Theta-Functions in Mathematical Model of Noise Quantization
DOI:
https://doi.org/10.14529/cmse180102Keywords:
distribution density, noise quantization, theta-functions JacobiAbstract
This article presents the new formula for two-dimensional density function probability of the noise quantization, which allows us write it with the help of mathematical expression, which consists of only theta-functions Jacobi. The method of obtaining this formula is given. The derivation is based on the fact that at a suitable change of variables some members of the double row are destroyed. It shows the principle of producing all of the formulas of this family. This principle is based on properties of symmetry theta-function. The symmetry of theta-functions allows us to express one theta-function by another theta-function and obtain other formulas consisting only of theta-functions Jacobi. This family of formulas allows us to obtain expressions for the organization of model experiments, supported by basic mathematical packages. They enable us to receive numerical characteristics random processes such the functions of parameters that give rise to their Gaussian random processes in an analytical form. Their use increases the rate of convergence of simulation results. These formulas enable us carry out the synthesis of the desired expression in an analytic form for functional transformations of random vectors and processes in signal process.
References
Balyasnikov B.M., Vorona M.C., Zavolokin V.V., Korshunov A.Y., Maksimenko M.D., Odinochenko N.M. Matematicheskaya model shuma kvantovaniya signalov, otrazhennyh ot protyazhennyh prostranstvennyh pomeh [A Mathematical Model of the Quantization of the Signals Reflected from Expended Spatial Interference]. Proceedings of the Mozhaisky Military Space Academy St.Petersburg, 2011. vol. 633, no. 2. pp. 131–138. (in Russian)
Tihonov V.I. Statistichcheskaya radioteknika [Statistical Radio Engineering]. Moscow: Sovetskoe Radio, 1982. 624 p. (in Russian)
Abramowitz M., Stegun I. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. National Bureau of Standards Applied Mathematics Series–55, 1964. 832 p.
Korn G., Korn T. Mathematical Handbook. For Scientists and Engineers. Definitions, Theorems and Formulas for Reference and Revive. Second, Enlargend and Revised Edition. McGraw–Hill Book Co., New York, San Francisco, Toronto, London, Sydney, 1968. 832 p.
Lawden D.F. Elliptic Function and Application. Springer Verlag New York 1989. 336 p. DOI: 10.1007/978-1-4757-3980-0
Bateman H., Erdelyi A., Higher Transcendental Functions: Elliptic and Automorphic Functions. Lame and Mathieu Functions. Vol. 3, McGraw–Hill Book Co., New York, 1955. 300 p.
