A Geometric Algebra Framework for Vector-Valued Signals and Invariant Systems
Guardado en:
| Udgivet i: | ProQuest Dissertations and Theses (2025) |
|---|---|
| Hovedforfatter: | |
| Udgivet: |
ProQuest Dissertations & Theses
|
| Fag: | |
| Online adgang: | Citation/Abstract Full Text - PDF |
| Tags: |
Ingen Tags, Vær først til at tagge denne postø!
|
MARC
| LEADER | 00000nab a2200000uu 4500 | ||
|---|---|---|---|
| 001 | 3246803815 | ||
| 003 | UK-CbPIL | ||
| 020 | |a 9798293811601 | ||
| 035 | |a 3246803815 | ||
| 045 | 2 | |b d20250101 |b d20251231 | |
| 084 | |a 66569 |2 nlm | ||
| 100 | 1 | |a Dalal, Mamta | |
| 245 | 1 | |a A Geometric Algebra Framework for Vector-Valued Signals and Invariant Systems | |
| 260 | |b ProQuest Dissertations & Theses |c 2025 | ||
| 513 | |a Dissertation/Thesis | ||
| 520 | 3 | |a Convolution and Fourier transform operators are key concepts of signal processing, providing essential tools for analyzing linear time-invariant (LTI) systems in the time-domain and frequency-domain, respectively. These fundamental operators have been instrumental in advancing the field of signal processing and image processing. The traditional LTI systems theory deals with complex-valued (CV) time series signals. The CV signals representation follows linearity with CV scalars exhibiting an inbuilt rotation-invariance (RI) property. To acknowledge the presence of RI in CV signals, the CV product in the conventional convolution definition is interpreted in terms of a scale rotation. It is important to note that replacing the CV scalars with real scalars results in a loss of the rotational invariance property. We introduce a linear rotation-invariant time-invariant (LRITI) system with vector-valued (VV) signals. We develop an analogous theory to characterize LRITI systems using VV signals with a new tool called geometric algebra (GA). We define the RI property for VV systems using GA where only real numbers are considered as scalars. To begin with the proposed GA-based formulation, we generalize the convolution operation for VV systems using rotor representation. In addition, we provide a compatible frequency-domain analysis for VV signals and LRITI systems. First, VV bivector exponential signals are shown to be eigen-functions of LRITI systems. A Fourier transform is defined, and we propose two generalized definitions of frequency response: the first valid for bivector exponential in an arbitrary plane and the second valid for a general signal decomposed into a set of totally orthogonal planes (TOPs). Finally, we establish a convolution property for the Fourier transform with respect to TOPs. Together, these results provide compatible time-domain and frequency-domain analyses, thereby enabling a more comprehensive analysis of VV signals and LRITI systems. | |
| 653 | |a Electrical engineering | ||
| 653 | |a Applied mathematics | ||
| 653 | |a Computer science | ||
| 653 | |a Mechanical engineering | ||
| 773 | 0 | |t ProQuest Dissertations and Theses |g (2025) | |
| 786 | 0 | |d ProQuest |t ProQuest Dissertations & Theses Global | |
| 856 | 4 | 1 | |3 Citation/Abstract |u https://www.proquest.com/docview/3246803815/abstract/embedded/7BTGNMKEMPT1V9Z2?source=fedsrch |
| 856 | 4 | 0 | |3 Full Text - PDF |u https://www.proquest.com/docview/3246803815/fulltextPDF/embedded/7BTGNMKEMPT1V9Z2?source=fedsrch |