Hypercomplex Coefficients Research Articles

Математичне моделювання представлень експоненціальної і логарифмічної функцій у гиперкомплексній числовій системі узагальнених кватерніонів

The process of mathematical modeling representations of exponential and logarithmic functions in hypercomplex number system using generalized quaternion defining a linear differential equation with hypercomplex coefficients is being considered. Some properties of these representations and their relationship with the concepts of exponents in specific non-commutative hypercomplex numerical systems dimension four have been considered. Refs: 21 titles.

Induced Representations and Hypercomplex Numbers

In the search for hypercomplex analytic functions on the half-plane, we review the construction of induced representations of the group G=SL(2,R). Firstly we note that G-action on the homogeneous space G/H, where H is any one-dimensional subgroup of SL(2,R), is a linear-fractional transformation on hypercomplex numbers. Thus we investigate various hypercomplex characters of subgroups H. The correspondence between the structure of the group SL(2,R) and hypercomplex numbers can be illustrated in many other situations as well. We give examples of induced representations of SL(2,R) on spaces of hypercomplex valued functions, which are unitary in some sense. Raising/lowering operators for various subgroup prompt hypercomplex coefficients as well. The paper contains both English and Russian versions. Keywords: induced representation, unitary representations, SL(2,R), semisimple Lie group, complex numbers, dual numbers, double numbers, Moebius transformations, split-complex numbers, parabolic numbers, hyperbolic numbers, raising/lowering operators, creation/annihilation operators

Realization of an orthogonal filter with hypercomplex coefficients

AbstractThis 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): 52–60, 2002

Realization of two transfer functions with complex coefficients by a transfer function with hypercomplex coefficients

AbstractThis paper shows that two complex coefficient transfer functions can be realized by a single hypercomplex coefficient network, utilizing all four outputs of the hypercomplex network. the realization is considered for the case where the two arbitrary complex coefficient transfer functions do and do not have a common denominator. It is shown that the transfer functions can be realized by the hypercomplex coefficient network of the same order in the former case and by the hypercomplex coefficient network with the half‐order in the latter case. the realization of the transfer functions with the common denominator is interesting, but the general method to derive the transfer functions is not known.From such a viewpoint, this paper proposes the discrete‐type Kautz approximation with the positive frequency as the domain of approximation, including the optimal pole determination algorithm. By this approach, two complex coefficient transfer functions with an n‐th‐order common denominator are derived from the two 2n‐th‐order real or complex coefficient transfer functions. the method can be considered as one that simultaneously approximates the amplitude and the phase. the proposed approximation method is an approximation only for the positive frequency domain, and the property of the analytic signal that the negative frequency component is zero can be utilized. As a result, the method proposed in this paper has the advantages that the degree of freedom in the filter design is enhanced and the sampling frequency can be halved. In addition, the network structure can be simplified.

Strictry Proper Digital Filters with Hypercomplex Coefficients and Reduction of Multiplications in Hypercomplex Algebra

In this paper, we consider strictly proper digital filters with hypercomplex coefficients, and we also consider reduction of multiplications in hypercomplex algebra.As for the former problem, the first-order digital filter with hypercomplex coefficients can realize arbitrary-order strictly proper digital filters with real coefficients less than 4, whose order of numerator is smaller than that of denominator by 1.As for the latter problem, if we apply Quadratic Residue Number System to hypercomplex algebra, we can reduce the number of multiplications to 1/4 of the original one. And the most important thing of this result is that we only need 4 multiplications per multiplier and that these 4 multiplications can be done at the same time.

Digital filters with hypercomplex coefficients

AbstractThis paper considers digital filters with hyper‐complex coefficients. Filters of this type can reduce the order of filters to one‐fourth the one with real coefficients. It is reported that the first‐order digital filter with hypercomplex coefficients can realize the fourth‐order digital filter with real coefficients which can be expressed by a second‐order all‐pass filter with complex coefficients.This paper considers also generalization including the foregoing special case and shows that the first‐order digital filter with hypercomplex coefficients can realize an arbitrary‐order digital filter with real coefficients less than 4.Arbitrary higher‐order digital filter with real coefficients can be realized by parallel connection or cascade connection of first‐order digital filters with hypercomplex coefficients. This paper considers the following three different realizations: (a) realization by parallel connection; (b) realization by cascade connection of real part; and (c) realization by cascade connection. We find that case (b) is the most low sensitive. Finally, stability criteria of hypercomplex digital filters of first‐order are suggested.

On the multiplication of reduced biquaternions and applications

We present two efficient algorithms for the multiplication of the so-called reduced biquaternions. The proposed algorithms may be used for the realization of digital filters with hypercomplex coefficients.