Singularity Behaviour of the Density, Information, and Entropy Functions Defining a Uniform Non-stationary Stochastic Process

318e88d7-9d15-44a1-be41-ff41f81ecaa820220428073743887wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON INFORMATION SCIENCE AND APPLICATIONS2224-34021790-083210.37394/23209http://wseas.org/wseas/cms.action?id=40462920222920221910.37394/23209.2022.19https://wseas.com/journals/isa/2022.phpSingularity Behaviour of the Density, Information, and Entropy Functions Defining a Uniform Non-stationary Stochastic ProcessStevanBerberElectrical, Computer, and Software Engineering Department The University of Auckland 5 Grafton Road, 1010 Auckland Central NEW ZEALANDPrecise definitions and derivatives of the time-dependent continuous and discrete uniform probability density functions and related information and entropy functions are investigated. A stochastic system is formed that can represent a uniform noise source having a time-dependent variance and forming a uniform non-stationary stochastic process. The information and entropy function of the system are defined, and their properties are investigated in the time domain, including the limit cases defined for infinite and zero values of the time-dependent variance. In particular, the singularity properties of the entropy function will be investigated when the time-dependent variance reaches infinity. Like in thermodynamics, where the physical entropy of a system increases all the time, the information entropy of the stochastic system in information theory is also expected to increase towards infinity when the variance increases. All investigations are conducted for both the continuous and discrete random variables and their density functions. The presented theory is of particular interest in analyzing the Gaussian density function having infinite variance and tending to a uniform density function.4282022428202211413112https://wseas.com/journals/isa/2022/a245109-007(2022).pdf10.37394/23209.2022.19.12https://wseas.com/journals/isa/2022/a245109-007(2022).pdf10.1002/j.1538-7305.1948.tb00917.xShannon, C. E. A Mathematical Theory of Communication. The Bell System Technical Journal, Vol. 27, pp. 379–423, 623–656, July, October, (1948). Glattfelder, J. B. Information–Consciousness– Reality, (Springer Nature Switzerland AG, 2019). Nicholson, B. S. at all. Time-information uncertainty relations in thermodynamics, Nature Physics, VOL 16, Dec. 2020, p. 1211- 1215. Bell, J. L. Continuity and infinitesimals, In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, Summer 2017 Edition. Montgomery, D.C., Runger, G. C. Applied Statistics and Probability for Engineers, (John Wiley and Sons, New York, Fourth Edition, 1994). Papoulis, A., Pillai, S. U. Probability, Random Variables and Stochastic Processes, (McGrawHill, New York, Fourth Edition, 2002). Peebles. P. Z. Probability, Random Variables, and Random Signal Principles, (McGraw-Hill, New York, Fourth Edition, 2001). Manolakis, D. G., Ingle, V. K., Kogan, S. M. Statistical and Adaptive Signal Processing, (Artech House. INC., 2005). Gray, M. R. Davisson, L. D. An Introduction to Statistical Signal Processing, (Cambridge University Press, 2004). Proakis, G. J. Digital Communications, (McGraw-Hill, New York, Fourth Edition, 2001). Berber, S. Discrete Communication Systems, (Oxford University Press, 2021). Haykin, S. Communication Systems, (John Wiley & Sons Ltd., Fourth Edition, 2001). 10.37394/23206Berber, S. The Exponential, Gaussian and Uniform Truncated Discrete Density Functions for Discrete-Time Systems Analysis, WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 16, Art. #26, 2017, pp. 226-238.