Hypercomplex Numbers—A Tool for Enhanced Efficiency and Intelligence in Digital Signal Processing

Zlatka Valkova-Jarvis (), Maria Nenova and Dimitriya Mihaylova
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Zlatka Valkova-Jarvis: Faculty of Telecommunications, Technical University of Sofia, 1000 Sofia, Bulgaria
Maria Nenova: Faculty of Telecommunications, Technical University of Sofia, 1000 Sofia, Bulgaria
Dimitriya Mihaylova: Faculty of Telecommunications, Technical University of Sofia, 1000 Sofia, Bulgaria

Mathematics, 2025, vol. 13, issue 3, 1-29

Abstract: Mathematics is the wide-ranging solid foundation of the engineering sciences which ensures their progress by providing them with its unique toolkit of rules, methods, algorithms and numerical systems. In this paper, an overview of the numerical systems that have currently found an application in engineering science and practice is offered, while also mentioning those systems that still await full and comprehensive applicability, recognition, and acknowledgment. Two possible approaches for representing hypercomplex numbers are proposed—based on real numbers and based on complex numbers. This makes it possible to justify the creation and introduction of numerical systems specifically suited to digital signal processing (DSP), which is the basis of all modern technical sciences ensuring the technological progress of mankind. Understanding the specifics, peculiarities, and properties of the large and diverse family of hypercomplex numbers is the first step towards their more comprehensive and thorough study, and hence their use in a number of high-tech intelligent applications in various engineering and scientific fields, such as information and communication technologies (ICT), communication and neural networks, cybersecurity and national security, artificial intelligence (АI), space and military technologies, industrial engineering and machine learning, astronomy, applied mathematics, quantum physics, etc. The issues discussed in this paper are, however, far from exhausting the scientific topics related to both hypercomplex numbers in general and those relevant to DSP. This is a promising scientific area, the potential of which has not yet been fully explored, but research already shows the enhanced computational efficiency and intelligent performance of hypercomplex DSP.

Keywords: real; complex; hypercomplex numbers; quaternions; biquaternions; bicomplex numbers; digital signal processing; Cayley–Dickson algebra (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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