Ya.A. Kalinovsky, Institute for information recording NAS of Ukraine
Yu.E. Boyarinova, Institute for information recording NAS of Ukraine, National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute»
Ya.V. Khitsko, National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute»
A.S. Sukalo, National University of Water Management and Environmental Management
Èlektron. model. 2018, 40(5):27-40
https://doi.org/10.15407/emodel.40.05.027
ABSTRACT
The method of multiplication of hypercomplex numbers is proposed, which provides a significant reduction in the volume of real operations. The method consists in the transition to weakly filled isomorphic hypercomplex number systems (HNS) in which a smaller number of real multiplications is required for hypercomplex multiplication. Such pairs of isomorphic HNS, as well as expressions for isomorphism operators, have been synthesized. The developed method should be used to construct fast linear convolutional algorithms.
KEYWORDS
hypercomplex numerical system, linear convolution, isomorphism, multiplication, complex numbers, double numbers.
REFERENCES
- Bleichut, (1984), Bystrye algoritmy tsyfrovoy obrabotki signalov [Rapid algorithms of digital signal processing], Mir, Moscow. USSR.
- Nussbaumer, (1985), Bystroe preobrazovanie Furie i algoritmy vychisleniya svyortok [Fast Fourier transform and computation of convolution algorithms], Radio and svyaz, Moscow, USSR.
- Drozd, A. and Kirichenko, V.V. (1980), Konechno-mernye algebry [Finite-dimensional algebras], Vyshcha shkola, Kiev, USSR.
- Sinkov, V. Boyarinova, Yu.E. and Kalinovsky, Ya.A. (2010), Konechno-mernye giperkompleksnye chislovye sistemy. Osnovy teorii. Primeneniya. [Finite-dimensional hypercomplex numerical systems. Fundamentals of the theory. Applications], Infodruk, Kiev, Ukraine.
- Kalinovsky, Ya.A. and Boyarinova, Yu.E. (2012), Vysoko-razmernye izomorfmye giperkompleksnye chislovye sistemy i ikh ispolzovanie dlya povysheniya effektivnosti vychislenii [High-dimensional isomorphic hypercomplex number systems and their use for increasing the efficiency of computations], Infodruk, Kiev, Ukraine.
- Kalinovsky, A. (2017), “Effective algorithms for solving the isomorphism equations for hypercomplex number systems using exponential representations”, Elektronnoe modelirovanie, Vol. 39, no. 1, pp. 75-90.
- Toyoshima, (2002), Computationally efficient implementation of hypercomplex digital filters, IEICE Trans. Fundamentals, E85-A, 8, pp. 1870-1876.
- Schutte, D. (1991), Digitalfilter zur Verarbeitung komplexer und hypercomplexer Signale, Dissertation. Paderborn, 1991.
- Schulz, , Seitz, J. and LustosadaCosta, J.P. (2011), Widely linear SIMO filtering for hypercomplex numbers, IEEE Information Theory Workshop, pp. 390-394. https://doi.org/10.1109/ITW.2011.6089486
- Kalinovsky, A., Boyarinova, Yu.E. and Khitsko, Ya.V. (2015), “Optimization of the total parametric sensitivity of reversible digital filters with coefficients in non-canonical hypercomplex number systems”, Elektronnoe modelirovanie, Vol. 37, no. 5, pp. 117-126.
- Kalinovsky, O. (2007), “The development of method in the theory of hypercomplex number systems for mathematical modeling and computing is calculated”, Dissertations of Dr Sc. (Tech.), 01.05.02, Kyiv, Ukraine.
- Kalinovsky, Ya.A. (2013), “Structure of the hypercomplex method for fast calculation of linear convolution of discrete sign”, Reestratsiya, zberigannya i obrobka dannykh, Vol. 15, no. 1, pp. 31-44.