Geometry of complex data

Sangston, K. James

doi:10.1109/taes.2016.150029

Cited by 7 publications

(4 citation statements)

References 109 publications

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“…Geometric algebra has its origins in the work of Clifford and Grassman in the 19th century and is considered as a unified language for mathematics and physics. It is based on the notion of an invertible product of vectors that captures the geometric relationship between two vectors, i.e., their relative magnitudes and the angle between them [23]. Some investigations have defined the properties of geometric algebra [24,25] applied to physics and engineering.…”

Section: Basic Definitions Of Geometric Algebramentioning

confidence: 99%

Quadrature Current Compensation in Non-Sinusoidal Circuits Using Geometric Algebra and Evolutionary Algorithms

2019

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Non-linear loads in circuits cause the appearance of harmonic disturbances both in voltage and current. In order to minimize the effects of these disturbances and, therefore, to control the flow of electricity between the source and the load, passive or active filters are often used. Nevertheless, determining the type of filter and the characteristics of their elements is not a trivial task. In fact, the development of algorithms for calculating the parameters of filters is still an open question. This paper analyzes the use of genetic algorithms to maximize the power factor compensation in non-sinusoidal circuits using passive filters, while concepts of geometric algebra theory are used to represent the flow of power in the circuits. According to the results obtained in different case studies, it can be concluded that the genetic algorithm obtains high quality solutions that could be generalized to similar problems of any dimension.

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Section: Basic Definitions Of Geometric Algebramentioning

confidence: 99%

Quadrature Current Compensation in Non-Sinusoidal Circuits Using Geometric Algebra and Evolutionary Algorithms

2019

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“…In particular, the projection angle has been used for radar signal detection, but the design details and performance analysis are not provided for specific detection scenarios. 20 This paper proposes and derives the geometric meaning of complex vectors' inner product based on the GA projection processing, which is projecting the vector onto the hyperplane. Then, the method is applied to two typical radar complex signal processing, i.e., discrete Fourier transform (DFT) 21 and matched filtering.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, the GA tools have been applied to signal processing, 12 , 14 , 16 radar, 17 image processing, 18 robotics, 19 and other fields. In particular, the projection angle has been used for radar signal detection, but the design details and performance analysis are not provided for specific detection scenarios 20 …”

Section: Introductionmentioning

confidence: 99%

Application of geometric algebraic projection operator in radar complex signal processing

Jiang

Fan

2023

J. Appl. Rem. Sens.

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.A variable projection angle is obtained based on the geometric algebra (GA) projection operator using the geometric meaning of complex vectors’ inner product. Considered that the inner product processing is widely used in radar applications, i.e., radar signal detection and matched filtering, a geometric detector, and a geometric matched filter are proposed based on the projection angle in the GA framework. Several simulations, as well as experiments, confirm that the proposed geometric detector overperforms the traditional constant false-alarm rate (CFAR) detector, especially in the uniform background, and in adaptions to the clutter edge and large dynamic change of signals. Moreover, the higher efficiency of the geometric detector is indicated by the analysis comparison of computation complexity, which indicated that the geometric detector possesses higher efficiency than the traditional CFAR detectors. The response characteristics of the geometric matched filter for intermittent sampling and repeater jamming (ISRJ) are also validated through theoretical and simulation analysis. The presented results provide a geometric perspective for solving the design of radar detectors and the suppression of ISRJ.

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“…However, standard AFs treat different dimensions as multi-channel signals, which may lose the structural information between different dimensions [15]. The GA-based adaptive filters (GAAF) have been used for simultaneous filtering of multi-dimensional signals for its faster convergence speed and suitability of multivector for multi-dimensional signal modeling.…”

mentioning

confidence: 99%

Geometric-algebra affine projection adaptive filter

Ren

Zhi

Zhang

2021

EURASIP J. Adv. Signal Process.

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Geometric algebra (GA) is an efficient tool to deal with hypercomplex processes due to its special data structure. In this article, we introduce the affine projection algorithm (APA) in the GA domain to provide fast convergence against hypercomplex colored signals. Following the principle of minimal disturbance and the orthogonal affine subspace theory, we formulate the criterion of designing the GA-APA as a constrained optimization problem, which can be solved by the method of Lagrange Multipliers. Then, the differentiation of the cost function is calculated using geometric calculus (the extension of GA to include differentiation) to get the update formula of the GA-APA. The stability of the algorithm is analyzed based on the mean-square deviation. To avoid ill-posed problems, the regularized GA-APA is also given in the following. The simulation results show that the proposed adaptive filters, in comparison with existing methods, achieve a better convergence performance under the condition of colored input signals.

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Geometry of complex data

Cited by 7 publications

References 109 publications

Quadrature Current Compensation in Non-Sinusoidal Circuits Using Geometric Algebra and Evolutionary Algorithms

Quadrature Current Compensation in Non-Sinusoidal Circuits Using Geometric Algebra and Evolutionary Algorithms

Application of geometric algebraic projection operator in radar complex signal processing

Geometric-algebra affine projection adaptive filter

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