Representation of discrete functions using mean of weighted discrete white noise WSS random processes

Authors
Mahdi Mahmoudi
Jafar Pourrostam
1. Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran
Abstract
There are several ways of representing functions using some other basis functions. One way is using methods like Fourier series, power series, Laurent series and expansion in Bessel functions. Basis functions in these methods are deterministic functions and don’t have any random parameters. In another way, we can use a linear combination of some basis functions to determine an approximation to a known function, such that the basis functions are also with known definition but have some random parameters chosen from a probability space. In this paper we use discrete random processes as a basis to represent deterministic discrete functions. The functions to be represented are discrete. The used random processes are discrete WSS random processes, or equivalently sequences of i.i.d. random variables. In this method for each random process a coefficient is related, and the mean of these weighted random processes are calculated. These coefficients are random variables that depend on both random process and the deterministic function. It is shown that this mean of weighted random processes converges to the deterministic function with probability 1 (almost surely) and also in MS sense. If we consider the mean of only a finite number of random processes, we obtain an approximation to the deterministic function. The error of this approximation is considered as a random variable and the mean of the error is calculated. A formula relating the energy of the deterministic function and the power of the coefficients, similar to the Parseval's theorem in Fourier analysis, is obtained. There is some characteristics like linearity in this representation.
Keywords
discrete random processes
random basis
random series
series expansion
[1] E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, Cambridge University Press, 2021.

[2] G. F. Simmons, Differential Equations with Applications and Historical Notes, CRC Press, 2017.

[3] B. Igelnik, Y. H. Pao, "Stochastic choice of basis functions in adaptive function approximation and the functional-link net", IEEE Trans. Neural Networks, vol. 6, no. 6, pp. 1320-1329, 1995.

[4] A. Rahimi, B. Recht, "Uniform Approximation of Functions with Random Bases", IEEE 46th Annual Allerton Conference on Communication, Control, and Computing, pp. 555-561, 2008.

[5] A. Papoulis, S. U. Pillai, Probability, Random Variables and Stochastic Processes, McGrawHill, 2002.

[6] H. Pishro Nik, Introduction to Probability, Statistics, and Random Processes, available at https://www.probabilitycourse.com ,Kappa Research, LLC, 2014.

[7] E. Kreyszig, Advanced Engineering Mathematics, Wiley, 1999.