Wireless Communication System Performance in M2M Nakagami-m Fading Channel

Abstract

In this paper, we investigate a wireless mobile communication system with a cooperative network operating over a mobile-to-mobile (M2M) fading channel in the presence of co-channel interference. Since there is no direct line of sight between the source (S) and the destination (D) due to various obstacles, the signal transmission takes place by using a relay (R). We modeled and represented the desired signal and co-channel interference by using the Nakagami-m fading distribution from the source (S) to the relay (R), as well as from the relay (R) to the destination (D). As the popularity of M2M communication has increased in 5G cellular networks, the performance analysis of the proposed system is of great importance. We derived closed-form expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the signal-to-interference ratio (SIR) at the input of the destination station. Based on the cumulative distribution function (CDF), we also evaluated the outage probability (Pout) of the proposed communication system. In addition, we derived a closed-form approximate expression for the level crossing rate (LCR) by using the Laplace approximation formula for the three-fold integral. Validity of the derived theoretical results we approved by Monte Carlo simulations.

1. Introduction

Modern mobile communication systems and wireless systems, in general, have to deliver a very high amount of data to users. An efficient way to accomplish this task in an impaired channel, with fading, interference, and noise, is to use cooperation between the users [1,2]. Mobile-to-mobile (M2M) communication has an implementation in different applications, such as ad hoc wireless networks, intelligent transportation systems, and relay-based cellular networks [3]. The popularity of M2M communications has increased in the 5G cellular networks [4]. In cooperative networks, the source and the destination may communicate directly and/or via one or multiple relays. A relay may amplify and forward (AF) or fully decode and forward (DF) the received information. However, AF relays, due to their simplicity and energy consumption, have a higher practical importance. The high number of users causing co-channel interference may significantly influence the system’s performance. In such a case, we can neglect the thermal noise, and the environment is called interference-limited.

Cooperative relay systems have a wide range of applications. With the increase in end users’ demands for reliable and fast information transfer, the use of these systems has also increased. In rural areas where there are not many obstacles, AF and DF relay systems serve to transmit large amounts of data using free space optics (FSO) communications. FSO communications provide a higher data rate than traditional radio frequency (RF) communications [5]. In addition, there is a frequent application of hybrid cooperation relay systems. These systems use FSO communications to transmit information from the source to the relay and RF communications to transfer data from the relay to the destination [6,7,8]. In contrast, in urban areas where we have many users, the signal encounters many obstacles, so we use RF communications to transmit information from the source to the relay and from the relay to the destination. Although RF systems cannot match FSO systems in terms of many performances, they still have an advantage in such environments [9].

Due to its wide application in practical cases, the performance analysis of these systems is of exceptional importance for the further development of modern communications. The outage probability, or the probability that the reception quality is below a certain threshold, is a significant performance measure of a communication system. The authors of [10] analyze dual-hop AF and DF relaying in a Rayleigh fading channel with multiple interferences at the relay and destination. However, the Nakagami-m fading channel is more general, and its great accordance with the experimental data is the main reason for its usage in literature for the analysis of wireless cooperative communication systems [3,11,12,13,14,15,16,17,18]. In [11,12], the authors consider dual-hop AF relaying in the Nakagami-m fading environment without interference. The authors of [3,13] consider dual-hop interference-limited relaying with multiple interferers at the relay, with AF [3] and DF [13] relays in the Nakagami-m channel, and give the approximate expressions for the outage probability. In [14], the authors analyze a lower bound on the outage probability for a universal environment, where multiple interferers are present at the source and destination, and in [16], the authors consider the lower bound on the outage probability of AF mobile-to-mobile (M2M) relaying system in Nakagami fading channel and relay selection. The authors of [11,18] analyze the outage probability of a dual-hop DF cooperative system in the Nakagami-m channel with different scenarios. Furthermore, this paper brings a conditional relaying scheme where a relay may help source-destination communication if a specified condition is verified. In contrast, a full-duplex DF relaying is the topic of [17] analysis of several interference and self-interference scenarios.

Aside from the outage probability, the level crossing rate (LCR) is another performance measure of a wireless communication system. We define LCR as the average number of level crossings in a positive or negative direction within the interval of one second [19]. The LCR of the non-cooperative interference-limited Rice/Nakagami fading channel is considered in [20] for single-channel reception and in [21] for the case of diversity reception. LCR of a dual-hop AF relaying system, in the absence of interference, for a double Gaussian channel, is analyzed in [22], and for a twofold Nakagami-m fading channel with the average fade duration, in [23]. In addition to the LCR, the probability density function (PDF) and the average fade duration are obtained in [24] from other dual-hop AF Rician fading channel. The interference was not present, and the paper analyzes mobile-to-mobile (M2M), mobile-to-fixed, and fixed-to-mobile channels. In [21,25], the authors consider the LCR of a multi-hop DF relaying with the multiple interferences at relays in Rayleigh/Rician/Nakagami-m and Nakagami-Hoyt fading channels, both noise and interference-limited.

The authors of [26,27,28,29,30,31] present first and second-order statistics for Weibull, Rayleigh, and Double Nakagami distributions in cooperative relay systems. In [26], the authors consider a mobile-to-mobile communication system with a relay. They use the Weibull short-term fading model to describe the random variable process in channels and analyze the system performance. For Weibull short-term fading, the authors calculated the probability density function, cumulative density function, level crossing rate, and average fade duration. Authors of [27] consider the product of two Rayleigh random variables and a Nakagami-m random variable. In that research, they did not consider cooperative systems with relay, but results for level crossing rate can implement in similar mobile-to-mobile systems. The research presented in [28] provides results for a communication system in which data transmission performs via relays. The obtained results are interesting because the data transmission system uses an RF link (Nakagami-m distribution) to transmit data from the source to the relay and an FSO link (Nakagami-m squared distribution) to transmit data from the relay to the destination. The analysis of statistical processes described by Rayleigh, Nakagami-m, Weibull, and α-μ distributions in multi-hop wireless communication systems is presented in [29]. What the closed-form analytical expressions for the PDF and CDF have in common is their representation through the Meijer function. Current research and analysis of communication systems often rely on Meijer functions. The solutions presented through these functions make numerical calculations and simulation models easy. Similar to [26], the authors in [30] present an analysis of system performance using the Rayleigh distribution. This paper also analyzes the mobile-to-mobile cooperative system’s PDF, CDF, and LCR.

In [32,33,34,35], the authors present results for PDF and CDF for three Nakagami-m distributions, while [20,21,23,27,35,36] show average fade duration (AFD). Furthermore, the authors of [32] and [35] consider a more complicated case for mobile-to-mobile cooperative communication systems. In their work, they analyze the performance of the triple Nakagami-m random process. In addition, through presented graphical results, the authors explain implementation in wireless communication systems. Although the results presented in [32] and [35] represent a more complicated case from a theoretical point of view, the lack of closed-form analytical expressions for the PDF, CDF, outage probability, and average fade duration makes the analysis of this system cumbersome. In addition to mobile-to-mobile cooperative communication systems that use relays for data transmission, the use of various diversity techniques is also high. The authors of [33] provide the implementation of the diversity technique for the Nakagami-m random variables process. Performance analysis of communication systems has an important role. Insight into system performance can tell us a lot about the properties and behavior of those systems. In such cases, it is necessary to confirm the validity of the obtained results through the simulation process. In [34], the authors propose a sum of a sinusoid as a Nakagami-m fading simulator.

In this paper, we consider a wireless communication system operating over M2M interference-limited Nakagami-m multi-path fading channel in the presence of Nakagami-m co-channel interference. We evaluate expressions for the probability density function (PDF), the cumulative distribution function (CDF), and the level crossing rate (LCR) of the output signal-to-interference ratio (SIR). This paper gives the exact closed-form expressions for PDF and CDF and a closed-form approximate expression for LCR. We confirm all results by Monte-Carlo simulation.

The authors believe that the consideration of this system and the results they presented deserves attention, considering its wide range of implementation in modern communication systems.

For example, although the system seems simple at first glance, that is its advantage. A drone wireless relay system for disaster recovery and locating persons trapped under debris is just one of the possible implementations. We can also successfully implement such system in physical layer security to avoid eavesdropping attacks. By placing a relay on the drone, we can avoid eavesdropping in cases where the location of the eavesdropper is between the source and the destination. Finally, we can implement such system in systems with the adaptive transmission. In cases where we use algorithms with optimization of transmitter power or data rate, this system can be used as a channel state for information or as a channel for prediction.

The paper consists of five sections: Introduction, System model, Statistics of M2M Fading Channels, Numerical Results, and Conclusion. In the Introduction section, we explain the importance of cooperative relay systems and give a literature review. The System model section describes relations between the source, relay, and destination, the conditions under which the system works, and the channel model with preset assumptions. Statistics of M2M Fading Channels section provides analytical expressions in closed-form for PDF, CDF, outage probability, and LCR. The numerical Results section explains and graphically represents obtained results.

2. System Model

We will consider a dual-hop cooperative relay system shown in Figure 1. Due to various obstacles that block the signal, we do not have direct communication between the source (S) and the destination (D). To establish communication between the source (S) and destination (D), we use the AF relay (R). However, the same considerations stand for the other direction, so bi-directional communications are fully supported. As already mentioned in the Introduction, the communication system is dominantly impaired by interference due to a large number of users. Therefore, thermal noise may be neglected, and the environment is interference-limited. The interference is present in both sections, from source (S) to relay (R) and from relay (R) to destination (D). Since source–relay (S-R) and relay–destination (R-D) communications are performed in different time slots, we may consider them independent. The desired signal and the interference fading channels are also considered to be uncorrelated.

For the system model we are considering, interference degrades signal quality significantly more than thermal noise. In such case, we will observe the signal quality at the destination node through the signal-to-interference ratio (SIR). In this system model, random variables that describe desired signal and interference envelopes in both sections (S-R, R-D) of the communication system have Nakagami-m distribution. Nakagami-m is a general distribution that reduces to Rayleigh distribution for m = 1 and hyper-Rayleigh or worse-than-Rayleigh for m = 0.5. For m → ∞, Nakagami-m fading reduces to the fading-free channel. We denote random variables that describe the desired signal with x₁ and x₂ and interferences with y₁ and y₂ for the first and the second section, respectively. Envelopes in the first and second sections are independent non-identically distributed Nakagami-m random variables with the following probability density functions [32]:

$$\begin{array}{l}{p}_{x_i}(x_i)=\frac{2}{\Gamma (m)}{\left(\frac{m_{x_i}}{{\Omega }_{x_i}}\right)}^{m_{x_i}}{x}_i^{2m_{x_i}-1}\exp \left(-\frac{m_{x_i}}{{\Omega }_{x_i}}{x}_i^2\right),\hspace{0.17em}\hspace{0.17em}{x}_i\ge 0,\hspace{0.17em}i=1,2\\ p_{y_i}(y_i)=\frac{2}{\Gamma (m)}{\left(\frac{m_{y_i}}{{\Omega }_{y_i}}\right)}^{m_{y_i}}{y}_i^{2m_{y_i}-1}\exp \left(-\frac{m_{y_i}}{{\Omega }_{y_i}}{y}_i^2\right),\hspace{0.17em}\hspace{0.17em}{y}_i\ge 0,\hspace{0.17em}i=1,2\end{array}$$

(1)

where m_x1, m_x2, and m_y1, m_y2 are the desired signal and the interference fading severity parameters, respectively. With Ω_x1 and Ω_x2, we denoted the average signal power of the desired signal, and with Ω_y1 and Ω_y2, we denoted the average signal power of interference. In the following analysis, we have made assumption that the fading severity parameter is an integer. This is a quite common assumption in the literature because of the severe difficulties in the mathematical analysis of these channels. However, if needed, results, in this case, may be easily obtained by numerical integration.

3. Statistics of M2M Fading Channels

3.1. PDF of the Output SIR

We can present the signal-to-interference ratio at the destination node as $z=z_1{z}_2$, where z₁ and z₂ are signal-to-interference ratios in the first and the second section, respectively, defined as [26]:

$$z_i=\frac{x_i}{y_i},\hspace{0.17em}i=1,2$$

(2)

To find the PDF of z, p_z(z), we first derive the expressions for the PDFs of z₁ and z₂, p_z1(z₁), and p_z2(z₂). Since i = 1,2 is the ratio of the random variables x_i and y_i, its PDF is defined as [30]:

$$p_{z_i}(z_i)={\displaystyle \underset{0}{\overset{\infty }{\int }}d{y}_i{y}_i{p}_{x_i}(y_i{z}_i)}{p}_{y_i}(y_i),\hspace{0.17em}\hspace{0.17em}i=1,2$$

(3)

After solving the integral in (3), as explained in Appendix A, we come to:

$$p_{z_i}(z_i)=\frac{2\Gamma \left(m_{x_i}+m_{y_i}\right)}{\Gamma \left(m_{x_i}\right)\Gamma \left(m_{y_i}\right)}{\left({\Omega }_{y_i}{m}_{x_i}\right)}^{m_{x_i}}{\left({\Omega }_{x_i}{m}_{y_i}\right)}^{m_{y_i}}\frac{z_i^{2m_{x_i}-1}}{{\left({\Omega }_{x_i}{m}_{y_i}+m_{x_i}{\Omega }_{y_i}{z}_i^2\right)}^{m_{x_i}+m_{y_i}}}$$

(4)

Since $z=z_1{z}_2$ we have [37]:

$$p_z(z)={\displaystyle \underset{0}{\overset{\infty }{\int }}d{z}_2\frac{1}{z_2}{p}_{z_1}(z/z_2)p_{z_2}(z_2)}$$

(5)

After substituting values for p_z1(z₁) and p_z2(z₂) in (5) and simplifying the obtained expression, we find:

$$\begin{array}{cc}\hfill p_z(z)& =\frac{4}{B\left(m_{x_1},m_{y_1}\right)B\left(m_{x_2},m_{y_2}\right)}{\left(\frac{{\Omega }_{x_1}{m}_{y_1}}{{\Omega }_{y_1}{m}_{x_1}}\right)}^{m_{y_1}}{\left(\frac{{\Omega }_{y_2}{m}_{x_2}}{{\Omega }_{x_2}{m}_{y_2}}\right)}^{m_{x_2}}{z}^{-1-2m_{y_1}}\hspace{0.17em}\hfill \\ & \times {\displaystyle \underset{0}{\overset{\infty }{\int }}d{z}_2\frac{z_2^{2m_{x_2}+2m_{y_1}-1}}{{\left(1+\frac{{\Omega }_{x_1}{m}_{y_1}}{m_{x_1}{\Omega }_{y_1}{z}^2}{z}_2^2\right)}^{m_{x_1}+m_{y_1}}{\left(1+\frac{m_{x_2}{\Omega }_{y_2}}{{\Omega }_{x_2}{m}_{y_2}}{z}_2^2\right)}^{m_{x_2}+m_{y_2}}}}\hfill \end{array}$$

(6)

Using Equation (3.259.3) from [38]:

$${\displaystyle \underset{0}{\overset{\infty }{\int }}{x}^{\lambda -1}{(1+\alpha x^p)}^{-\mu }{(1+\beta x^p)}^{-\nu }dx}=\frac{1}{p}{\alpha }^{-\lambda /p}B\left(\frac{\lambda }{p},\mu +\nu -\frac{\lambda }{p}\right){}_{2}F_1\left(\nu ,\frac{\lambda }{p};\mu +\nu ;1-\frac{\beta }{\alpha }\right)$$

(7)

where $B(a,b)$ is the beta function and ${}_{2}F_1\left(a,b;c;d\right)$ is the hypergeometric function, the final expression for PDF is

$$\begin{array}{cc}\hfill p_z(z)& =\frac{2B\left(m_{x_2}+m_{y_1},m_{x_1}+m_{y_2}\right)}{B\left(m_{x_1},m_{y_1}\right)B\left(m_{x_2},m_{y_2}\right)}{\left(\frac{{\Omega }_{y_1}{\Omega }_{y_2}}{{\Omega }_{x_1}{\Omega }_{x_2}}\frac{m_{x_1}{m}_{x_2}}{m_{y_1}{m}_{y_2}}\right)}^{m_{x_2}}{z}^{2m_{x_2}-1}\hfill \\ & \times {}_{2}F_1(m_{x_2}+m_{y_2},m_{x_2}+m_{y_1};m_{x_1}+m_{y_1}+m_{x_2}+m_{y_2};\hspace{0.17em}1-z^2\frac{{\Omega }_{y_1}{\Omega }_{y_2}}{{\Omega }_{x_1}{\Omega }_{x_2}}\frac{m_{x_1}{m}_{x_2}}{m_{y_1}{m}_{y_2}})\hfill \end{array}$$

(8)

Parameters from Equations (6) and (7) have the following relation:

$$\begin{array}{c}\lambda =2m_{x_2}+2m_{y_1}\hfill \\ \alpha =\frac{{\Omega }_{x_1}{m}_{y_1}}{m_{x_1}{\Omega }_{y_1}{z}^2}\hfill \\ \beta =\frac{m_{x_2}{\Omega }_{y_2}}{{\Omega }_{x_2}{m}_{y_2}}\hfill \\ \mu =m_{x_1}+m_{y_1}\hfill \\ \nu =m_{x_2}+m_{y_2}\hfill \\ p=2\hfill \end{array}$$

3.2. CDF and Outage Probability of the Output SIR

The cumulative distribution function of z is defined as [26]:

$$F_z(z)={\displaystyle \underset{0}{\overset{z}{\int }}dt{p}_z(t)}$$

(9)

By inserting Equation (6) into (9) we obtain:

$$\begin{array}{cc}\hfill F_z(z)& =\frac{4{\left({\Omega }_{y_1}{m}_{x_1}\right)}^{m_{x_1}}{\left({\Omega }_{x_1}{m}_{y_1}\right)}^{m_{y_1}}{\left({\Omega }_{y_2}{m}_{x_2}\right)}^{m_{x_2}}{\left({\Omega }_{x_2}{m}_{y_2}\right)}^{m_{y_2}}}{B\left(m_{x_1},m_{y_1}\right)B\left(m_{x_2},m_{y_2}\right)}\hfill \\ & \times {\displaystyle \underset{0}{\overset{z}{\int }}{\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{z_2^{2m_{x_2}+2m_{y_1}-1}}{{\left({\Omega }_{x_2}{m}_{y_2}+{\Omega }_{y_2}{m}_{x_2}{z}_2^2\right)}^{m_{x_2}+m_{y_2}}}\frac{t^{2m_{x_1}-1}}{{\left({\Omega }_{x_1}{m}_{y_1}{z}_2^2+{\Omega }_{y_1}{m}_{x_1}{t}^2\right)}^{m_{x_1}+m_{y_1}}}}d{z}_{2}dt}\hfill \end{array}$$

(10)

The double integral in (10) may be first integrated over t and then over z₂ because the integrand satisfies the condition of Fubini’s theorem [39], i.e., the absolute value of the integrand is < ∞ within the limits of the integration. Therefore,

$$\begin{array}{cc}\hfill F_z(z)& =\frac{4{\left({\Omega }_{y_1}{m}_{x_1}\right)}^{m_{x_1}}{\left({\Omega }_{x_1}{m}_{y_1}\right)}^{m_{y_1}}{\left({\Omega }_{y_2}{m}_{x_2}\right)}^{m_{x_2}}{\left({\Omega }_{x_2}{m}_{y_2}\right)}^{m_{y_2}}}{B\left(m_{x_1},m_{y_1}\right)B\left(m_{x_2},m_{y_2}\right)}\hfill \\ & \times {\displaystyle \underset{0}{\overset{\infty }{\int }}d{z}_2}\frac{z_2^{2m_{x_2}+2m_{y_1}-1}}{{\left({\Omega }_{x_2}{m}_{y_2}+{\Omega }_{y_2}{m}_{x_2}{z}_2^2\right)}^{m_{x_2}+m_{y_2}}}{\displaystyle \underset{0}{\overset{z}{\int }}dt\frac{t^{2m_{x_1}-1}}{{\left({\Omega }_{x_1}{m}_{y_1}{z}_2^2+{\Omega }_{y_1}{m}_{x_1}{t}^2\right)}^{m_{x_1}+m_{y_1}}}}\hfill \end{array}$$

(11)

Let us define:

$$J={\displaystyle \underset{0}{\overset{z}{\int }}dt\frac{t^{2m_{x_1}-1}}{{\left({\Omega }_{x_1}{m}_{y_1}{z}_2^2+{\Omega }_{y_1}{m}_{x_1}{t}^2\right)}^{m_{x_1}+m_{y_1}}}}$$

(12)

After variable substitution $s={\Omega }_{x_1}{m}_{y_1}{z}_2^2+{\Omega }_{y_1}{m}_{x_1}{t}^2$, Equation (12) becomes:

$$J=\frac{1}{2{\Omega }_{y_1}^{m_{x_1}}{m}_{x_1}^{m_{x_1}}}{\displaystyle \underset{{\Omega }_{x_1}{m}_{y_1}{z}_2^2}{\overset{{\Omega }_{x_1}{m}_{y_1}{z}_2^2+{\Omega }_{y_1}{m}_{x_1}{z}^2}{\int }}ds\frac{{\left(s-{\Omega }_{x_1}{m}_{y_1}{z}_2^2\right)}^{m_{x_1}-1}}{s^{m_{x_1}+m_{y_1}}}}$$

(13)

Having in mind the fading parameter m is an integer, after expanding by using Equation (1.111) from [38]:

$${\left(s-{\Omega }_{x_1}{m}_{y_1}{z}_2^2\right)}^{m_{x_1}-1}={\displaystyle \sum _{k=0}^{m_{x_1}-1}\left(\begin{array}{c}{m}_{x_1}-1\\ k\end{array}\right)s^{m_{x_1}-k-1}{(-1)}^k{z}_2^{2k}{\Omega }_{x_1}^k{m}_{y_1}^k}$$

(14)

and changing the order of the integral and sum, the following is obtained:

$$J=\frac{1}{2{\Omega }_{y_1}^{m_{x_1}}{m}_{x_1}^{m_{x_1}}}{\displaystyle \sum _{k=0}^{m_{x_1}-1}\left(\begin{array}{c}{m}_{x_1}-1\\ k\end{array}\right){(-1)}^k{z}_2^{2k}{\Omega }_{x_1}^k{m}_{y_1}^k}{\displaystyle \underset{{\Omega }_{x_1}{m}_{y_1}{z}_2^2}{\overset{{\Omega }_{x_1}{m}_{y_1}{z}_2^2+{\Omega }_{y_1}{m}_{x_1}{z}^2}{\int }}ds{s}^{-(m_{y_1}+k+1)}}$$

(15)

which becomes

$$J=\frac{1}{2{\Omega }_{y_1}^{m_{x_1}}{m}_{x_1}^{m_{x_1}}}\left(S_1-S_2\right)$$

(16)

where

$$\begin{array}{c}{S}_1=\frac{1}{{\Omega }_{x_1}^{m_{y_1}}{m}_{y_1}^{m_{y_1}}{z}_2^{2m_{y_1}}}\frac{2^{1-m_{x_1}}(m_{x_1}-1)!}{(2m_{x_1}-1)!!}\\ S_2={\displaystyle \sum _{k=0}^{m_{x_1}-1}\left(\begin{array}{c}{m}_{x_1}-1\\ k\end{array}\right)\frac{{(-1)}^k}{m_{y_1}+k}}\hspace{0.17em}\frac{z_2^{2k}{\Omega }_{x_1}^k{m}_{y_1}^k}{{\left({\Omega }_{x_1}{m}_{y_1}{z}_2^2+{\Omega }_{y_1}{m}_{x_1}{z}^2\right)}^{m_{y_1}+k}}\end{array}$$

(17)

Now, the CDF becomes

$$F_z(z)=\frac{2{\left({\Omega }_{x_1}{m}_{y_1}\right)}^{m_{y_1}}{\left({\Omega }_{y_2}{m}_{x_2}\right)}^{m_{x_2}}{\left({\Omega }_{x_2}{m}_{y_2}\right)}^{m_{y_2}}}{B\left(m_{x_1},m_{y_1}\right)B\left(m_{x_2},m_{y_2}\right)}\left(J_1-J_2\right)$$

(18)

where

$$\begin{array}{l}{J}_1={\displaystyle \underset{0}{\overset{\infty }{\int }}d{z}_2}\frac{z_2^{2m_{x_2}+2m_{y_1}-1}}{{\left({\Omega }_{x_2}{m}_{y_2}+{\Omega }_{y_2}{m}_{x_2}{z}_2^2\right)}^{m_{x_2}+m_{y_2}}}{S}_1\\ J_2={\displaystyle \underset{0}{\overset{\infty }{\int }}d{z}_2}\frac{z_2^{2m_{x_2}+2m_{y_1}-1}}{{\left({\Omega }_{x_2}{m}_{y_2}+{\Omega }_{y_2}{m}_{x_2}{z}_2^2\right)}^{m_{x_2}+m_{y_2}}}{S}_2\end{array}$$

(19)

After several mathematical operations, J₁ simplifies to

$$J_1=\frac{2^{1-m_{x_1}}(m_{x_1}-1)!}{(2m_{x_1}-1)!!}\frac{1}{{\Omega }_{x_1}^{m_{y_1}}{m}_{y_1}^{m_{y_1}}}\frac{1}{{\left({\Omega }_{x_2}{m}_{y_2}\right)}^{m_{x_2}+m_{y_2}}}{\displaystyle \underset{0}{\overset{\infty }{\int }}d{z}_2}\frac{z_2^{2m_{x_2}-1}}{{\left(1+\frac{{\Omega }_{y_2}{m}_{x_2}}{{\Omega }_{x_2}{m}_{y_2}}{z}_2^2\right)}^{m_{x_2}+m_{y_2}}}$$

(20)

which may be solved using Equation (3.251.11) from [38]

$${\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{x^{\mu -1}}{{(1+\beta \cdot x^p)}^{\nu }}dx}=\frac{1}{p}{\beta }^{-\mu /p}B\left(\frac{\mu }{p},\nu -\frac{\mu }{p}\right)$$

(21)

as

$$J_1=\frac{2^{-m_{x_1}}(m_{x_1}-1)!}{(2m_{x_1}-1)!!}\frac{B\left(m_{x_2},m_{y_2}\right)}{{\Omega }_{x_1}^{m_{y_1}}{m}_{y_1}^{m_{y_1}}{\left({\Omega }_{x_2}{m}_{y_2}\right)}^{m_{y_2}}}\frac{1}{{\left({\Omega }_{y_2}{m}_{x_2}\right)}^{m_{x_2}}}$$

(22)

Using Equation (3.259.3) from [38], integral J₂ may be expressed as

$$\begin{array}{cc}\hfill J_2& =\frac{{\left({\Omega }_{y_1}{m}_{x_1}\right)}^{m_{x_2}}}{{\left({\Omega }_{x_2}{m}_{y_2}\right)}^{m_{x_2}+m_{y_2}}{\left({\Omega }_{x_1}{m}_{y_1}\right)}^{m_{x_2}+m_{y_1}}}\frac{z^{2m_{x_2}}}{2}{\displaystyle \sum _{k=0}^{m_{x_1}-1}\left(\begin{array}{c}{m}_{x_1}-1\\ k\end{array}\right)\frac{{(-1)}^k}{m_{y_1}+k}}B\left(m_{x_2}+m_{y_1}+k,m_{y_2}\right)\hfill \\ & \times {}_{2}F_1\left(m_{x_2}+m_{y_2},m_{x_2}+m_{y_1}+k;m_{x_2}+m_{y_1}+m_{y_2}+k;1-z^2\frac{{\Omega }_{y_1}{\Omega }_{y_2}}{{\Omega }_{x_1}{\Omega }_{x_2}}\frac{m_{x_1}{m}_{x_2}}{m_{y_1}{m}_{y_2}}\right)\hfill \end{array}$$

(23)

By substituting Equations (22) and (23) in (18), after some mathematical manipulations, we obtain the expression for CDF:

$$\begin{array}{cc}\hfill F_z(z)& =\frac{B\left(m_{x_1},m_{x_1}\right)}{B\left(m_{x_1},m_{y_1}\right)}-\frac{1}{B\left(m_{x_1},m_{y_1}\right)B\left(m_{x_2},m_{y_2}\right)}{\left(\frac{{\Omega }_{y_1}{\Omega }_{y_2}}{{\Omega }_{x_1}{\Omega }_{x_2}}\frac{m_{x_1}{m}_{x_2}}{m_{y_1}{m}_{y_2}}\right)}^{m_{x_2}}\hfill \\ & \times z^{2m_{x_2}}{\displaystyle \sum _{k=0}^{m_{x_1}-1}\left(\begin{array}{c}{m}_{x_1}-1\\ k\end{array}\right)\frac{{(-1)}^k}{m_{y_1}+k}}B\left(m_{x_2}+m_{y_1}+k,m_{y_2}\right)\hfill \\ & \times {}_{2}F_1\left(m_{x_2}+m_{y_2},m_{x_2}+m_{y_1}+k;m_{x_2}+m_{y_1}+m_{y_2}+k;1-z^2\frac{{\Omega }_{y_1}{\Omega }_{y_2}}{{\Omega }_{x_1}{\Omega }_{x_2}}\frac{m_{x_1}{m}_{x_2}}{m_{y_1}{m}_{y_2}}\right)\hfill \end{array}$$

(24)

Finally, the outage probability is defined as the probability that the signal-to-interference ratio at the destination mobile station is lower than a certain threshold z_th:

$$P_{out}\left(z_{th}\right)=\mathrm{Pr}\left[z<z_{th}\right]=F_z\left(z_{th}\right)$$

(25)

3.3. LCR of the Output SIR

The LCR is evaluated as the average value of the first derivative of the SIR or the ratio of two products of Nakagami-m random processes. This average value is evaluated by using the joint probability density function of the considered ratio and its first derivative. The signal-to-interference ratio at the input of the destination station can be expressed as the ratio:

$$z=\frac{x}{y}=\frac{x_1{x}_2}{y_1{y}_2}$$

(26)

The first derivative of z is:

$$\dot{z}=\frac{x_2}{y_1{y}_2}{\dot{x}}_1+\frac{x_1}{y_1{y}_2}{\dot{x}}_2-\frac{x_1{x}_2}{y_1^2{y}_2}{\dot{y}}_1-\frac{x_1{x}_2}{y_1{y}_2^2}{\dot{y}}_2$$

(27)

It is well known that the first derivative of the Nakagami-m random variable follows zero-mean Gaussian distribution [40]. Therefore, ${\dot{x}}_1$, ${\dot{x}}_2$, ${\dot{y}}_1$ and ${\dot{y}}_2$, have zero-mean Gaussian distribution with the following variances [40]:

$${\sigma }_{{\dot{x}}_i}^2={\pi }^2{f}_m^2\frac{{\Omega }_{x_i}}{m_{x_i}},\hspace{0.17em}{\sigma }_{{\dot{y}}_i}^2={\pi }^2{f}_m^2\frac{{\Omega }_{y_i}}{m_{y_i}},\hspace{0.17em}i=1,2$$

(28)

where f_m is the maximum Doppler frequency.

Since a linear transformation of a Gaussian random variable is also a Gaussian random variable, $\dot{z}$ follows a conditional Gaussian distribution. The mean of $\dot{z}$ is

$$\overline{\dot{z}}=\frac{x_2}{y_1{y}_2}\overline{{\dot{x}}_1}+\frac{x_1}{y_1{y}_2}\overline{{\dot{x}}_2}-\frac{x_1{x}_2}{y_1^2{y}_2}\overline{{\dot{y}}_1}-\frac{x_1{x}_2}{y_1{y}_2^2}\overline{{\dot{y}}_2}=0$$

(29)

because $\overline{{\dot{x}}_1}=\overline{{\dot{x}}_2}=\overline{{\dot{y}}_1}=\overline{{\dot{y}}_2}=0$.

The variance of $\dot{z}$ is

$${\sigma }_{\dot{z}}^2={\pi }^2{f}_m^2\frac{x_2^2}{y_1^2{y}_2^2}\frac{{\Omega }_{x_1}}{m_{x_1}}\left(1+z^2\frac{y_1^2{y}_2^2}{x_2^4}\frac{{\Omega }_{x_2}}{{\Omega }_{x_1}}\frac{m_{x_1}}{m_{x_2}}+z^2\frac{y_2^2}{x_2^2}\frac{{\Omega }_{y_1}}{{\Omega }_{x_1}}\frac{m_{x_1}}{m_{y_1}}+z^2\frac{y_1^2}{x_2^2}\frac{{\Omega }_{y_2}}{{\Omega }_{x_1}}\frac{m_{x_1}}{m_{y_2}}\right)$$

(30)

The joint probability density function of $z,\dot{z},x_2,y_1,$ and $y_2$ is

$$p_{z\dot{z}{x}_2{y}_1{y}_2}(z,\dot{z},x_2,y_1,y_2)=p_{\dot{z}}(\dot{z}|z,x_2,y_1,y_2)p_{z{x}_2{y}_1{y}_2}(z,x_2,y_1,y_2)$$

(31)

where

$$p_{z{x}_2{y}_1{y}_2}(z,x_2,y_1,y_2)=p_{x_2}(x_2)p_{y_1}(y_1)p_{y_2}(y_2)p_z(z|x_2,y_1,y_2)$$

(32)

The conditional joint probability density function of z is

$$p_z(z|x_2,y_1,y_2)=\left|\frac{d{x}_1}{dz}\right|f_{x_1}\left(\frac{z{y}_1{y}_2}{x_2}\right)$$

(33)

After substituting Equations (32) and (33) into (31), the expression for $p_{z\dot{z}{x}_2{y}_1{y}_2}(z,\dot{z},x_2,y_1,y_2)$ becomes

$$p_{z\dot{z}{x}_2{y}_1{y}_2}(z,\dot{z},x_2,y_1,y_2)=\frac{y_1{y}_2}{x_2}{p}_{x_1}\left(\frac{z{y}_1{y}_2}{x_2}\right)p_{x_2}(x_2)p_{y_1}(y_1)p_{y_2}(y_2)p_{\dot{z}}(\dot{z}|z,x_2,y_1,y_2)$$

(34)

The joint probability density function of z and $\dot{z}$ is

$$p_{z\dot{z}}(z,\dot{z})={\displaystyle \underset{0}{\overset{\infty }{\int }}d{x}_2}{\displaystyle \underset{0}{\overset{\infty }{\int }}d{y}_1}{\displaystyle \underset{0}{\overset{\infty }{\int }}d{y}_2}\frac{y_1{y}_2}{x_2}{p}_{x_1}\left(\frac{z{y}_1{y}_2}{x_2}\right)p_{x_2}(x_2)p_{y_1}(y_1)p_{y_2}(y_2)p_{\dot{z}}(\dot{z}|z,x_2,y_1,y_2)$$

(35)

The LCR is defined by [26]

$$N_z(z)={\displaystyle \underset{0}{\overset{\infty }{\int }}d\dot{z}}\dot{z}{p}_{z\dot{z}}(z\dot{z})$$

(36)

Finally, the level crossing rate of signal-to-interference ratio process at the input of the destination station is:

$$N_z(z)={\displaystyle \underset{0}{\overset{\infty }{\int }}d{x}_2}{\displaystyle \underset{0}{\overset{\infty }{\int }}d{y}_1}{\displaystyle \underset{0}{\overset{\infty }{\int }}d{y}_2}\frac{y_1{y}_2}{x_2}{p}_{x_1}\left(\frac{z{y}_1{y}_2}{x_2}\right)p_{x_2}(x_2)p_{y_1}(y_1)p_{y_2}(y_2){\displaystyle \underset{0}{\overset{\infty }{\int }}d\dot{z}}\dot{z}{p}_{\dot{z}}(\dot{z}|z,x_2,y_1,y_2)$$

(37)

Using

$${\displaystyle \underset{0}{\overset{\infty }{\int }}d\dot{z}}\dot{z}{p}_{\dot{z}}(\dot{z}|z,x_2,y_1,y_2)=\frac{1}{\sqrt{2\pi }{\sigma }_{\dot{z}}}{\displaystyle \underset{0}{\overset{\infty }{\int }}\dot{z}{e}^{-\frac{{\dot{z}}^2}{2{\sigma }_{\dot{z}}^2}}d\dot{z}}=\frac{{\sigma }_{\dot{z}}}{\sqrt{2\pi }}$$

(38)

we have

$$N_z(z)={\displaystyle \underset{0}{\overset{\infty }{\int }}d{x}_2}{\displaystyle \underset{0}{\overset{\infty }{\int }}d{y}_1}{\displaystyle \underset{0}{\overset{\infty }{\int }}d{y}_2}\frac{y_1{y}_2}{x_2}{p}_{x_1}\left(\frac{z{y}_1{y}_2}{x_2}\right)p_{x_2}(x_2)p_{y_1}(y_1)p_{y_2}(y_2)\frac{{\sigma }_{\dot{z}}}{\sqrt{2\pi }}$$

(39)

After substituting the probability density function (1) into (39), the expression for the level crossing rate becomes

$$\begin{array}{cc}\hfill N_z(z)& =\frac{16}{\Gamma (m_{x_1})\Gamma (m_{x_2})\Gamma (m_{y_1})\Gamma (m_{y_2})}{\left(\frac{m_{x_1}}{{\Omega }_{x_1}}\right)}^{m_{x_1}}{\left(\frac{m_{x_2}}{{\Omega }_{x_2}}\right)}^{m_{x_2}}{\left(\frac{m_{y_1}}{{\Omega }_{y_1}}\right)}^{m_{y_1}}{\left(\frac{m_{y_2}}{{\Omega }_{y_2}}\right)}^{m_{y_2}}\hfill \\ & \times z^{2m_{x_1}-1}\frac{1}{\sqrt{2\pi }}\pi f_m\sqrt{\frac{{\Omega }_{x_1}}{m_{x_1}}}{\displaystyle \underset{0}{\overset{\infty }{\int }}d{x}_2}{\displaystyle \underset{0}{\overset{\infty }{\int }}d{y}_1}{\displaystyle \underset{0}{\overset{\infty }{\int }}d{y}_2}\hfill \\ & \times \sqrt{1+z^2\frac{y_1^2{y}_2^2}{x_2^4}\frac{{\Omega }_{x_2}}{{\Omega }_{x_1}}\frac{m_{x_1}}{m_{x_2}}+z^2\frac{y_2^2}{x_2^2}\frac{{\Omega }_{y_1}}{{\Omega }_{x_1}}\frac{m_{x_1}}{m_{y_1}}+z^2\frac{y_1^2}{x_2^2}\frac{{\Omega }_{y_2}}{{\Omega }_{x_1}}\frac{m_{x_1}}{m_{y_2}}}\hfill \\ & \times \exp (-\frac{m_{x_1}}{{\Omega }_{x_1}}\frac{z^2{y}_1^2{y}_2^2}{x_2^2}-\frac{m_{x_2}}{{\Omega }_{x_2}}{x}_2^2-\frac{m_{y_1}}{{\Omega }_{y_1}}{y}_1^2-\frac{m_{y_2}}{{\Omega }_{y_2}}{y}_2^2\hfill \\ & +2(m_{x_2}-m_{x_1})\ln x_2+2(m_{x_1}+m_{y_1}-1)\ln y_1+2(m_{x_1}+m_{y_2}-1)\ln y_2)\hfill \end{array}$$

(40)

The three-fold integral in the above expression is solved by using the Laplace approximation formula (theorem) for the three-fold integral [41]:

$${\displaystyle \underset{0}{\overset{\infty }{\int }}dx}{\displaystyle \underset{0}{\overset{\infty }{\int }}dy}{\displaystyle \underset{0}{\overset{\infty }{\int }}dz}g(x,y,z)\exp (-\lambda f(x,y,z))={\left(\frac{2\pi }{\lambda }\right)}^{3/2}\frac{g(x_0,y_0,z_0)}{\sqrt{B(x_0,y_0,z_0)}}\exp (-\lambda f(x_0,y_0,z_0))$$

(41)

where x₀, y₀, and z₀ are solutions of the following set of equations:

$$\begin{array}{l}\frac{\partial f(x_0,y_0,z_0)}{\partial x_0}=0\\ \frac{\partial f(x_0,y_0,z_0)}{\partial y_0}=0\\ \frac{\partial f(x_0,y_0,z_0)}{\partial z_0}=0\end{array}$$

(42)

And

$$B(x_0,y_0,z_0)=\left|\begin{array}{ccc}\frac{{\partial }^{2}f(x_0,y_0,z_0)}{\partial x_0^2}& \frac{{\partial }^{2}f(x_0,y_0,z_0)}{\partial x_0\partial y_0}& \frac{{\partial }^{2}f(x_0,y_0,z_0)}{\partial y_0\partial z_0}\\ \frac{{\partial }^{2}f(x_0,y_0,z_0)}{\partial x_0\partial y_0}& \frac{{\partial }^{2}f(x_0,y_0,z_0)}{\partial y_0^2}& \frac{{\partial }^{2}f(x_0,y_0,z_0)}{\partial y_0\partial z_0}\\ \frac{{\partial }^{2}f(x_0,y_0,z_0)}{\partial x_0\partial z_0}& \frac{{\partial }^{2}f(x_0,y_0,z_0)}{\partial y_0\partial z_0}& \frac{{\partial }^{2}f(x_0,y_0,z_0)}{\partial z_0^2}\end{array}\right|$$

(43)

For the considered case, the constant λ = 1 and functions f and g are

$$g(x_2,y_1,y_2)=\sqrt{1+z^2\frac{y_1^2{y}_2^2}{x_2^4}\frac{{\Omega }_{x_2}}{{\Omega }_{x_1}}\frac{m_{x_1}}{m_{x_2}}+z^2\frac{y_2^2}{x_2^2}\frac{{\Omega }_{y_1}}{{\Omega }_{x_1}}\frac{m_{x_1}}{m_{y_1}}+z^2\frac{y_1^2}{x_2^2}\frac{{\Omega }_{y_2}}{{\Omega }_{x_1}}\frac{m_{x_1}}{m_{y_2}}}$$

(44)

$$\begin{array}{cc}\hfill f(x_2,y_1,y_2)& =\frac{m_{x_1}}{{\Omega }_{x_1}}\frac{z^2{y}_1^2{y}_2^2}{x_2^2}+\frac{m_{x_2}}{{\Omega }_{x_2}}{x}_2^2+\frac{m_{y_1}}{{\Omega }_{y_1}}{y}_1^2+\frac{m_{y_2}}{{\Omega }_{y_2}}{y}_2^2-\hfill \\ & -2(m_{x_2}-m_{x_1})\ln x_2-2(m_{x_1}+m_{y_1}-1)\ln y_1-2(m_{x_1}+m_{y_2}-1)\ln y_2\hfill \end{array}$$

(45)

4. Numerical Results

This section presents numerical results that indicate the influence of different fading parameters on the PDF, the outage probability, and LCR. The validity of the theoretical results is confirmed with the results obtained by the Monte-Carlo simulation with one million simulation steps. It is assumed, for all the following figures, that the Nakagami-m short-term fading severity parameter is equal for the desired signal and interference, in both sections, i.e., $m_{x_1}=m_{x_2}=m_{y_1}=m_{y_2}=m$. The other assumption is the equality of the desired signal and the interference power for both sections, i.e., ${\Omega }_{x_1}={\Omega }_{x_2}={\Omega }_x$ and ${\Omega }_{y_1}={\Omega }_{y_2}={\Omega }_y$. These assumptions are introduced for the sake of a more simple, clean graphical representation of the results and do not limit the generality of the results.

Figure 2 depicts the PDF versus signal-to-interference ratio for various values of fading severity parameters and different average signal-to-interference power ratios (SIR) Ω_x/Ω_y.

An increase in the value of the severity fading parameter has the effect of narrowing the range of the PDF on the destination node, as well as increasing the value of the maximum. An increase in the ratio of the average signal power parameters Ωx/Ωy results in the expansion of the range of the PDF on the destination node, as well as a decrease in the maximum value. From Figure 2 we can see a good agreement between the theoretical and simulation results.

Figure 3, Figure 4 and Figure 5 depict the outage probability as a function of different parameters. Again, these figures show a good agreement between the theoretical and simulation results, this time for CDF.

Figure 3 depicts the outage probability of the system versus the outage threshold, z_th. From this figure, we can see that the outage probability is higher for higher values of the outage threshold. In contrast, the influence of the fading severity parameter is twofold. For lower values of z_th, the outage probability is lower for higher m. Likewise, for higher values of z_th, the outage probability is higher for higher m. Further, from Figure 3 we can see that the outage probability is equal for m = 1 = 3 at z_th = 0.33 for Ω_x/Ω_y = 1/3, m = 1 = 3 at at z_th = 1 for Ω_x/Ω_y = 1, and m = 1 = 3 at z_th = 3 for Ω_x/Ω_y = 3. This can be confirmed if we put Ω_x/Ω_y = z_th in Equation (24). In this case, the outage probability is always equal to 0.5 and does not depend on m, z_th, or Ω_x/Ω_y.

Figure 4 and Figure 5 show the outage probability as a function of m and Ω_x/Ω_y, respectively. From Figure 4, we can see that the outage threshold has a larger influence on the outage probability than the fading severity parameter m. Then, from Figure 5, we can see the influence of SIR on outage probability. As the ratio between the power of the desired signal and the interference increases, the outage probability decreases. Outage probability depends on the outage threshold that is selected. SNR over this level guarantees the reliable performance of the transmission system.

Figure 6, Figure 7 and Figure 8 show the influence of fading and signal parameters on the LCR. Aside from the Laplace approximation results, we present the results obtained by the numerical integration of Equation (40). These results are also confirmed with the Monte-Carlo simulation results, based on the sum-of-sinusoids Nakagami-m channel model.

The normalized LCR of signal-to-interference ratio process at the output of the M2M fading channel, as a function of LCR threshold z, is shown in Figure 6 for different values of fading severity parameter m. The signal-to-interference ratio is Ωx/Ωy = 3. The results show that the LCR is lower for higher m. In addition, for higher m, the threshold has a higher influence on the LCR. There is a good agreement between the exact (numerical integration), approximation, and simulation results, with the simulation results being slightly off for low z.

Figure 7 shows the LCR as a function of fading severity parameter m, for different values of signal-to-interference ratio and z = 10 dB. We can see that the influence of parameter m on the level crossing rate is higher for lower values of the signal-to-interference ratio.

The normalized LCR, obtained by the Laplace approximation versus signal-to-interference ratio, for several values of Nakagami-m severity parameters m and LCR thresholds z, is depicted in Figure 8.

As we can see, the level crossing rate increases for lower values of the signal-to-interference ratio, achieves maximum, and decreases for higher values of the signal-to-interference ratio. The influence of the signal-to-interference ratio on the level crossing rate is higher for lower values of the signal-to-interference ratio.

5. Conclusions

In this paper, we considered a wireless communication system with a cooperative network operating over mobile-to-mobile fading channels in the presence of co-channel interference. Communication between source and destination takes place via relay. The mobile-to-mobile fading channel has two sections (from source to relay and from relay to destination), and the desired signal and co-channel interference are present in both. In addition, the desired signal and co-channel interference in both sections experience Nakagami-m short-term fading. All these systems have practical applications such as drone wireless relay systems for disaster recovery and physical layer security to avoid eavesdropping attacks and adaptive transmission.

We evaluated the probability density function and cumulative distribution function of the signal-to-interference ratio at the input of the destination station. These theoretical results are shown as closed-form expressions. All the results are verified with Monte-Carlo simulations. As shown, the theoretical results are in great accordance with the simulation results. Analytical expressions in closed form for PDF and CDF represent the basis for calculating other system performances such as the outage probability, channel capacity, bit-error rate (BER), level crossing rates, average fade duration, and many others. The closed-form expressions for PDF and CDF presented in this work are not limited only to mobile-to-mobile communication systems with relays. These results are useful for performance analyses in systems with adaptive transmission, physical layer security, etc.

The outage probability as a measure of the performance of the considered system allows us to see the stability and reliability of the system. The results show that the outage probability increases when the power of the desired signal decreases or the power of co-channel interference increases. Moreover, if the average signal-to-interference power ratio is equal to the outage threshold, the outage probability is Pout = 0.5, regardless of the fading severity parameter m and the actual values of z_th and Ωx/Ωy. The conclusion is that the outage probability of the system depends much more on the threshold level than on the severity parameter and the ratio of the power of the desired signal and interference. It is important because, with threshold level selection, we can control system reliability.

In addition, the closed-form expression for the level crossing rate was derived using the Laplace approximation theorem. The level-crossing rate results are also verified with the Monte-Carlo simulation. The obtained results showed that the level crossing rate increases for lower values of the signal-to-interference ratio and decreases for higher values of the signal-to-interference ratio.