1. Introduction

Mobile-to-mobile (M2M) communication technology has attracted significant research interest in recent years. It is widely employed in many popular wireless communication systems such as mobile ad-hoc networks and vehicle-to-vehicle networks [1]. When both the transmitter and receiver are in motion, the double-Rayleigh fading model has been shown to be applicable [2]. Extending this model by characterizing the fading between each pair of transmit and receive antennas as Nakagami distributed, the double-Nakagami fading model has also been considered [3]. The moment generating, probability density, cumulative distribution, and moment functions of the N-Nakagami distribution have been developed in closed form using Meijer’s G-function [4].

Cooperative diversity has been proposed as a promising solution for the high data-rate coverage required in M2M communication networks. Based on amplify-and-forward (AF) relaying, the pairwise error probability (PEP) for cooperative inter-vehicular communication (IVC) systems over double-Nakagami fading channels had been derived in [5]. The exact symbol error rate (SER) and asymptotic SER expressions for M2M networks with decode-and-forward (DF) relaying had been obtained using the widely employed moment generating function (MGF) approach over double-Nakagami fading channels [6].

Incremental relaying is a promising technique to conserve channel resources by restricting the relaying process based on certain conditions [7]. Typically, the instantaneous signal-to-noise ratio (SNR) of the source-destination link is used as an indication of the channel reliability. If the quality of the direct link between the source and destination is sufficiently high, it is not necessary to forward a signal from a relay. Otherwise, a relay should participate to improve the quality of the received signals. In [8], closed form expressions for the error probability, outage probability (OP) and average achievable rate for incremental DF and AF relaying with a single relay over Rayleigh fading channels were derived. In [9], a cooperative transmission technique with distributed opportunistic incremental DF (DOIDF) relaying was presented. The OP of DOIDF was derived, and results were given which showed that DOIDF can achieve the same space diversity order as multiple-input single-output (MISO) and single-input multiple-output (SIMO) systems. An incremental best relay scheme for multiple relays was proposed in [10], and adaptive modulation was applied to the proposed scheme to satisfy the spectral efficiency and the bit error rate (BER) requirements. Closed form expressions for the error rate, OP and average channel capacity were obtained in [11] for incremental best relay cooperative diversity networks with DF and AF relaying over independent and non-identical Rayleigh fading channels. Based on incremental AF relaying scheme,the exact closed form expressions for the lower bound on OP of multiple-mobile-relay-based M2M cooperative networks with relay selection over N-Nakagami fading channels were derived in [12].

Optimum power allocation is a key technique to realize the full potentials of relay-assisted transmission. A novel approach for OP analysis of the multiple-node AF relay network was provided in [13],and optimal power allocation was studied based on the derived OP bound. In [14], game theory was applied to network problems, in most cases to solve the resource allocation problems.To address the misbehavior problem, continuous-time protocols were proposed in [15],and matrix exponential was used to model the queueing process. In [16], a new continuous-time model for carrier sense multiple access (CSMA) wireless networks was proposed,which taken into account non-saturated queues.A new method for the study of the competition and cooperation relationships among nodes in a network using a CSMA protocol implemented with an exponential backoff process was provided in [17].

However, to the best of our knowledge, the average bit error probability (BEP) performance of mobile-relay-based M2M cooperative networks with incremental DF (IDF) relaying over N-Nakagami fading channels has not yet been investigated. We present the analysis for the N-Nakagami case which subsumes the double-Nakagami results in [5,6] as special cases. Exact average BEP expressions are derived here for IDF relaying over N-Nakagami fading channels. The average BEP is optimized based on the power-allocation parameter. We resort to numerical methods to solve this optimization problem. In order to show the accuracy of the analytical results, the average BEP performance under different conditions is evaluated through numerical simulations. Results are presented which show that the fading coefficient, the number of cascaded components, the relative geometrical gain, and the power allocation parameter have a significant influence on the average BEP performance.

The rest of the paper is organized as follows. The IDF relaying M2M cooperative network model is presented in Section 2. Section 3 provides exact average BEP expressions for IDF relaying. In Section 4, the average BEP is optimized based on the power allocation parameter.In Section 5, Monte Carlo simulation results are presented to verify the analysis. Finally, some concluding remarks are given in Section 6.

2. System Model

We consider a mobile-relay-based cooperation model, namely a single mobile source (MS) node, a single mobile relay (MR) node, and a single mobile destination (MD) node. The nodes operate in half-duplex mode, and are equipped with a single pair of transmit and receive antennas.

According to [5], let dSD, dSR, and dRD represent the distances of the MS to MD, MS to MR, and MR to MD links, respectively. Assuming the path loss between the MS and MD is unity, the relative gain of the MS to MR and MR to MD links are defined as GSR=(dSD/dSR)v and GRD = (dSD/dRD)v, respectively, where v is the path loss coefficient [18]. We further define the relative geometrical gain as μ= GSR/GRD, which indicates the location of the relay with respect to the source and destination [5]. When the relay is close to the destination node, μ is negative, and when the relay is close to the source node, μ is positive. When the relay has the same distance to the source and destination nodes, μ is 0 dB.

Let h=hk, k∈{SD, SR, RD}, represent the complex channel coefficients of the MS to MD, MS to MR, and MR to MD links, respectively, which follow an N-Nakagami distribution. h is assumed to be the product of statistically independent, but not necessarily identically distributed, N random variables[4]

where N is the number of cascaded components, and ai is a Nakagami distributed random variable with probability density function (PDF)

where Γ(·) is the Gamma function, m is the fading coefficient, and Ω is a scaling factor.

The PDF of h is given by [4]

where G[·] is Meijer’s G-function .

Let y=|hk|2, k∈{SD, SR, RD}, so that ySD=| hSD|2, ySR=|hSR|2, and yRD=|hRD|2. The corresponding cumulative density functions (CDF) of y can be derived as [4]

By taking the first derivative of (4) with respect to y, the corresponding PDF can be obtained as[4]

The IDF relaying process can be described as follows. During the first time slot, the MS broadcasts a signal to the MD and MR. The received signals rSD and rSR at the MD and MR during this time slot can be written as [6]

where x denotes the transmitted signal, nSD and nSR are zero-mean complex Gaussian random variables with variances N0/2 per dimension. Here, E is the total energy which is used by both the source and relay nodes during two time slots. K is the power allocation parameter that controls the fraction of power reserved for the broadcast phase. If K=0.5, equal power allocation (EPA) is used.

During the second time slot, the MR decides whether to decode and forward the signal to the MD by comparing the instantaneous SNR γSD to a threshold Rt. Let γSD denote the instantaneous SNR of the MS to MD link. If γSD>Rt, the MD will broadcast a ‘success’ message to the MS and MR. Then the MS will transmit the next message, and the MR remains silent. The output SNR at the MD can then be calculated as

where

If γSD

where nRD is a conditionally zero-mean complex Gaussian random variable with variance N0/2 per dimension.

If selection combining (SC) is used at the MD, the output SNR can be calculated as

where

3. Average BEP of IDF Relaying M2M cooperative networks

In this section, exact average BEP expressions for IDF relaying M2M cooperative networks are derived. The average unconditional error probability of the signal at the MD using incremental DF relaying can be written as [8]

where Pdiv(e) is the average probability that an error occurs in the combined diversity transmission from the MS and MR to the MD. Pdirect(e) is the probability of error at the destination given that the destination decides that the relay does not forward the source signal.

The CDF of the SNR γSD can be expressed as [4]

where

So that

Next, Pdirect(e) is given by

where

a and b are constants that depend on the modulation employed, e.g. for BPSK a = 0.5 and b = 1, and for QPSK a = 0.5 and b = 0.5. We then have

Combining (19) and (20), Pdirect(e) can be expressed as

G1 can be expressed as

To evaluate the integral in (22), the following integral function can be employed [19]

where 0≤m≤q, 0≤n≤p, 0≤s≤v, and 0≤t≤u, are integers, ai, bi, ci and di are arbitrary real values, and

Equation (22) is then given by

G2 can be expressed as

Numerical techniques can easily be used to evaluate the integral in G2.

Pdiv(e) is given by [20]

where PSR(e) is the probability of error at the relay, Px(e) is the probability that an error occurs at the destination given that the relay decoding was unsuccessful, and Pcom(e) is the probability that an error occurs at the destination given that the relay decoding was successful.

Then PSR(e) can be expressed as

where

When the relay makes a decoding error and forwards an incorrect signal to the destination, Px(e) is given by [20]

where

To evaluate G3, consider

Then from (23), we obtain

To evaluate G4, consider

where

So that

G5 can easily be evaluated via numerical integration.

Pcom(e) can be expressed as

where

Combining (39) and (40), we obtain

G8 is given by

G6, G7 and G9 can also be evaluated via numerical integration. Finally, substituting (16), (17), (18) and (39) in (13), we obtain the exact average BEP expressions for IDF relaying M2M cooperative networks.

4. Average BEP-Optimized Power Allocation

To further improve the performance, we aim to optimally allocate the power between broadcasting and relaying phases. For optimization of the power allocation, we consider the average BEP as our objective function. The resulting average BEP needs to be minimized with respect to the power-allocation parameter K (0 ≤ K ≤ 1).

Fig. 1 presents the effect of the power allocation parameter K on the average BEP performance of IDF relaying M2M cooperative networks over N-Nakagami fading channels. The number of cascaded components is N=2, the fading coefficient is m=2, the relative geometrical gain is μ=0 dB, and the threshold is Rt=4 dB. These results show that the average BEP performance is improved with an increase in the SNR. For example, with K=0.4, the average BEP is 1×10-2 for SNR=10 dB, 3×10-4 for SNR=20 dB, and 1.5×10-5 for SNR=30 dB. When SNR=10 dB, the optimum value of K is 0.68, when SNR=20 dB, the optimum value of K is 0.77, and when SNR=30 dB, the optimum value of K is 0.88. This indicates that equal power allocation (EPA) is not always the best approach.

Fig. 1.The effect of the power-allocation parameter K on the average BEP performance

From Fig. 1, it can readily be checked that these expressions are convex functions with respect to K.The convexity of the average BEP functions under consideration guarantees that the local minimum found through optimization will indeed be a global minimum. But,a closed-form solution is not available. We resort to numerical methods to solve this optimization problem. The optimum power allocation (OPA) values can be obtained a priori for given values of operating SNR and propagation parameters. The OPA values can be stored for use as a lookup table in practical implementation.

In Table 1, we present optimum values of K for N-Nakagami fading channels. We assume that the number of cascaded components is N=2, the fading coefficient is m=1, the relative geometrical gain is μ=-10 dB, and the threshold is Rt=0 dB.

Table 1.OPA parameters K

Fig. 2 presents the effect of the EPA and OPA on the average BEP performance with BPSK modulation. For OPA, the values of K are used in Table 1 for BPSK modulations. For EPA, K=0.5. These results show that the average BEP performance of OPA is better than that of EPA. For example, with SNR=10 dB, the average BEP is 2×10-2 for OPA, while 2.6×10-2 for EPA.

Fig. 2.The effect of EPA and OPA on the average BEP performance

5. Numerical Results

In this section, we present Monte-Carlo simulations and numerical methods to confirm the derived analytical results. The simulation results are obtained for BPSK modulations. Additionally, random number simulation was done to confirm the validity of the analytical approach. All the computations were done in MATLAB and some of the integrals were verified through MAPLE. The links between MS to MD, MS to MR and MR to MD are modeled as N-Nakagami distribution. The total energy is E =1. The thresholds considered are Rt=-4 dB, 0 dB, and 4 dB. The fading coefficient is m=1,2,3, the number of cascaded components is N=2,3,4, and the relative geometrical gain is μ=20dB, 5dB,-20dB, respectively. In Fig. 4, 5, 6, 7, we only want to present the effect of Rt, m, N, μ on the average BEP, respectively, so the value of K should remain unchanged. For different values of Rt, m, N, μ, the optimized values of K are different. We resort to numerical methods to obtain the optimized values of K.

Fig. 3 presents the average BEP performance of IDF relaying M2M cooperative networks over N-Nakagami fading channels with BPSK modulation. The relative geometrical gain is μ=0 dB, the power allocation parameter is K=0.5, and the threshold is Rt =4 dB. The following cases are considered based on the number of cascaded components N and the fading coefficient m:

Fig. 3.The average BEP performance over N-Nakagami fading channels.

Case 1: mSD = 1, mSR = 1, mRD=1 and NSD =3, NSR= NRD=3.

Case 2: mSD=1, mSR =1, mRD =1 and NSD =2, NSR=NRD=2.

Case 3: mSD=2, mSR =2, mRD =2 and NSD =2, NSR=NRD=2.

Fig. 3 shows that the numerical results coincide with the theoretical results, which verifies the accuracy of the analysis. As the SNR increases, the average BEP performance improves, as expected. For example, in Case 2, when the SNR=10 dB the average BEP is 4×10-2, and when SNR=15 dB the average BEP is 9×10-3.

Fig. 4 presents the effect of the threshold Rt on the average BEP performance of IDF relaying M2M cooperative networks. The relative geometrical gain is μ=0 dB, the power allocation parameter is K=0.5, and the number of cascaded components is N=2. The thresholds considered are Rt=-4 dB, 0 dB, and 4 dB. These results show that the average BEP performance is improved as Rt increases. For example, with SNR=12 dB, the average BEP is 1.8×10-2 when Rt =-4 dB, 9×10-3 when Rt =0 dB, and 3.5×10-3 when Rt =4 dB. As Rt increases, the probability that the relay forwards the source signal increases, and the resulting performance is better than that with just direct transmission, so the average BEP performance improves. For fixed Rt, an increase in the SNR decreases the average BEP, as expected.

Fig. 4.The effect of the threshold Rt on the average BEP performance.

Fig. 5 presents the effect of the number of cascaded components N on the average BEP performance of IDF relaying M2M cooperative networks over N-Nakagami fading channels. The number of cascaded components is N=2, 3, and 4, which denotes 2-Nakagami, 3-Nakagami, and 4-Nakagami fading channels, respectively. The fading coefficient is m=2, the relative geometrical gain is μ=0 dB, the threshold is Rt=4 dB, and the power allocation parameter is K=0.7. These results show that the average BEP performance is degraded as N is increased. For example, when SNR=10 dB, the ABEP is 6.5×10-3 for N=2, 1.5×10-2 for N=3, and 2.8×10-2 for N=4. This is because the fading severity of the cascaded channels increases as N is increased. For fixed N, an increase in the SNR decreases the average BEP, as expected.

Fig. 5.The effect of the number of cascaded components N on the average BEP performance.

Fig. 6 presents the effect of the relative geometrical gain μ on the average BEP performance of IDF relaying M2M cooperative networks over N-Nakagami fading channels. The number of cascaded components is N=2, and the fading coefficient is m=2. The values of the relative geometrical gain considered are μ=20 dB, 5 dB, and -20 dB. The threshold is Rt=4 dB, and the power allocation parameter is K=0.8. These results show that the average BEP performance is improved as μ is reduced. For example, when SNR=10 dB, the average BEP is 2.6×10-2 for μ=20 dB, 1.3×10-2 for μ=5 dB, and 3×10-3 for μ=-20 dB. This indicates that the best location for the relay is near the destination. For fixed μ, an increase in the SNR results in a decrease in the average BEP, as expected.

Fig. 6.The effect of the relative geometrical gain μ on the average BEP performance.

Fig. 7 presents the effect of the fading coefficient m on the average BEP performance of IDF relaying M2M cooperative networks over N-Nakagami fading channels. The number of cascaded components is N=2, and the fading coefficients considered are m=1,2, and 3. The relative geometrical gain is μ=0 dB, the threshold is Rt=4 dB, and the power allocation parameter is K=0.7. These results show that the average BEP performance is improved as the fading coefficient m is increased. For example, when SNR=15dB, the ABEP is 7.2×10-3 for m=1, 8.7×10-4 for m=2, and 4×10-4 for m=3. This is because the fading severity of an N-Nakagami channel is less for a larger m. For fixed m, an increase in the SNR again reduces the average BEP.

Fig. 7.The effect of the fading coefficient m on the average BEP performance.

6. Conclusion

Exact average bit error probability (BEP) expressions for IDF relaying M2M cooperative networks over N-Nakagami fading channels were derived. Simulation results were presented which confirm the analysis. The results given showed that the fading coefficient m, the number of cascaded components N, the relative geometrical gain μ, and the power allocation parameter K have a significant effect on the average BEP performance. The expressions derived here are simple to compute and thus complete and accurate performance results can easily be obtained with negligible computational effort. In the future, we will consider the impact of correlated channels on the average BEP performance of IDF relaying M2M cooperative networks.