Joint Transmit Antenna Selection and Power Allocation for ISDF Relaying Mobile-to-Mobile Sensor Networks

The outage probability (OP) performance of multiple-relay incremental-selective decode-and-forward (ISDF) relaying mobile-to-mobile (M2M) sensor networks with transmit antenna selection (TAS) over N-Nakagami fading channels is investigated. Exact closed-form OP expressions for both optimal and suboptimal TAS schemes are derived. The power allocation problem is formulated to determine the optimal division of transmit power between the broadcast and relay phases. The OP performance under different conditions is evaluated via numerical simulation to verify the analysis. These results show that the optimal TAS scheme has better OP performance than the suboptimal scheme. Further, the power allocation parameter has a significant influence on the OP performance.


Introduction
To meet the increasing demands for high-data-rate services, mobile-to-mobile (M2M) communications has attracted significant interest from both industry and academia [1]. The M2M system architecture is described in [2]. In M2M communication systems, mobile users can directly communicate with each other without using a base station. This requires half the resources of traditional cellular communications, and thus improves the spectral efficiency and reduces the traffic load of the core network [3]. M2M communications can also be used to increase the data rate, reduce energy costs, reduce transmission delays, and extend the coverage area. Due to these advantages, M2M communications is an excellent choice for inter-vehicular communications, mobile sensor networks, and mobile heterogeneous networks [4]. M2M technologies have been proposed for home network applications [5]. In contrast to conventional fixed-to-mobile (F2M) cellular systems, both the transmitter and receiver in M2M systems can be in motion. Further, they are equipped with low elevation antennas. Thus, the widely employed Rayleigh, Rician, and Nakagami fading channels are not applicable to M2M communications systems [6]. Experimental results and theoretical analysis have shown that M2M channels can be described by cascaded fading channels [7]. The cascaded Rayleigh (also named as N-Rayleigh), fading channel was presented in [8]. The N-Rayleigh fading channel with N = 2, denoted the double-Rayleigh fading model, was considered in [9]. The N-Rayleigh fading channel It is assumed that the antennas at the MS and MD have the same distance to the relay nodes. Using the approach in [11], the relative gain of the MS to MD link is GSD = 1, the relative gain of the MS to MRl link is GSRl = (dSD/dSRl) v , and the relative gain of the MRl to MD link is GRDl = (dSD/dRDl) v , where v is the path loss coefficient, and dSD, dSRl, and dRDl are the distances of the MS to MD, MS to MRl, and MRl to MD links, respectively [28]. To indicate the location of MRl with respect to the MS and MD, the relative geometrical gain μl = GSRl/GRDl is defined. When MRl is closer to the MD, μl is less than 1, and when MRl is closer to the MS, μl is greater than 1. When MRl has the same distance to the MS and MD, μl is 1 (0 dB).
Let MSi denote the ith transmit antenna at MS and MDj denote the jth receive antenna at MD. Further, let h = hk, kSDij, SRil, RDlj represent the complex channel coefficients of the MSi to MDj, MSi to MRl, and MRl to MDj links, respectively. If the ith antenna at the MS is selected, during the first time slot the received signal rSDij at MDj is given by It is assumed that the antennas at the MS and MD have the same distance to the relay nodes. Using the approach in [11], the relative gain of the MS to MD link is G SD = 1, the relative gain of the MS to MR l link is G SRl = (d SD /d SRl ) v , and the relative gain of the MR l to MD link is where v is the path loss coefficient, and d SD , d SRl , and d RDl are the distances of the MS to MD, MS to MR l , and MR l to MD links, respectively [28]. To indicate the location of MR l with respect to the MS and MD, the relative geometrical gain µ l = G SRl /G RDl is defined. When MR l is closer to the MD, µ l is less than 1, and when MR l is closer to the MS, µ l is greater than 1. When MR l has the same distance to the MS and MD, µ l is 1 (0 dB).
Let MS i denote the ith transmit antenna at MS and MD j denote the jth receive antenna at MD. Further, let h = h k , kP{SD ij , SR il , RD lj } represent the complex channel coefficients of the MS i to MD j , MS i to MR l , and MR l to MD j links, respectively. If the ith antenna at the MS is selected, during the first time slot the received signal r SDij at MD j is given by KEh SDij x`n SDij (1) and the received signal r SRil at MR l by where x denotes the transmitted symbol, and n SRil and n SDij are additive white Gaussian noise (AWGN) with zero mean and variance N 0 /2. During the two time slots, E is the total energy used by the MS and MR, and K is the power allocation parameter. During the second time slot, by comparing the instantaneous SNR γ SDij to a threshold γ P , only the best MR decides whether to be active.
If γ SDij > γ P , then MS i and the best MR will receive a 'success' message. MS i then transmits the next message, and the best MR remains silent. The corresponding received SNR at MD j is If γ SDij < γ P , then MS i and the best MR will receive a 'failure' message. By comparing the instantaneous SNR γ SRi to a threshold γ T , the best MR decides whether to decode and forward the signal to the MD j , where γ SRi represents the SNR of the link between MS i and the best MR. The best MR is selected based on the following decision rule where γ SRil represents the SNR of the MS i to MR l link, and If γ Sri < γ T , then MS i will transmit the next message, and the best MR will not be used for cooperation. The corresponding received SNR at MD j is If γ Sri > γ T , then the best MR decodes and forwards the signal to MD j . The corresponding received signal at MD j is where n RDj is AWGN with zero mean and variance N 0 /2. If MD j uses selection combining (SC), the received SNR is given by where γ RDj represents the SNR of the link between the best MR and MD j . Using SC at the MD, the received SNR is where The optimal TAS scheme selects the transmit antenna w that maximizes the received SNR at the MD, namely The suboptimal TAS scheme selects the transmit antenna g that maximizes the instantaneous SNR of the direct link MS i to MD j , namely

System Model
The links in the system are subject to independent and identically distributed N-Nakagami fading, so that h follows the N-Nakagami distribution given by [10] h " where N is the number of cascaded components, and a t is a Nakagami distributed random variable with PDF f paq " 2m m Ω m Γpmq a 2m´1 exp´´m Ω a 2¯( 15) Γ(¨) is the Gamma function, m is the fading coefficient, and Ω is a scaling factor. Using the approach in [10], the PDF of h is given by where G[¨] is Meijer's G-function. Let y = |h k | 2 represent the square of the amplitude of h k . The corresponding CDF and PDF of y are [10] Fpyq " 1

The OP of the Optimal TAS Scheme
The OP of the optimal TAS scheme can be expressed as Sensors 2016, 16, 249 6 of 13

γ th > γ P
If γ th > γ P , the OP of the optimal TAS scheme can be expressed as where γ th is the threshold for correct detection at the MD. G 1 is given by G 2 can be written as and G 3 can be expressed as 3.2. γ th < γ P If γ th < γ P , the OP of the optimal TAS scheme can be expressed as where G 11 can be written as and G 22 is given by

The OP of the Suboptimal TAS Scheme
If γ th > γ P , the OP of the suboptimal TAS scheme can be expressed as GG 1 is given by GG 2 can be written as and GG 3 can be expressed as If γ th < γ P , the OP of the suboptimal TAS scheme can be expressed as where GG 11 can be written as and GG 22 can be expressed as Figure 2 presents the effect of the power allocation parameter K on the OP performance. The parameters are N = 2, m = 2, µ = 0 dB, N t = 2, L = 2, N r = 2, γ th = 5 dB, γ T = 2 dB, and γ P = 2 dB. These results show that the OP performance improves as the SNR is increased. For example, when K = 0.7, the OP is 2.3ˆ10´2 with SNR = 10 dB, 1.3ˆ10´4 with SNR = 15 dB, and 2.4ˆ10´7 with SNR = 20 dB. The optimum value of K is 0.86 with SNR = 10 dB, 0.92 with SNR = 15 dB, and 0.96 with SNR = 20 dB. This indicates that equal power allocation (EPA) is not the best scheme.

Optimal Power Allocation
Unfortunately, it is very difficult to derive a closed-form expression for K. Because the OP expressions are very complex, numerical methods are used to solve the optimization problem. The optimum power allocation (OPA) values were obtained for given values of SNR and system parameters. Table 1 presents the optimum values of K for three values of relative geometrical gain µ = 5 dB, 0 dB,´5 dB. The other parameters are N = 2, m = 2, N t = 2, L = 2, γ th = 5 dB, γ T = 2 dB, and γ P = 6 dB. For example, with a low SNR and µ = 5 dB, nearly all of the power should be used in the broadcast phase. As the SNR increases, the optimum value of K is reduced, so that at SNR = 20 dB only half of the power should be used in the broadcast phase.   Figure 3 presents the effect of the relative geometrical gain μ on the OP performance using the values of K given in Table 1. These results show that the OP performance improves as μ decreases. For example, when SNR = 10 dB, the OP is 3.6  10 −3 for μ = 5 dB, 4.5  10 −5 for μ = 0 dB, and 1.9  10 −7 for μ = −5 dB. This indicates that the best location for the relay is near the destination. For fixed μ, an increase in the SNR reduces the OP, as expected.

Numerical Results
In this section, simulation results are presented to confirm the analysis given previously. The Monte Carlo simulations were done using MATLAB, and the analytical results were verified using MAPLE. The total energy is E = 1, the fading coefficient is m = 1, 2, 3, the number of cascaded components is N = 2, 3, 4, the number of mobile relays is L = 2, the number of receive antennas is Nr = 2, the relative geometrical gain is μ = 0 dB, and the number of transmit antennas is Nt = 1, 2, 3.    Figure 3 presents the effect of the relative geometrical gain µ on the OP performance using the values of K given in Table 1. These results show that the OP performance improves as µ decreases. For example, when SNR = 10 dB, the OP is 3.6ˆ10´3 for µ = 5 dB, 4.5ˆ10´5 for µ = 0 dB, and 1.9ˆ10´7 for µ =´5 dB. This indicates that the best location for the relay is near the destination. For fixed µ, an increase in the SNR reduces the OP, as expected.   Figure 3 presents the effect of the relative geometrical gain μ on the OP performance using the values of K given in Table 1. These results show that the OP performance improves as μ decreases. For example, when SNR = 10 dB, the OP is 3.6  10 −3 for μ = 5 dB, 4.5  10 −5 for μ = 0 dB, and 1.9  10 −7 for μ = −5 dB. This indicates that the best location for the relay is near the destination. For fixed μ, an increase in the SNR reduces the OP, as expected.

Numerical Results
In this section, simulation results are presented to confirm the analysis given previously. The Monte Carlo simulations were done using MATLAB, and the analytical results were verified using MAPLE. The total energy is E = 1, the fading coefficient is m = 1, 2, 3, the number of cascaded components is N = 2, 3, 4, the number of mobile relays is L = 2, the number of receive antennas is Nr = 2, the relative geometrical gain is μ = 0 dB, and the number of transmit antennas is Nt = 1, 2, 3.

Numerical Results
In this section, simulation results are presented to confirm the analysis given previously. The Monte Carlo simulations were done using MATLAB, and the analytical results were verified using MAPLE. The total energy is E = 1, the fading coefficient is m = 1, 2, 3, the number of cascaded components is N = 2, 3, 4, the number of mobile relays is L = 2, the number of receive antennas is N r = 2, the relative geometrical gain is µ = 0 dB, and the number of transmit antennas is N t = 1, 2, 3. Figures 4 and 5 present the OP performance of the optimal TAS scheme with γ th = 5 dB, γ T = 2 dB and γ P = 6 dB, and γ th = 5 dB, γ T = 2 dB and γ P = 3 dB, respectively. The other parameters are N = 2, m = 2, K = 0.5, N t = 1, 2, 3, L = 2, N r = 2, and µ = 0 dB. This shows that the analytical results match the simulation results. The OP improves as the number of transmit antennas is increased. For example, when γ th = 5 dB, γ T = 2 dB, γ P = 3 dB, and SNR = 10 dB, the OP is 1.1ˆ10´1 when N t = 1, 1.3ˆ10´2 when N t = 3, and 1.4ˆ10´3 when N t = 3. For fixed N t , an increase in the SNR decreases the OP. Figures 4 and 5 present the OP performance of the optimal TAS scheme with γth = 5 dB, γT = 2 dB and γP = 6 dB, and γth = 5 dB, γT = 2 dB and γP = 3 dB, respectively. The other parameters are N = 2, m = 2, K = 0.5, Nt = 1, 2, 3, L = 2, Nr = 2, and μ = 0 dB. This shows that the analytical results match the simulation results. The OP improves as the number of transmit antennas is increased. For example, when γth = 5 dB, γT = 2 dB, γP = 3 dB, and SNR = 10 dB, the OP is 1.1  10 −1 when Nt = 1, 1.3  10 −2 when Nt = 3, and 1.4  10 −3 when Nt = 3. For fixed Nt, an increase in the SNR decreases the OP.   6 and 7 present the OP performance of the suboptimal TAS scheme with γth = 5 dB, γT = 2 dB and γP = 6 dB, and γth = 5 dB, γT = 2 dB and γP = 3 dB, respectively. The other parameters are N = 2, m = 2, K = 0.5, Nt = 1, 2, 3, L = 2, Nr = 2, and μ = 0 dB. This also shows that the analytical results match the simulation results. As expected, the OP improves as the number of transmit antennas is increased. For example, when γth = 5 dB, γT = 2 dB, γP = 6 dB, SNR = 12 dB, and Nt = 1, the OP is 4.1  10 −2 when Nt = 1, 4.7  10 −3 when Nt = 2, and 5.3  10 −4 when Nt = 3, For fixed Nt, an increase in the SNR decreases the OP, as expected. Figure 8 compares the OP performance of the optimal and suboptimal TAS schemes for different numbers of antennas Nt. The parameters are N = 2, m = 2, K = 0.5, μ = 0 dB, Nt = 2, 3, L = 2, Nr = 2, γth = 5 dB, γT = 2 dB, and γP = 3 dB. In all cases, for a given value of Nt the optimal TAS scheme has better OP performance. As predicted by the analysis, the performance gap between the two TAS schemes decreases as Nt is increased. When the SNR is low, the OP performance gap  Figures 4 and 5 present the OP performance of the optimal TAS scheme with γth = 5 dB, γT = 2 dB and γP = 6 dB, and γth = 5 dB, γT = 2 dB and γP = 3 dB, respectively. The other parameters are N = 2, m = 2, K = 0.5, Nt = 1, 2, 3, L = 2, Nr = 2, and μ = 0 dB. This shows that the analytical results match the simulation results. The OP improves as the number of transmit antennas is increased. For example, when γth = 5 dB, γT = 2 dB, γP = 3 dB, and SNR = 10 dB, the OP is 1.1  10 −1 when Nt = 1, 1.3  10 −2 when Nt = 3, and 1.4  10 −3 when Nt = 3. For fixed Nt, an increase in the SNR decreases the OP.   6 and 7 present the OP performance of the suboptimal TAS scheme with γth = 5 dB, γT = 2 dB and γP = 6 dB, and γth = 5 dB, γT = 2 dB and γP = 3 dB, respectively. The other parameters are N = 2, m = 2, K = 0.5, Nt = 1, 2, 3, L = 2, Nr = 2, and μ = 0 dB. This also shows that the analytical results match the simulation results. As expected, the OP improves as the number of transmit antennas is increased. For example, when γth = 5 dB, γT = 2 dB, γP = 6 dB, SNR = 12 dB, and Nt = 1, the OP is 4.1  10 −2 when Nt = 1, 4.7  10 −3 when Nt = 2, and 5.3  10 −4 when Nt = 3, For fixed Nt, an increase in the SNR decreases the OP, as expected. Figure 8 compares the OP performance of the optimal and suboptimal TAS schemes for different numbers of antennas Nt. The parameters are N = 2, m = 2, K = 0.5, μ = 0 dB, Nt = 2, 3, L = 2, Nr = 2, γth = 5 dB, γT = 2 dB, and γP = 3 dB. In all cases, for a given value of Nt the optimal TAS scheme has better OP performance. As predicted by the analysis, the performance gap between the two TAS schemes decreases as Nt is increased. When the SNR is low, the OP performance gap γ T = 2 dB and γ P = 6 dB, and γ th = 5 dB, γ T = 2 dB and γ P = 3 dB, respectively. The other parameters are N = 2, m = 2, K = 0.5, N t = 1, 2, 3, L = 2, N r = 2, and µ = 0 dB. This also shows that the analytical results match the simulation results. As expected, the OP improves as the number of transmit antennas is increased. For example, when γ th = 5 dB, γ T = 2 dB, γ P = 6 dB, SNR = 12 dB, and N t = 1, the OP is 4.1ˆ10´2 when N t = 1, 4.7ˆ10´3 when N t = 2, and 5.3ˆ10´4 when N t = 3, For fixed N t , an increase in the SNR decreases the OP, as expected.
between the optimal TAS scheme with Nt = 2 and the suboptimal TAS scheme with Nt = 3 is negligible. As the SNR increases, the OP performance gap also increases.

Conclusions
In this paper, exact closed-form OP expressions were derived for ISDF relaying M2M networks with TAS over N-Nakagami fading channels. Performance results were presented which show that between the optimal TAS scheme with Nt = 2 and the suboptimal TAS scheme with Nt = 3 is negligible. As the SNR increases, the OP performance gap also increases.

Conclusions
In this paper, exact closed-form OP expressions were derived for ISDF relaying M2M networks with TAS over N-Nakagami fading channels. Performance results were presented which show that  Figure 8 compares the OP performance of the optimal and suboptimal TAS schemes for different numbers of antennas N t . The parameters are N = 2, m = 2, K = 0.5, µ = 0 dB, N t = 2, 3, L = 2, N r = 2, γ th = 5 dB, γ T = 2 dB, and γ P = 3 dB. In all cases, for a given value of N t the optimal TAS scheme has better OP performance. As predicted by the analysis, the performance gap between the two TAS schemes decreases as N t is increased. When the SNR is low, the OP performance gap between the optimal TAS scheme with N t = 2 and the suboptimal TAS scheme with N t = 3 is negligible. As the SNR increases, the OP performance gap also increases.
Sensors 2016, 16, 249 11 of 13 between the optimal TAS scheme with Nt = 2 and the suboptimal TAS scheme with Nt = 3 is negligible. As the SNR increases, the OP performance gap also increases.

Conclusions
In this paper, exact closed-form OP expressions were derived for ISDF relaying M2M networks with TAS over N-Nakagami fading channels. Performance results were presented which show that

Conclusions
In this paper, exact closed-form OP expressions were derived for ISDF relaying M2M networks with TAS over N-Nakagami fading channels. Performance results were presented which show that the optimal TAS scheme has better OP performance than the suboptimal scheme. It was also shown that the power allocation parameter K can have a significant effect on the OP performance. The given expressions can be used to evaluate the OP performance of inter-vehicular networks, mobile wireless sensor networks, and mobile heterogeneous networks.