Outage Probability Performance Prediction for Mobile Cooperative Communication Networks Based on Artificial Neural Network

This paper investigates outage probability (OP) performance predictions using transmit antenna selection (TAS) and derives exact closed-form OP expressions for a TAS scheme. It uses Monte-Carlo simulations to evaluate OP performance and verify the analysis. A back-propagation (BP) neural network-based OP performance prediction algorithm is proposed and compared with extreme learning machine (ELM), locally weighted linear regression (LWLR), support vector machine (SVM), and BP neural network methods. The proposed method was found to have higher OP performance prediction results than the other prediction methods.


Introduction
Mobile applications have grown explosively in recent years, leading to an increased interest in mobile communication systems [1][2][3]. Relay-assisted mobile cooperative communication is an inevitable trend for future mobile networks, especially with regard to reliability [4][5][6]. In a study by the authors of [4], a multi-antenna decode-and-forward relay was used to assist a two-user non-orthogonal multiple access system and ensure secure transmission. For trusted and untrusted scenarios, secure relaying was considered in non-orthogonal multiple access [5]. A beamforming design was investigated for cooperative secure transmission in cognitive two-way relay networks [6].
As a promising technology, multiple-input-multiple-output (MIMO) can improve the performance of mobile cooperative communication. Massive MIMO was combined with general orthogonal precoding for high mobility scenarios in [7]. A minimum mean square error (MMSE) detector was used for channel estimation of massive MIMO systems [8]. In [9], the pairwise error probability (PEP) of the massive multiuser MMSE-MIMO systems was derived. However, the hardware complexity of MIMO also increased. Transmit antenna selection (TAS) is a practical option to reduce hardware complexity. It has also been attracting the attention of academicians [10][11][12]. Based on the asymptotic theory on order statistics, the authors of Ref. [13] derived the asymptotic upper capacity bounds of massive MIMO system with TAS over Rayleigh fading channels. A TAS strategy was investigated for full-duplex distributed antenna systems in [14]. Based on Monte Carlo Tree Search (MCTS), a self-supervised 1.
For the TAS scheme, we derive the novel exact closed-form mathematical OP expressions.

2.
Based on the derived OP expressions, a back-propagation (BP) neural network-based OP performance prediction algorithm is proposed. We use the OP theoretical results to generate training data. We test extreme learning machine (ELM), locally weighted linear regression (LWLR), support vector machine (SVM), and BP neural network methods.

3.
Through Monte-Carlo simulations, we verify the derived OP expressions. Compared with ELM, LWLR, and SVM methods, the Monte-Carlo results verify that our method can consistently achieve higher prediction results.
This paper is organized as follows: Section 2 provides the related work. The system model is presented in Section 3. The OP performance of the TAS scheme is investigated in Section 4. Based on the BP neural network, we propose an OP performance prediction algorithm in Section 5. OP performance is evaluated in Section 6. Section 7 offers concluding remarks.

Related Work
Cooperative communication has been proposed to improve the performance of mobile communication networks. The system performance of MIMO AF cooperative networks over the shadowed-Rician fading model was investigated in [22]. The OP performance of mobile cooperative networks was investigated in [23]. The Stackelberg game was exploited for a hybrid satellite-terrestrial cooperative network, and the harmonic mean function was used to select the best relay node in [24]. The error performance of the spatial modulation system over spatially correlated Rayleigh channels was presented in [25].
TAS is widely employed in cooperative networks to reduce hardware complexity. In [26], a novel TAS strategy was proposed for full-duplex AF relaying over the Rayleigh model. To reduce multiuser interference, the TAS mechanism was introduced into constant envelope pre-coding [27]. In [28], a TAS-maximal ratio combining relay system investigated the effect of channel estimation error. In [29], to reduce hardware complexity, two TAS schemes were proposed for secure transmission using precoding-aided spatial modulation.
Traditionally, performance evaluation is achieved by mathematical superposition, approximation, and fitting. These methods are faced with oversimplified real-world issues. Machine learning techniques have overcome these issues and are widely used in performance prediction [30]. In [31], a LWLR method was proposed to predict the damping ratio of a dominant mode online.SVM regression model was used to propose localizing algorithms for large-scale wireless sensor networks in [32]. A novel evolutionary algorithm was proposed for data classification problem with ELM in [33]. Because of good nonlinear prediction ability, a BP neural network model is more suitable for performance prediction. By using a BP neural network, Ref. [34] proposed a monitoring method of total seed mass. In [35], the BP network was used to obtain a direct readout of the applied force. To predict the telecommunication customer churn, Ref. [36] used a particle classification method to optimize the BP network. The authors of Ref. [37] employed BP network for a high accuracy channel estimation in secure cooperative transmission. In [38], the BP network was used to predict the Rayleigh fading channel. Figure 1 presents the system model, which includes a mobile source (MS) node with N t antennas, a mobile destination (MD) node with N r antennas, and L mobile relay (MR) nodes. The nodes operate in half-duplex mode. The LMR nodes utilize their individual uplink channel state information (CSI) to select the best MR. The best MR node participates in collaboration. The MD node calculates the received signal-to-noise ratio (SNR) from the best MR node. The MD node orders the received SNR from N t antennas, and then feedbacks the selected antenna to the MS node. optimize the BP network. The authors of Ref. [37] employed BP network for a high accuracy channel estimation in secure cooperative transmission. In [38], the BP network was used to predict the Rayleigh fading channel. Figure 1 presents the system model, which includes a mobile source (MS) node with Nt antennas, a mobile destination (MD) node with Nr antennas, and L mobile relay (MR) nodes. The nodes operate in half-duplex mode. The LMR nodes utilize their individual uplink channel state information (CSI) to select the best MR. The best MR node participates in collaboration. The MD node calculates the received signal-to-noise ratio (SNR) from the best MR node. The MD node orders the received SNR from Nt antennas, and then feedbacks the selected antenna to the MS node. The channel coefficient h = hk, k∈{SDij, SRil, RDlj},i∈(1,Nt),j∈(1,Nr),l∈(1,L). The amplitude of h follows N-Nakagami distribution [19].The total power is E.

System Model
In the first timeslot, MSi transmits the signal a. MDj and MRl receive the signals rSDij and rSRil as: where GSDij = 1 is the relative gain of MS→MD, GSRilis the relative gain of MSi→MRl. nSRil and nSDij have mean 0 and variance N0/2. K is the power allocation parameter, which controls power distribution between the MS and MR. For MS, the power is KE. For MR, the power is (1−K)E, K∈(0,1).
In the first timeslot, MS i transmits the signal a. MD j and MR l receive the signals r SDij and r SRil as: where G SDij = 1 is the relative gain of MS→MD, G SRil is the relative gain of MS i →MR l . n SRil and n SDij have mean 0 and variance N 0 /2. K is the power allocation parameter, which controls power distribution between the MS and MR. For MS, the power is KE. For MR, the power is (1−K)E, K∈(0,1).
In the second timeslot, with AF method, MD j receives the signal as: where G RDlj is the relative gain of MR l →MD j , n RDlj has mean 0 and variance N 0 /2. c ilj is given as [39]: where N 0 is noise power. MD j calculates the received signal-to-noise ratio (SNR) as: where γ SRDij is the SNR of MS→MR→MD link,γ SDij is the SNR of MS→MD link,γ SRil is the SNR of MS i →MR l link, andγ RDlj is the SNR of MR l →MD j link.
It is difficult to obtain the closed-form solution to γ SRDij . With the help of [40,41], we obtain an upper bound of γ SRDij as: MD calculates the received SNR as: where γ ij = max(γ SDij , γ upij ). We select g of TAS scheme as:

The OP of Optimal TAS Scheme
We obtain the OP as: where γ th is a given threshold,γ up is the upper bound ofγ SRD. The V 1 is evaluated as: where the G[·] is the Meijer G function, which is given as [19].
Next, V 2 is evaluated as: where

Input and Output Selection
By the derived closed-form OP expressions, we can see that OP performance is affected significantly by m, N, G and K. We use m, N, G, K and other parameters as indicators. The input X includes 17 indicators, the output y is the corresponding OP performance obtained by Equation (13). The 17 . X i is given as:

BP Neural Network Structure
The BP neural network is a kind of multi-layer pre-feedback artificial neural network. It changes its internal states according to the inputs, and produce outputs depending on the inputs and activation function. Figure 2 shows the BP neural network. It has three layers, namely the input layer, the hidden layer, and the output layer. For the input layer, there are 17 neurons. For the hidden layer, there are q neurons. For the output layer, there is 1 neuron. The network is formed by connecting the neurons in different layers, resulting in a directed and weighted graph. For the input and hidden layers, w ij is the weight coefficient, b j is the bias value. For the hidden and output layers, v j is the weight coefficient, θ is the bias value. For the hidden layer, input is given as: The output is given as: where f(x) is the activation function. For the output layer, input is given as: The output is given as: ( ) The output error EE is given as: where y h is the output for h-th data, and d h is the desired output. Figure 3 shows the flowchart of the OP performance prediction algorithm. For the hidden layer, input is given as:

The Flowchart of OP Performance Prediction Algorithm
The output is given as: where f (x) is the activation function. For the output layer, input is given as: The output is given as: The output error EE is given as: where y h is the output for h-th data, and d h is the desired output. Figure 3 shows the flowchart of the OP performance prediction algorithm.

Metric
We use mean squared error (MSE) to evaluate the performance of different methods. MSE is computed as follows: where PP is the number of testing data.

Numerical Results
In this section, E = 1.μ = GSR/GRD. Figure 4 presents the OP performance of the TAS scheme. Table 1 gives the parameters employed. From Figure 4, we see that the Monte-Carlo results and analytical results are similar. The OP is improved as Nt increased.

Metric
We use mean squared error (MSE) to evaluate the performance of different methods. MSE is computed as follows: where PP is the number of testing data.

Numerical Results
In this section, E = 1.µ = G SR /G RD . Figure 4 presents the OP performance of the TAS scheme. Table 1 gives the parameters employed. From Figure 4, we see that the Monte-Carlo results and analytical results are similar. The OP is improved as N t increased.    Figure 5 presents the effect of Nt on the OP performance. Table 2 gives the parameters employed. From Figure 5, with Nt increased, the OP decreases. When SNR=12dB, the OP is 8.6 × 10 −2 with Nt = 3, 3.6 × 10 −2 with Nt = 4, and 1.5 × 10 −2 with Nt = 5.    Figure 5 presents the effect of N t on the OP performance. Table 2 gives the parameters employed. From Figure 5, with N t increased, the OP decreases. When SNR=12dB, the OP is 8.6 × 10 −2 with N t = 3, 3.6 × 10 −2 with N t = 4, and 1.5 × 10 −2 with N t = 5.   Figure 6 presents the effect of K on OP performance. Table 3 gives the parameters employed. From Figure 6, with SNR increased, the OP decreases. When K = 0.5, the OP is 3.8 × 10 −2 with SNR = 10 dB, 2.0 × 10 −4 with SNR = 15 dB, and 9.0 × 10 −8 with SNR = 20 dB.     Figure 6 presents the effect of K on OP performance. Table 3 gives the parameters employed. From Figure 6, with SNR increased, the OP decreases. When K = 0.5, the OP is 3.8 × 10 −2 with SNR = 10 dB, 2.0 × 10 −4 with SNR = 15 dB, and 9.0 × 10 −8 with SNR = 20 dB.   Figure 6 presents the effect of K on OP performance. Table 3 gives the parameters employed. From Figure 6, with SNR increased, the OP decreases. When K = 0.5, the OP is 3.8 × 10 −2 with SNR = 10 dB, 2.0 × 10 −4 with SNR = 15 dB, and 9.0 × 10 −8 with SNR = 20 dB.   In Figures 7-10, we compare the BP neural network with LWLR [42], SVM [43], and ELM [44] methods. (1) The LWLR [42] model is as follows: where θ denotes the coefficient vector of the linear equation, τ is the bandwidth parameter. (2) The SVM [43] model is as follows where w is the adjustable weight, w 2 is the Euclidean norm of the vector, ε i is the slack variable, and c is the penalty parameter. KK() is the kernel function, which has an important parameter g. (3) ELM [44]: ELM has the same framework as the BP neural network. The input weight of ELM is subject to random assignment by a certain distribution function, and the output weight is directly calculated via the least squares method. The hidden layer has q neurons. Compared with the BP neural network, the training and recognition processes of ELM are rapid. In Figures 7-10, we compare the BP neural network with LWLR [42], SVM [43], and ELM [44] methods.
(1) The LWLR [42] model is as follows: where θ denotes the coefficient vector of the linear equation, τ is the bandwidth parameter.
(2) The SVM [43] model is as follows where w is the adjustable weight, 2 w is the Euclidean norm of the vector, εi is the slack variable, and c is the penalty parameter. KK() is the kernel function, which has an important parameter g.
(3) ELM [44]: ELM has the same framework as the BP neural network. The input weight of ELM is subject to random assignment by a certain distribution function, and the output weight is directly calculated via the least squares method. The hidden layer has q neurons. Compared with the BP neural network, the training and recognition processes of ELM are rapid.     The parameters for the four different methods are given in Table 4. The number of train sets is 950, the number of test sets is 50. From Figures 7-10, we see that the MSE of the BP neural network is 0.0018862, which is lower than that of the LWLR, SVM, and ELM methods. Compared to the LWLR, SVM, and ELM methods, our method can consistently achieve higher OP performance prediction results.  Table 5 shows the running time and MSE comparison for the four methods. In Table 5, we see that, compared to ELM, BP has a longer running time, but its performance is better than ELM. In addition, compared to SVM and LWLR, BP has a shorter running time and a smaller MSE. This is because the LWLR is not suitable for complex nonlinear data, the SVM has difficulty solving multiclass prediction problems, and the weights of ELM are generated randomly and maintained through the whole training process. The BP algorithm has strong nonlinear analytical abilities and robustness for multi-class prediction problems. A comprehensive comparison shows that BP is the best. In Figure 11, we obtain the training state and see how the gradient changes with increase in the number of iterations. The parameters for the four different methods are given in Table 4. The number of train sets is 950, the number of test sets is 50. From Figures 7-10, we see that the MSE of the BP neural network is 0.0018862, which is lower than that of the LWLR, SVM, and ELM methods. Compared to the LWLR, SVM, and ELM methods, our method can consistently achieve higher OP performance prediction results.  Table 5 shows the running time and MSE comparison for the four methods. In Table 5, we see that, compared to ELM, BP has a longer running time, but its performance is better than ELM. In addition, compared to SVM and LWLR, BP has a shorter running time and a smaller MSE. This is because the LWLR is not suitable for complex nonlinear data, the SVM has difficulty solving multi-class prediction problems, and the weights of ELM are generated randomly and maintained through the whole training process. The BP algorithm has strong nonlinear analytical abilities and robustness for multi-class prediction problems. A comprehensive comparison shows that BP is the best. In Figure 11, we obtain the training state and see how the gradient changes with increase in the number of iterations.  The regression results are shown in Figure 12. In each plot, the relationship between the targets and outputs is indicated by correlation coefficient R. In Figure 12, R is 0.98994, which indicates that our method has a good prediction capability.  The regression results are shown in Figure 12. In each plot, the relationship between the targets and outputs is indicated by correlation coefficient R. In Figure 12, R is 0.98994, which indicates that our method has a good prediction capability. The regression results are shown in Figure 12. In each plot, the relationship between the targets and outputs is indicated by correlation coefficient R. In Figure 12, R is 0.98994, which indicates that our method has a good prediction capability.

Conclusions
In this paper, we derived exact closed-form OP expressions for AF relaying. To verify our proposed analysis, the theoretical results obtained were compared with Monte-Carlo simulation results.
The effect of K and N t on OP performance was also investigated. To predict OP performance, a BP neural network-based OP performance prediction algorithm was proposed. When compared with the LWLR, SVM, and ELM methods, the BP neural network-based method was found to consistently have higher OP performance prediction results. The MSE of the BP neural network was 0.0018862, which is lower than the MSE of the LWLR, SVM, and ELM methods. The proposed algorithm can be used to predict the OP performance of vehicular communication systems employed in inter-vehicular communications, intelligent highway applications, and mobile ad-hoc applications.
In the future, the impact of correlated N-Nakagami channels on OP performance will be evaluated. The long-short term memory (LSTM) model will be considered to predict OP performance. Compared to the BP algorithm, the LSTM could offer more details on time-series and capture short-and long-term memory, adaptively reflecting environmental categories.