Impact of co‐channel interference on performance of dual‐hop wireless ad hoc networks over α−μ fading channels

In this work, we investigate the performance of a dual‐hop cooperative network over α−μ fading channels with the presence of co‐channel interference (CCI) at both the relay and destination nodes. Amplify‐and‐forward (AF) relaying is considered in the relay node. The upper bound of the signal‐to‐interference‐plus‐noise ratio (SINR) of the dual‐hop relay link is used to determine the system performance. The probability density function (PDF) and the cumulative distribution function (CDF) of the upper bound of the SINR are analyzed. The system performance is determined in terms of the outage and error probabilities. Numerical results are used to present the performance analysis of the system.

and Zhong et al 3 studied the impact of CCI on AF dual-hop relay network performance, assuming Rayleigh fading channels, while both AI-Qahtani et al 4 and Ikki and Aissa 5 assumed the Nakagami-m fading channels. In Salhab et al, 6 the authors investigations assumed different fading channels for the interferes, where they considered Rayleigh, Nakagami-m, and Rician fading channels. Ilhan 7 investigated the dual-hop relaying system presence of co-channel interference assuming DF relaying scheme and Nakagami-m fading is adopted. Another work that investigated AF dual-hop relay network performance is represented in Nauryzbayev et al, 8 where α−μ fading channel is assumed and the performance investigation in terms of the ergodic capacity and ergodic outage probability; however, the interference presence where not considered in this research. Moreover, Nauryzbayev et al 9 and Amer and Al-Dharrab 10 derived the outage probability for both AF and DF relaying techniques in dual-hop networks in α−μ environment, without considering the interference presence at any of the nodes. In addition, the work in Magableh et al 11 investigated dual-hop AF relay network performance in terms of end-to-end capacity and outage capacity, assuming α−μ environment subject to CCI interference.
Motivated by the preceding, we investigated the error and outage performance of the dual-hop AF relaying network over α−μ fading channels in the presence of CCI affecting both the relay and destination nodes. The α−μ fading channel is considered since it is the general fading model for small-scale fading, and it represents the multipath fading channel model, in which the received signal consists of clusters of multipath waves that propagate in a nonhomogeneous environment. 12 The α parameter is used to represent the nonlinear behavior of the propagation medium, while the μ parameter represents the multipath-clusters number. The Nakagami-m and Rayleigh fading can be derived from the α−μ fading. In this study, the derived formulas are used to derive other fading channel models such as Rayleigh, Nakagami-m, Chi, and one-sided Gaussian where the α value equals 2.
Moreover, we derive the probability density function (PDF) and the cumulative distribution function (CDF) of the signal-to-interference-and-noise ratio (SINR) of dual-hop transmission with AF relaying. The obtained results are used to derive an analytical expression of the error and outage probability for the considered system. The rest of the paper is organized as follows: in Section 2, the system model is introduced. Then in Section 3, the SINR analysis is introduced. The system performance in terms of outage and average error probabilities is conducted and derived in Section 4. Numerical results are presented and discussed in Section 5. Finally, conclusions are drawn in Section 6.

| SYSTEM MODEL
Consider a dual-hop wireless communication system is shown in Figure (1), where the source node S communicates with a destination node D through a relay node R. The received signals at the relay node R and destination node D are corrupted by CCI signals from N and L co-channel interferer's denoted as fx h j g N j = 1 with energy of E h j and fx g k g L k = 1 with energy of E g k , respectively. The fading coefficients for all the links are assumed to be α−μ fading channel. AF relaying scheme is considered due to it's simplicity compared to other relaying sachems, where the signal received by the relay node is amplified before being forwarded to the destination node, the amplification factor of AF scheme is proportional to the inverse gain of the channel. The destination node D is assumed to have perfect knowledge of the channel state information (CSI), and with reference to Ikki and Aissa, 2 eq. 1 the received signal at the relay node R is given by F I G U R E 1 Dual-hop relay network with co-channel interference at the relay and destination nodes where h SR is the α−μ channel fading coefficient of the S ! R link, E S is the energy of the transmitted signal, d S is the desired data with unit energy, h j is the α−μ channel fading coefficient of the j th ! R link, E h j is the energy of the j th interferer at R, and d j is the j th co-channel interferers data with unit energy at the relay node. The additive-white-Gaussian noise (AWGN) at the relay node is denoted as n SR with a zero-mean and N o variance~CN(0,N o ).
The signal received at the destination node D from the relay node is expressed as where E R represents the energy of the relay node transmitted signal, h RD is the α−μ channel fading coefficient of the R ! D link, E g k is the energy of the interference signal at the destination node, g k is the α−μ fading coefficient of the interference channel at the destination, and d k is the kth co-channel interferer's data with unit energy at the destination, n SR denotes the AWGN at the destination node with a zero-mean and N o variance~CN(0,N o ). And y R is the relayed signal, which and can be represented as where the AF gain factor G AF is set to maintain the output energy from the relay to E R . The gain factor in the presence of CCI is computed as After substituting Equations (1) and (3) into Equation (2) then with some algebraic simplifications, the received signal at the destination node D can be expressed as

| SINR ANALYSIS
In this section, we will derive the expression for the SINR of the proposed dual-hop relaying communication system, where the SINR at the destination node D can be written as The destination node can be further simplified after introducing the effective signal-to-noise ratio (SNR) expressions γ eff , as where γ ef f SR = γ SR 1 + represent the effective SINR's at the relay and the destination node, respectively. The SNR at the relay and destinations nodes are expressed as γ SR = E S |h SR | 2 /N o and γ RD = E R |h RD | 2 /N o , respectively. The interference affecting the relay and destination nodes is denoted as The statistical characteristics (i.e., the PDF and the CDF) of the SINR of the proposed dual-hop network are developed in the following sections, in addition to the CCI density function presented in the system nodes.

| Co-channel interference probability density function
CCI occurs when a frequency band is used by multiple users at the same time. For cellular networks, CCI occurs by the reuse of frequency in the neighboring cells, which accordingly will affect the performance of the network. CCI may degrade the performance of the system; hence, studying its impact is very important and helps in developing mitigation techniques to reduce the performance degradation. CCI can be modeled as the sum of N independent-not-identicallydistributed (INID) α−μ variates, (i.e., P N i = 1 γ h I ), but in this section, the derivation of the formulas will be based on the independent-identically-distributed (IID) assumption. The SNR for a single interferer link can be expressed as Assuming α−μ fading channel, the α−μ PDF for a single interferer link is represented by The moment-generating function (MGF) approach will be used to develop a mathematical formula that can be used to express the PDF of the N INID α−μ variates. The MGF is defined mathematically as eqs. 3.3-6 and 3.3-7 in Peebles Peyton 13 : The α−μ PDF represented in Equation (9) can then substituted in Equation (10). The next step is to use and introduce the Meijer G-Function of the exponential function which is given as in Prudnikov and Marichev, 14 eq. 8.4.3.1 Finally, the resulting integral can be solved using eq. 2.24.3.1 in Prudnikov and Marichev, 14 with defining the parameters: (p = n = 0, q = m = 1, μ = 1/2 and c * = 1/2) also a new fraction is to be defined as well l k such that l k = α 2 À Á with the gcdðl, kÞ = 1 ð Þ ; great common divisor) by this the noninteger values of α would be included in the calculations. The MGF is formulated as where Δðk, aÞ = a k , a + 1 k , …, a + k − 1 k , and the Meijer G-function is defined generally in Wolfram, 15 eq. 07.34.02.0001.01 Applying the MGF approach, the density function of P N i = 1 γ h I can be derived and mathematically expressed using where M γ Ri ðsÞ represents the MGF of the ith link which is expressed in Equation (11), and ℒ − 1 f:g is the inverse Laplace transformation. Substituting the derived MGF, the PDF of the sum of N-IID interferers is then can be evaluated as Equation (13) is not analytically tractable. However, under some conditions, for example when l = k = 1, this results in α = 2, and then the density function can be analytically expressed as Using the Meijer G-function expression given as in Wolfram, 15 eq 07.34.03.0271.01 finally, the PDF of can be written as This formula can be used to generate different IID CCI models such as the Rayleigh fading model as shown in Ikki and Aissa, 2 eq. 14 where α = 2 and μ = 1 and the Nakagami-m fading model as shown in Ikki and Aissa, 16 eq. 14 where α = 2.

| Effective SINR statistical characteristics
In this section, we will derive the statistical characteristics of the effective SINR γ ef f SR and γ ef f RD of the links S ! R and R ! D, respectively, in terms of the PDF and the CDF when both links are subject to α−μ fading channel. The PDF of γ ef f SR can be derived using eqs. 6-60 in Papoulis and Pillai, 17 where W v,μ (z) is the Whittaker function which is defined in Wolfram, 15 ð18Þ The CDF F γ ef f ðγÞ is presented as Substituting Equation (17) in Equation (19), then by using eq. 4.6 in Papoulis and Pillai 17 and then applying eq. 2.19.5.12 in Prudnikov and Marichev, 14 the CDF of γ SR ef f is evaluated as The CDF of γ RD ef f is evaluated as This result can be used to generate other special cases such as when μ SR =1 and μ h I = 1, the CDF of γ SR ef f is given in Ikki and Aissa, 2 eq. 17 while setting μ RD = 1 and μ g I = 1 can generate the CDF given in Ikki and Aissa, 2 eq. 19.

| Dual-hop relay SINR PDF and CDF
To derive the SINR for the relay link in Equation (7), we have used a tight upper bound to simplify the analysis of the network performance, that is, hence, f γ up ðγÞ can be expressed as Assuming identical fading links, that is, N = L = l interferes at the relay and destination nodes, the PDF of γ up is represented by applying the Whittaker properties in Olver et al, 19 eq. 13.14.31 and the Whittaker W function in Olver et al, 19 eq. 13.14.3 then finally using the identities in Olver et al, 19 eqs. 13.14.5 and 13.2.8 the PDF of γ up is obtained as where (a) j is the Pochhammer symbol and is defined as Γ(a + j)/Γ(a). The result in Equation (25) can be used to generate eq. 20 in Ikki and Aissa, 2 under the conditions: L = N, μ = μ I = 1, (γ hI = γ gI = γ I ) and (γ h = γ g = γ). While for non-identical links, Equation (25) can be can be used to generate eq. 21 in Ikki and Aissa. 2 The CDF of the dual-hop network is derived for the upper bound, that is, the minimum of two random variables as Assuming identical links, and substituting both Equations (20) and (21) into Equation (26), the CDF of the dualhop relay network is expressed as Γðμ −kÞ γμ γ after applying the Lagrange's identity with some mathematical manipulation, the CDF is derived as This formula can be used to generate different CDF such as Rayleigh fading channels where μ = 1. For identical links and N = L, the CDF can be expressed as , is the average SIR at the Relay node. For non-identical links, the CDF can be expressed as: where Λ = γ SR γ h I is the average signal-to-interference ratio (SIR) at the Relay node and Υ = γ RD γ g I is the average SIR at the destination node.

| Dual-hop relay SINR MGF
The MGF is very useful for deriving the performance metrics of the network, which is given as substituting the PDF of the dual-hope given in Equation (24) while assuming identical fading links, and using, 20 eq. 2.3.6.9 the MGF can be expressed after some mathematical manipulation as where the U (a, b, x) is the confluent hypergeometric function of the second kind and is defined generally as in Wolfram, 15 eq. 07.33.02.0001.01

| Outage probability analysis
In this section, the outage probability of the dual-hop relay network will be analyzed. The outage probability is an important performance measure, as the analysis of the outage probability is essential to characterize the error performance and reliability. Outage probability is defined as the probability at which the SINR falls below a threshold value (γ Th ), or mathematically; which can be evaluated after substituting the PDF and evaluating the integration as ð34Þ Equation (34) can be used to generate many different formulas such as Ikki and Aissa 2 , eq. 21 which represents identical fading channel with μ = 1, and as in Ikki and Aissa 2 , eq. 23 for non-identical fading channel with μ = 1, and many other formulas for other fading channels such that with μ = m which corresponds to the Nakagami-m fading model.

| Average error probability analysis
The average error probability is given by where P e (γ) is the conditional error probability for a given γ. For coherent binary-phase-shift-keying (BPSK) modulation, the average bit error rate (BER) is expressed as using the alternative form of the Q-function expressed in terms of the MGF as in Goldsmith, 21 eq. 6.44 and applying the upper bound given in Simon and Alouini, 22 eq. 9.27 we have developed several formulas for BER of BPSK modulation for different cases as • For identical fading channel coefficient with α = 2, μ = 1 L = N, μ sr = μ rd = μ Ih = μ Ig = 1, γ sr = γ rd = γ and γ Ih = γ Ig = γ I , the average error probability is as where Λ = is the average SIR at the relay or the destination node. • For non-identical fading channel links with L 6 ¼ N, μ sr = μ rd = μ Ih = μ Ig = 1, γ sr 6 ¼ γ rd and γ Ih 6 ¼ γ Ig , the error probability is expressed as The coefficients are defined as • For identical fading channel links with L = N, μ sr = μ rd = m, μ Ih = μ Ig = m I , γ sr = γ rd = γ and γ Ih = γ Ig = γ I , the error probability for BPSK modulation is given as

| NUMERICAL RESULTS
In this section, we illustrate the expressions developed in Section 4 for the error and outage performance of the considered dual-hop AF relaying system, and illustrate the effect of interference on system performance. We consider different values of the parameter μ of the assumed α−μ fading channel model while α is fixed with the value of α = 2, and in these results, the fading parameters are assumed equal for all the dual-hop links and for the interferers links. The dual-F I G U R E 4 Dual-hop outage probability for α = 2, μ = m = 2 and N = L = 5. SNR, signal-to-noise ratio F I G U R E 5 Average error probability for identical fading channel with α = 2 and μ = 1. SNR, signal-to-noise ratio hop performance characteristics are plotted in terms of the outage and error probability as a function of the average SNR in decibel. Figures (2), (3), and (4) depict the outage performance of a dual-hop AF relaying system where α−μ fading channels are assumed with the presence of CCI interference at the relay and destination nodes. In Figure (2), the outage performance is illustrated for fading parameters α = 2 and μ = 1, with equal power interferers INR = 3 dB for different but equal number of interferers at the relay and destination nodes. While in Figure (3) the outage is characterized with fading parameters α = 2 and μ = 2 with equal power interferers INR = 12 dB and different but equal number of interferers at the relay and destination nodes. Figure (4) shows the outage with fading parameters of α = 2 and μ = 2 with different values of power interferers. Moreover, the system performance degradation due to the increase in number of interferers is obvious in these curves.