Mathematical Problems in Engineering

Volume 2018, Article ID 4967613, 9 pages

https://doi.org/10.1155/2018/4967613

## Efficient Numerical Methods for Analysis of Square Ratio of *κ*-*μ* and *η*-*μ* Random Processes with Their Applications in Telecommunications

Correspondence should be addressed to Stefan R. Panić; moc.oohay@cnpnafets

Received 12 October 2017; Accepted 30 January 2018; Published 11 March 2018

Academic Editor: Bo Shen

Copyright © 2018 Gradimir V. Milovanović et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We will provide statistical analysis of the square ratio of and random processes and its application in the signal-to-interference ratio (SIR) based performance analysis of wireless transmission subjected to the influence of multipath fading, modelled by fading model, and undesired occurrence of co-channel interference (CCI), distributed as random process. First contribution of the paper is deriving exact closed expressions for the probability density function (PDF) and cumulative distribution function (CDF) of square ratio of and random processes. Further, a verification of accuracy of these PDF and CDF expressions was given by comparison with the corresponding approximations obtained by the high-precision quadrature formulas of Gaussian type with respect to the weight functions on . The computational procedure of such quadrature rules is provided by using the constructive theory of orthogonal polynomials and the MATHEMATICA package OrthogonalPolynomials created by Cvetković and Milovanović (2004). Capitalizing on obtained expression, important wireless performance criteria, namely, outage probability (OP), have been obtained, as functions of transmission parameters. Also, possible performance improvement is observed through a glance at SC (selection combining) reception employment based on obtained expressions.

#### 1. Introduction

Let and be mutually independent Gaussian random processes, with variances . Modelling mean values of Gaussians, with and , and assuming initially to be an integer, the resulting random process is obtained as the following function of Gaussians: which follows distribution, whose PDF is given in the following form (cf. [1]): with and is the zero-the order modified Bessel function of the first kind [2, Eq. 8.445].

As a general probability distribution model, this model includes some classical distribution models as its particular cases, for example, Rician, and Nakagami-*m* distribution models as special cases (as the One-Sided Gaussian and the Rayleigh distributions since they also represent special cases of Nakagami-*m*).

Let and , , be Gaussian random processes, with . Assuming initially to be an integer, resulting random process obtained as a function of Gaussians follows distribution, whose PDF is given in the following form (cf. [1]):with defined as , the ratio of arbitrary variances of independent Gaussians defined as , and distribution parameters and defined as , , . As a general probability distribution model, this model also includes some classical distribution models as its particular cases, for example, Nakagami- (Hoyt), one-sided Gaussian, Rayleigh, and Nakagami-.

Multipath fading is physical phenomena that occurs as randomly delayed components of desired signal combine in constructive or destructive manner at the reception [1]. In the case when a line of sight (LOS) component between the transmitter and the receiver is present, the most general fading model which describes the short-term signal variation is kappa-mu () fading model [3, 4]. Fading model, defined as the function of parameter , related to the dominant/scattered components powers quotient, and parameter , related to the number of propagating clusters, easily reduces to other fading models, by setting corresponding values for parameters and . When , the model transforms to the Rice model with arbitrary factor, [4]. By assigning , the observed model transforms to the Nakagami- model [4]. Co-channel interference (CCI) signal which is transmitted at the same frequency as the carrier signal is also exposed to multipath fading phenomenon [5]. Due to complicated propagation of interfering signal, CCI at the reception is usually modelled with random process that describes small scale signal variations in general non-line-of-sight condition. Recently proposed [6] eta-mu () fading distribution meets those conditions. As mentioned in [7], model is presented in function of the parameter , related to the in-phase/in-quadrature components scattered wave powers quotient of each cluster, and parameter , related to the number of multipath clusters in the environment. As a general distribution, this model spans through other some well-known fading models as its particular cases, that is, Hoyt model and Nakagami- model. Namely, Nakagami- distribution can be obtained in an exact manner by assigning parameter values and (cf. [6]).

Observing interference-limited fading environment, where corruptive effects of thermal noise can be ignored, it is necessary to determine properties of signal-to-interference ratio (SIR) in order to carry out effective performance analysis of observed system. Indeed, SIR has been measured in base and mobile stations by using SIR estimators [8]. In order to determine behavior of instantaneous SIR random process, we must determine square ratio of and random processes, , and analyze its properties. Here, exact closed expressions for the probability density function (PDF) and cumulative distribution function (CDF) of square ratio of and random processes are shown. Further, a verification of accuracy of these PDF and CDF expressions was given by comparing with the corresponding approximations obtained by the high-precision quadrature formulas of Gaussian type with respect to the weight functions on . The computational procedure of such quadrature rules is provided by using Gautschi’s constructive theory of orthogonal polynomials (see [9, 10]) and the MATHEMATICA package OrthogonalPolynomials created by Cvetković and Milovanović (see [11, 12]). Based on obtained expressions, rapidly converging infinite-series expression for wireless communication systems performance measure, outage probability (OP) is also presented, and OP is considered for different values of transmission parameters. At the end, possible performance improvement will be considered through a glance at SC (selection combining) reception employment.

The paper is organized as follows. System model and closed-form of the PDF and CDF expressions are presented in Section 2. Section 3 is devoted to an alternative approach based on the constructive theory of orthogonal polynomials and the corresponding quadrature rules of Gaussian type. System performances and numerical results are presented in Sections 4 and 5, respectively. Finally, concluding comments are given in Section 6.

#### 2. System Model and Closed-Form of the PDF and CDF Expressions

The desired signal follows distributed random process modelled as (cf. [3]) where with , denoting desired signal average power, while being the th order modified Bessel function of the first kind [2, Eq. 8.445]. Due to physical separation between the co-channel interferer and target receiver, which is very large in practice, it is very unlikely that direct LOS condition (s) exist, for CCI signal. The most general fading model, which does not include LOS condition is fading model, is chosen here for modelling random envelope fluctuations of CCI signal with corresponding PDF [6]: where stands for the average power of CCI, while denotes Gamma function [2, Eq. 8.310.1]. Here and are interfering signal parameters, written in the function of parameter as follows [6]:

Now, we will observe random envelope statistics in observed interference-limited system. Here the novelty and significance of this model must be pointed out again. Namely, for the first time scenario is observed when desired signal envelope variations are caused by fading occurrence (most general fading model which includes LOS component existence), while CCI signal envelope variations are caused by fading occurrence (most general fading model that does not include LOS component). Previously, various other scenarios have been observed in [1, 13–15]. In [1, 14] LOS component existence in CCI signal has been assumed, which is not so common case, because of large physical distance between the interferer and receiver. In [13] non-LOS model was observed for desired signal, while Gamma approximation of random variables sum has been observed. Finally, neither in general nor in comprehensive study, such as [15], has this model been considered. In the case when CCI is much stronger than Gaussian noise in same channel, PDF for the instantaneous SIR, , can be determined according to [16] in the form

After substituting (6) and (8) into (10), we can derive PDF of SIR in the following closed-form:where and stands for the average SIR.

Namely, with respect to well-known series representation of modified Bessel function (cf. [2, Eq. 8.445]) substituting (6) and (8) in (10) results in By using the well-known definition of the Gamma function [2, Eq. 8.310.1]: (10) can be reduced into (11). A similar procedure has been used in [7]. By taking finite numbers of terms in the last expression for we obtain its approximation in a closed-form: The closed-form expression (16) converges rapidly, since only (10–15) terms are needed to be summed in each sum in order to reach an accuracy of five significant decimal digits, for observed set of system parameters. Otherwise, in a case of slowly convergent series, we need some methods for accelerating their convergence (cf. [17–19]).

Following [16], the cumulative distribution function (CDF) can be obtained as

After substituting (11) into (17), by applying the same mathematical transformations given in Appendix of [7], we can obtain the following closed-form expression of rapid convergence: where and is the incomplete Beta function [2, Eq. 8.38] (for some extended special functions see [20]). Detailed convergence analysis of this expression is provided in Table 1.