Hypercomplex numbers in digital signal processing

Schütte, H.; Wenzel, Jakob

doi:10.1109/iscas.1990.112431

Cited by 55 publications

(28 citation statements)

References 2 publications

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“…As it is often (but not always) the case, engineers were precursors and a number of papers have appeared in the recent years studying digital signal processing and associated filters in the setting of bicomplex numbers (which they often called reduced biquaternions). We mention as a sample [42][43][44][45][46][47], and, for a study in the setting of hyperbolic (and multi-hyperbolic numbers), [48]. The motivation in these papers to use the bicomplex numbers is to have augmented parallelism and more efficient computations.…”

Section: Introductionmentioning

confidence: 99%

Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis

Alpay

Luna-Elizarrarás²,

Shapiro³

et al. 2014

SpringerBriefs in Mathematics

117

118

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Contents Introduction Chapter 1. Bicomplex and hyperbolic numbers 1.1. Bicomplex numbers 1.2. Conjugations and moduli. 1.3. The Euclidean norm on BC 1.4. Idempotent decompositions 1.5. A partial order on D and a hyperbolic-valued norm Chapter 2. Bicomplex functions and matrices 2.1. Bicomplex holomorphic functions 2.2. Bicomplex matrices Chapter 3. BC-modules 3.1. BC-modules and involutions on them 3.2. Constructing a a BC-module from two complex linear spaces. Chapter 4. Norms and inner products on BC-modules 4.1. Real-valued norms on bicomplex modules 4.2. D-valued norm on BC-modules 4.3. Bicomplex modules with inner product. 4.4. Inner products and cartesian decompositions 4.5. Inner products and idempotent decompositions. 4.6. Complex inner products on X induced by idempotent decompositions. 4.7. The bicomplex module BC n . 4.8. The ring H(C) of biquaternions as a BC-module Chapter 5. Linear functionals and linear operators on BCmodules 5.1. Bicomplex linear functionals 5.2. Polarization identities 5.3. Linear operators on BC-modules.

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Section: Introductionmentioning

confidence: 99%

Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis

Alpay

Luna-Elizarrarás²,

Shapiro³

et al. 2014

SpringerBriefs in Mathematics

117

118

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“…It has four components, i.e. (1) where , , , and are real and , , and satisfy (2) From (2), the multiplication of quaternions is not commutative. Owning to this, many operations, such as Fourier transforms [47] and convolutions, are different from those of the complex algebra [25] and the eigenvalues of a quaternion matrix boil down to two categories, left and right eigenvalues [5] ( 3) In ( a quaternion matrix , then every element of the set is also an eigenvalue of [5].…”

Section: Introductionmentioning

confidence: 99%

“…Owning to this, many operations, such as Fourier transforms [47] and convolutions, are different from those of the complex algebra [25] and the eigenvalues of a quaternion matrix boil down to two categories, left and right eigenvalues [5] ( 3) In ( a quaternion matrix , then every element of the set is also an eigenvalue of [5]. On the other hand, the concept of reduced biquaternions (RBs) was first introduced by Schütte and Wenzel [2]. The major difference between RBs and quaternions is the multiplication rules, which are commutative for RBs.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalues and Singular Value Decompositions of Reduced Biquaternion Matrices

Pei¹,

Chang²,

Ding³

et al. 2008

IEEE Trans. Circuits Syst. I

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In this paper, the algorithms for calculating the eigenvalues, the eigenvectors, and the singular value decompositions (SVD) of a reduced biquaternion (RB) matrix are developed. We use the SVD to approximate an RB matrix in the least square sense and define the pseudoinverse matrix of an RB matrix. Moreover, the RB SVD is employed to implement the SVD of a color image. The computational complexity for the SVD of an RB matrix is only one-fourth of that for the SVD of conventional quaternion matrices. Therefore, many useful image-processing methods using the SVD can be extended to a color image without separating the color image into three channels. The numbers of the eigenvalues of an RB matrix, the th roots of an RB, and the zeros of an RB polynomial with degree are all finite and equal to 2 , not infinite as those of conventional quaternions.Index Terms-Quaternion, reduced biquaternion (RB), singular value decomposition (SVD) and eigenvalue of reduced biquaternion (RB) matrix.

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“…They have recently been introduced into digital signal processing. In 1990, Schütte and Wenzel [5] proposed the application of Reduced Biquaternions (RBs). RBs are a variety of hypercomplex numbers proposed by Hamilton in the 19th century.…”

Section: Introductionmentioning

confidence: 99%

“…Because the commutative property holds for multiplication, they form a commutative ring. Henceforth in this paper, the term hypercomplex number implies these RBs [5].…”

Section: Introductionmentioning

confidence: 99%

Realization of an orthogonal filter with hypercomplex coefficients

Okuda

Kamata

Takahashi

2001

Electron Comm Jpn Pt III

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SUMMARYThis paper proposes an orthogonal digital filter to simultaneously realize a pair of complex-coefficient digital filters with power complementarity by means of hypercomplex-coefficient all-pass digital filters. Hypercomplex filters can realize equivalent characteristics with only half as great a filter order as complex-coefficient digital filters and only one-quarter as great an order as real-coefficient digital filters. A method of realization of the hypercomplex-coefficient all-pass transfer function is discussed as well as a method of determining a transfer function with power complementarity for a certain complex-coefficient digital filter. Further, an example of the hypercomplex-coefficient transfer function is used and the effectiveness of the proposed method is shown. © 2001 Scripta Technica, Electron Comm Jpn Pt 3, 85(3): 5260, 2002

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Hypercomplex numbers in digital signal processing

Cited by 55 publications

References 2 publications

Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis

Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis

Eigenvalues and Singular Value Decompositions of Reduced Biquaternion Matrices

Realization of an orthogonal filter with hypercomplex coefficients

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