienceThe multi-input multioutput (MIMO) systems involving multirelational signals generated from distributed sources have been emerging as the most generalized model in practice. The existing work for characterizing such a MIMO system is to build a corresponding transform tensor, each of whose entries turns out to be the individual z-transform of a discrete-time impulse response sequence. However, when a MIMO system has a global feedback mechanism, which also involves multirelational signals, the aforementioned individual z-transforms of the overall transfer tensor are quite difficult to formulate. Therefore, a new mathematical framework to govern both feedforward and feedback MIMO systems is in crucial demand. In this work, we define the tensor z-transform to characterize a MIMO system involving multirelational signals as a whole rather than the individual entries separately, which is a novel concept for system analysis. To do so, we extend Cauchy’s integral formula and Cauchy’s residue theorem from scalars to arbitrary-dimensional tensors, and then, to apply these new mathematical tools, we establish to undertake the inverse tensor z-transform and approximate the corresponding discrete-time tensor sequences. Our proposed new tensor z-transform in this work can be applied to design digital tensor filters including infinite-impulse-response (IIR) tensor filters (involving global feedback mechanisms) and finite-impulse-response (FIR) tensor filters. Finally, numerical evaluations are presented to demonstrate certain interesting phenomena of the new tensor z-transform.http://dx.doi.org/10.1155/2024/6614609 |
| spellingShingle | Shih Yu Chang Hsiao-Chun Wu Tensor z-Transform Journal of Applied Mathematics |
| title | Tensor z-Transform |
| title_full | Tensor z-Transform |
| title_fullStr | Tensor z-Transform |
| title_full_unstemmed | Tensor z-Transform |
| title_short | Tensor z-Transform |
| title_sort | tensor z transform |
| url | http://dx.doi.org/10.1155/2024/6614609 |
| work_keys_str_mv | AT shihyuchang tensorztransform AT hsiaochunwu tensorztransform |