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Performance analysis and power allocation for multihop multibranch amplifyandforward cooperative networks over generalized fading channels
EURASIP Journal on Wireless Communications and Networking volume 2013, Article number: 160 (2013)
Abstract
In this article, efficient power allocation strategies for multihop multibranch amplifyandforward networks are developed in generalized fading environments. In particular, we consider the following power optimization schemes: (i) minimizing of the all transmission powers subject to an outage constraint; and (ii) minimizing the outage probability subject to constraint on total transmit powers. In this study, we first derive asymptotically tight approximations for the statistics of the received signaltonoise ratio (SNR) in the system under study with maximal ratio combining and selection combining receiver. With the statistical characterization of the received SNR, we then carry out a thorough performance analysis of the system. Finally, the asymptotic expression of the outage probability is used to formulate the original optimization problems using geometric programming (GP). The GP can readily be transformed into nonlinear convex optimization problem and therefore solved efficiently and globally using the interiorpoint methods. Numerical results are provided to substantiate the analytical results and to demonstrate the considerable performance improvement achieved by the power allocation.
1 Introduction
Recently, it has been shown that the throughput, coverage, and battery life of resourceconstrained wireless ad hoc networks can be increased through the use of multihop relay transmission [1, 2]. The main idea is that communication is achieved by relaying the information from the source to the destination through the use of many intermediate terminals in between. In a multihop multibranch transmission system, a source communicates with the destination through several multihop branches, each of which consists of multiple intermediate relay nodes. As a result, the destination node can receive multiple independent copies of the same signal and can achieve diversity without the need to install multiple antennas at the source node or the destination node. On the other hand, emerging wireless applications, e.g., wireless sensor and ad hoc networks, have an increasing demand for small devices having limited battery lifetimes. For a more efficient use of the power resources, the problem of optimally distributing the power among the source node and the relay nodes has drawn great attention from wireless service providers and academia.
Over the past decade, a considerable effort in the literature has been devoted to the performance analysis of cooperative relay systems. In particular, the performance of dualhop amplify and forward (AF) networks has widely been analyzed in [3–5] and [6–10] for single and multiple relay scenarios, respectively. Multihop multibranch AF network has also been investigated in a few recent works [11–14]. Specifically, Ribeiro et al. [11] studied the symbol error probability of these networks for a class of fading models, whose probability density functions (pdf) have nonzero values at the origin, including Rayleigh and Rician fading channels, when the average signaltonoise ratio (SNR) is sufficiently large. Renzo et al. [13] exploit the Moment Generating Function (MGF)based approach for performance analysis of multihop multibranch networks over fading channels. Amarasuriya et al. [14] proposed a new class of upper bounds on the endtoend SNR of a multihop system and then derived an asymptotic expression for the symbol error rate (SER) of the multihop multibranch setup in independent and identically distributed (i.i.d.) Nakagamim fading conditions. Common to all aforementioned studies is the simplified assumption that all hops and also all diversity paths have the same fading conditions. However, due to the wide spatial distribution of the relay nodes in a practical wireless system, the hops may undergo different kinds of fading conditions. In the ensuing text, we refer to this setup as generalized fading environments. Note, moreover, that resource allocation is assumed to be fixed in these works. In other words, all theses works provide complicated bounds on the performance metrics such as outage probability [11], SER [14], or numerical methods [13], which render the practical solutions for the resource allocation problem impossible.
Management of available radio resources plays a key role in improving the performance of wireless networks. Many research efforts have been devoted to investigate the performance improvement of relay networks by optimally allocate the radio resources [15–17]. It is worth mentioning that dualhop relaying scheme is typically considered in the aforementioned studies and poweroptimized multihop relaying is only studied in [17]. To the best of the authors’ knowledge, the ultimate benefit of power control in multihop multibranch networks has not been studied in the existing literature. One main goal of this article is to fill this important gap.
In this article, we develop efficient power allocation frameworks for multihop multibranch networks in generalized fading environments. In particular, our power allocation schemes aimed at: (i) minimizing the transmitter powers subject to an outage constraint; and (ii) minimizing the outage probability subject to constraint on total transmit powers. Thanks to the asymptotically tight approximation of the outage performance, that we develop for both maximal ratio combining (MRC) and selection combining (SC) receivers, we can formulate the original optimization problems using geometric programming (GP). The GP can readily be transformed into nonlinear convex optimization problem and therefore solved efficiently and globally using the interiorpoint methods [18, 19].
The remainder of this article is organized as follows. Section 2 describes the system model. Section 3 studies the asymptotic performance evaluation of the multihop system. Section 4 presents the asymptotic analysis of the multihop multibranch system. The problem formulation for power optimization is given in Section 5. Simulation and numerical results are presented in Section 6, followed by the conclusions in Section 7.
2 System model
Consider a generalized cooperative system with M diversity branches and ${\left\{{N}_{i}\right\}}_{i=1}^{M}$ hops for each branch as shown in Figure 1. We denote R_{i,j} (1≤i≤M and 1≤j≤N_{ i }−1) as the j th relay in the i th branch and h_{ i n } (1≤n≤N_{ i }) as the channel coefficient for the n th hop in the i th branch. We assume that the distance between relay clusters (hop) is much larger than the distance between the nodes in any one cluster. Therefore, the channel gains of the hops are independently but not necessarily identically distributed (i.n.i.d).
When AF relaying is employed, the relay node R_{i,j} amplifies the signal received from the preceding terminal by a factor A_{ i j } given by
where ${\sigma}_{n}^{2}$ is the power of additive white Gaussian noise (AWGN) ^{a} and P_{i 0} is the source transmission power in branch i. Let we denote the instantaneous SNR of the j th hop of the i th branch by ${\gamma}_{\mathit{\text{ij}}}={h}_{\mathit{\text{ij}}}{}^{2}{P}_{i(j1)}/{\sigma}_{n}^{2}$. The received SNR of the i th branch is given by [20]
which can be well approximated by [20]
especially for sufficiently large values of SNR.
While a number of different distributions are possible for fading amplitudes, we choose here the generalized Gamma distribution, whose pdf is [21]
where Γ(·) is the gamma function defined in [22], Eq.(8.310.1), m is the fading parameter, υ is the shape parameter and Ω:=β^{υ}m is the powerscaling parameter. In what follows, we will use the shorthand notation $X\sim \mathcal{G}(a,b)$ to denote that X follows generalized Gamma distribution with parameters a and b. With a proper choice of three parameters m, β and υ the generalized Gamma distribution can represent a wide variety of distributions including the Rayleigh (m=υ=1), Nakagamim (υ=1), Weibull (m=1), lognormal (m→∞,υ=0), and AWGN (m→∞,υ=1) cases. We also mention that although the Rician pdf cannot exactly be represented by a generalized Gamma, it indeed constitutes a very good approximation if the shape parameter υ=1 and the relationship $m\approx \frac{{(K+1)}^{2}}{2K+1}$ between the Rician factor K and the fading figure m holds [23]. The pdf of γ_{ i j } then can be expressed as [24]
where ξ_{ i j }=Γ(m_{ i j }+1/υ_{ i j })/Γ(m_{ i j }) and ${\stackrel{\u0304}{\gamma}}_{\mathit{\text{ij}}}=\mathbb{E}\left\{\right{h}_{\mathit{\text{ij}}}{}^{2}\}{P}_{i(j1)}/{\sigma}_{n}^{2}$, with $\mathbb{E}\{\xb7\}$ being the expectation operator.
3 Asymmetric multihop system
In this section, we study the performance of the asymmetric multihop systems. We first derive the asymptotic statistics of the received SNR at the destination. Then, we obtain closedform expressions for the outage probability and the average SER of the system under the high SNR assumption.
3.1 Statistics of the endtoend SNR
To analyze the performance of the multihop system, we need statistical characterization of its endtoend SNR. In this section, we derive the cumulative distribution function (cdf), pdf and MGF of the received SNR.
Although the expression given in (3) for γ_{ i } is more mathematically tractable than the one given in (3), the statistics of γ_{ i } in (3) are unknown for an arbitrary number of hops. ^{b} In order to keep a tractable analysis, we use the upper bound of γ_{ i } in (3) as [25],
The following proposition summarizes the results on statistics of γ_{ i } in the highSNR regime.
Proposition 1.
Let ${\gamma}_{\mathit{\text{ij}}}\sim \mathcal{G}\left({m}_{\mathit{\text{ij}}},{\left(\frac{{\stackrel{\u0304}{\gamma}}_{\mathit{\text{ij}}}}{{\xi}_{\mathit{\text{ij}}}}\right)}^{{\upsilon}_{\mathit{\text{ij}}}}\right)$, j=1,…,N_{ i }, be the independent hop SNRs for the i th branch. The asymptotic cdf of $\overline{{\gamma}_{i}}$ is then given by
The pdf and the MGF of $\overline{{\gamma}_{i}}$ are, respectively, given by
Proof
See Proof of Proposition 1 in Appendix. □
3.2 Performance analysis
With the statistical characterization of the received SNR derived in previous section, we can carry out a thorough performance analysis of the multihop system. We focus in what follows on outage probability and average SER performance measures.
3.2.1 Outage probability
The outage probability is one of the most commonly used performance measures in wireless systems. The outage probability in a multihop AF system is defined as the probability that the endtoend instantaneous received SNR falls below a predetermined threshold γ_{ t h }. This threshold is a protection value of the SNR, above which the quality of service is satisfactory. Therefore, the outage probability is given by the Pr(γ_{ i }<γ_{ t h }), which can easily be calculated by evaluating the cdf of γ_{ i } at γ_{ t h }. Consequently, asymptotic expression for the outage probability of the considered system over asymmetric fading channels can be obtained using (7) as
For the special case of i.i.d Nakagamim fading where m_{ i j }=m_{ i }, ${\stackrel{\u0304}{\gamma}}_{\mathit{\text{ij}}}={\stackrel{\u0304}{\gamma}}_{i}$, and υ_{ i j }=1, (7) is reduced to
which is consistent with the result obtained in [14].
3.2.2 SER
In addition to the outage probability, the average SER, is another standard performance criterion of cooperative diversity systems. The derived MGF can be used to evaluate the average SER of the multihop AF system under MPSK and MQAM. The average SER of MPSK can be written as [26]
where g_{ p s k }= sin2(π/M). For the square MQAM signals that have constellation size M=2^{k} with an even k, the average SER is given [26] as
where g_{ q a m }=3/2(M−1).
Closedform solutions for (12) and (13) in the general case seem analytically intractable. ^{c} However, using the available software packages such as Mapel and Mathematica this evaluation can be performed easily for a required degree of accuracy. The numerical results and simulation results are discussed in Section 6.
4 Asymmetric multihop multibranch system
In this section, we study the performance of the multihop multibranch AF systems in generalized fading channels. The destination node combines the received signals from different paths. Specifically, we examine two different combining techniques: MRC and SC [26]. With MRC, the received signals from multiple diversity branches are cophased, weighted, and combined to maximize the output SNR. ^{d} MRC provides the maximum performance improvement relative to all other combining techniques by maximizing the SNR of the combined signal. However, MRC also has the highest complexity of all combining techniques since it requires knowledge of the fading amplitude in each signal branch. As such we consider MRC as an important theoretical benchmark to quantify the performance of the considered network. SC is often used in practice as an alternative technique because of its reduced complexity relative to the optimum MRC scheme. In its conventional form, SC diversity only processes one of the diversity branches, specifically, the one determined by the receiver to have the highest SNR. The most important reason behind the popularity of the SC is the simplicity in implementation and decrease in resource requirement and complexity at the receiver, while still achieving full diversity.
4.1 Statistics of combined SNR
To analyze the performance of MRC and SC, we need statistical characterization of their combined SNR. In this section, we derive the cdf, pdf, and MGF of the combined SNR with MRC and SC.
4.1.1 MRC
MRC is the optimum combining scheme in the absence of interference [26], Ch.11. The total SNR at the output of the MRC combiner is simply given by
To obtain the statistics of ${\overline{\gamma}}_{\mathit{\text{MRC}}}$, i.e., sum of several independent variables, we need the following lemma.
Lemma 1
Let us consider a finite set of an arbitrary and independent nonnegative random variables (RV) X={x_{1},…,x_{ M }}, whose pdf’s, ${p}_{{x}_{k}}(\xb7)$k=1,…,M, tends to ${p}_{{x}_{k}}={a}_{k}{\gamma}^{{t}_{k}}+o\left({\gamma}^{{t}_{k}+\u03f5}\right)$ for γ→0^{+} and ϵ>0. If the RV z_{ k } is defined as
then the cdf of z_{ k } can be expressed as
where λ_{0}=0 and D_{0}=1.
Proof
See Proof of Lemma 1 in Appendix. □
The following propositions summarize the results for the cdf and MGF of ${\overline{\gamma}}_{\mathit{\text{MRC}}}$ for highSNR regime.
Proposition 2
Let ${\overline{\gamma}}_{i}$, i=1,…,M, be independent branch SNRs. The asymptotic cdf and MGF of the ${\overline{\gamma}}_{\mathit{\text{MRC}}}$ in generalized fading environments are, respectively, given by
Proof
Using Lemma 1 and employing the result for the product of two series [27], Eq. (10) gives, after some manipulation, the desired result in (16a).
The MGF of ${\overline{\gamma}}_{\mathit{\text{MRC}}}$ can directly be found from
where $\mathcal{\mathcal{L}}(\xb7)$ denotes the Laplace transform. Therefore, substituting the cdf given in (16a) into (17) and using $\mathcal{\mathcal{L}}\left({x}^{\nu}\right)=\Gamma (\nu +1)/{s}^{\nu +1}$, the MGF given in (16b) is achieved. □
4.1.2 SC
Instead of using MRC, which requires exact knowledge of the all CSIs, a system may use SC which simply requires SNR measurements. Indeed, SC is considered as the least complicated receiver [26], Ch. 11. The total SNR at the output of the SC combiner is given by
The following proposition gives the cdf and the MGF of ${\overline{\gamma}}_{\mathit{\text{SC}}}$.
Proposition 3
Let ${\overline{\gamma}}_{i}$, i=1,…,M be independent branch SNRs. The asymptotic cdf and the MGF of the ${\overline{\gamma}}_{\mathit{\text{SC}}}$ in generalized fading environments are, respectively, given by
Proof
If the branches fade independently, the cdf of the ${\overline{\gamma}}_{\mathit{\text{SC}}}$ is given
Using the result in [27], Eq.(10) for the product of two series, the desired result given in (21b) is derived.
The MGF of ${\overline{\gamma}}_{\mathit{\text{SC}}}$ can be obtained following the same procedure used to obtain (16b). □
4.2 Performance analysis
In this section, the outage probability and the SER are derived for the MRC and SC receivers.
4.2.1 Outage probability
Using (16a) and (21b), the asymptotic outage probability of the system with MRC and SC receivers can readily be obtained as
4.2.2 SER
Note that the asymptotic SER is found by substituting our results for the asymptotic MGF in (16b) and (19b) into (12) and (13), respectively, for the MRC and SC receiver with MPSK and MQAM. However, seeking a closedform solution to (12) and (13) is intractable due to the integration over θ. To avoid this integration, we invoke the accurate approximations in [28], Eq. (34) and [28], Eq. (36) to get the asymptotic SER for MPSK and MQAM, respectively.
4.2.3 Diversity order
By defining the diversity order as $d=\underset{\stackrel{\u0304}{\gamma}\to \infty}{lim}log\left({P}_{\mathit{\text{out}}}\right)/log\left(\stackrel{\u0304}{\gamma}\right)$, one can easily check that MRC and SC receiver attain diversity order
Note that although both MRC and SC schemes achieve the same diversity order, the MRC scheme achieves an additional coding gain.
5 Power allocation for multihop multibranch cooperative system
In this section, two effective transmit power allocation schemes are described. The power allocation scheme which tends to minimize the total power of the system is developed in Section 5.2. A suboptimal scheme is proposed in Section 5.3 aimed at minimizing the outage probability. In the sequel, a brief introduction of GP for application to be discussed in the next two sections on power control problems is given.
5.1 GP
GP is wellinvestigated class of nonlinear, nonconvex optimization problems, which can be turned into a convex optimization problem [18]. Hence, a local optimum of a GP problem is also a global optimum, which can always be calculated efficiently using interiorpoint methods [29]. The polynomial time complexity of the interiorpoint methods, their high speed in practice, and availability of largescale software solvers make GP more appealing ^{e} (please see GP in Appendix for details on GP). We show that the corresponding optimization problems can be formulated as GP and thus optimal power allocation (OPA) can be obtained using the convex optimization techniques.
5.2 Minimizing the total transmit power
We consider the problem of minimizing the total transmitter power subject to bounds on individual powers and outage constraint. Note that to improve the system performance, the transmitting nodes can transmit at their maximum available power which cause themselves to run out of energy rapidly. This also implies that the number of available relay nodes will decrease quickly, which leads to lower throughput and higher transmission power for each node. However, by considering the QoS requirements, the channel qualities and OPA at the source and relay nodes, some of the transmitting nodes save their power and prolong their lifetime. In order to minimize the total transmit power of all nodes, subject to constraints on the individual transmitter powers and subject to a maximum allowed outage probability ϵ at the destination, we form the optimization problem as ^{f}
where P_{i 0}=P_{0}, i=1,…,M is the source power; ${P}_{\mathit{\text{ij}}}^{min}$ and ${P}_{\mathit{\text{ij}}}^{max}$ are, respectively, the minimum and maximum transmission power for the corresponding node which can be the same or different for rely nodes. Note that ${P}_{\mathit{\text{out}}}^{\mathit{\text{MRC}}}$ in (23) is a nonlinear function of the powers, which yields a posynomial upper bound inequality constraint for the optimization problem in (23). With MRC receiver at the destination, the optimization problem in (23) can be expressed as
where ${\psi}_{i{j}_{i}}=\frac{\Gamma ({\upsilon}_{i{j}_{i}}{m}_{i{j}_{i}}+1)}{\Gamma ({m}_{i{j}_{i}}+1)}{\left(\frac{\Gamma ({m}_{i{j}_{i}}+1/{\upsilon}_{i{j}_{i}})}{\Gamma ({m}_{i{j}_{i}}+1)}\right)}^{{\upsilon}_{i{j}_{i}}{m}_{i{j}_{i}}}$. Each of the terms ${\stackrel{\u0304}{\gamma}}_{i{j}_{i}}$ is a posynomial in P_{ i j } and the product of posynomials is also a posynomial [18]. Moreover, the inequality constraints (24b) and (24c) are monomial and the constraint in (24e) is a posynomial. Therefore, the optimization problem in (24) is a GP in the variables P_{ i j }, i=1,…,M, j=0,…,N_{ i }−1. By using the interiorpoint methods for GP we can solve the power allocation problem in (24).
5.3 Minimizing the outage probability
In this section, we explore the power allocation policy aimed at minimizing the outage probability. The problem formulation (24) can readily be modified to minimize the asymptotic outage probability as
where ${P}_{T}=\sum _{i=1}^{M}\sum _{j=1}^{{N}_{i}}{P}_{\mathit{\text{ij}}}^{max}$ is the total available power. It is obvious that the optimization problem (25) belongs to the class of GP problems and can efficiently be solved by using the interiorpoint methods.
5.3.1 Analytical results for a singlerelay cooperative network
In this section, we provide an analytical approximation of optimum power allocation for a threenode cooperative network, with a source S, a relay R, and a destination D. This analysis provides some insight for the formulated problem in section 5.3. We denote by P_{1} and P_{2} the transmitted power from source and relay, respectively. The optimization problem in (25) is then given by
where C_{1} and C_{2} are positive constants, capturing the fading effects of the links. Since the fading parameters generally take noninteger values, solving (26) does not yield closedform expressions for P_{1} and P_{2}. Nevertheless, the optimization problem defined in (26) includes, as special case, the Rayleigh fading environment. In this case, the power allocation problem is reduced to
where $\kappa =\frac{{\Omega}_{\mathit{\text{SR}}}}{{\Omega}_{\mathit{\text{RD}}}}$, with ${\Omega}_{\mathit{\text{SR}}}=\mathbb{E}\left(\right{h}_{\mathit{\text{SR}}}{}^{2})$ and ${\Omega}_{\mathit{\text{RD}}}=\mathbb{E}\left(\right{h}_{\mathit{\text{RD}}}{}^{2})$, is a measure for the quality of the S–R link compared to R–D link. Denoting the optimal source and relay powers by ${P}_{1}^{\ast}$ and ${P}_{2}^{\ast}$, respectively, and defining, ${\alpha}^{\ast}=\frac{{P}_{1}^{\ast}}{{P}_{2}^{\ast}}$, the OPA can be obtained from (27) as ${P}_{1}^{\ast}={P}_{T}/(1+{\alpha}^{\ast})$ and ${P}_{2}^{\ast}={\alpha}^{\ast}{P}_{T}/(1+{\alpha}^{\ast})$, with
From (28) we observe that

1.
When the relay is close to the destination, optimum value of P _{1} is ∼P _{ T }, and that of P _{2} is ∼0. These values indicate that it is better to spend most of the power in broadcast phase.

2.
When the relay is located midway between the source and destination, optimum value of P _{1} is ∼(2/3) P _{ T } which means that 66% of power should be spent in the broadcast phase and 33% of power should be dedicated to the relay terminal in the relaying phase. These values indicate that it is better to spend most of the power in broadcast phase.

3.
when relay is close to the source, P _{1} and P _{2} are found to be ∼0.5 P _{ T } indicating that equal power allocation (EPA) is nearly optimal.
Note that the same observations have been reported in [30, 31] for a threenode cooperative network.
5.4 Discussion on the implementation of power allocation schemes
The two proposed power allocation schemes are computed in a centralized manner at the destination. Centralized implementation of power allocation schemes requires a central controller to collect the information of all wireless links in order to find an optimal solution, and distribute the solution to the corresponding wireless nodes. Hence, information exchange plays a crucial role in implementing the resource optimization process. Useful information can be the full channel state information, or partial channel state information (e.g., average channel realizations), or some other quantized/codebookbased limitedrate feedback information.
The implementation of our proposed power allocation schemes requires that the destination has the information about the channel statistics rather than the instantaneous CSIs. Since the firstorder and secondorder statistics vary much slower than the instantaneous CSIs, the overhead is significantly reduced. The remaining, but most challenging task is keeping the amount of feedback overhead information, exchanged within the network, at a reasonable level. For this purpose, the destination determines the power coefficients. These coefficients are then quantized at the receiver and sent back to the transmitters over a lowrate feedback link [32]. Therefore, the signaling overhead is much lower than that of the conventional centralized methods.
6 Simulation results
In this section, we provide numerical results corroborating the analysis developed in the previous sections. It is assumed that the relays and the destination have the same value of noise power. We plot the performance curves in terms of outage probability and average SER versus the normalized average SNR per hop. We also set γ_{ t h }=3 dB.
Figure 2 plots the exact and asymptotic outage probability of a generic cooperative system in the context of various scenarios. Specifically, the outage probability of two and three branches AF network have been plotted for MRC and SC receivers. Moreover, the outage performance of direct transmission is also depicted as the baseline for comparison. In our simulations, the direct channel (first branch) is assumed to be Rayleigh fading channel corresponding to m_{11}=1. By contrast, the dualhop link (second branch) is assumed to be consisted of Nakagamim fading channels associated with m_{21}=1.1 and m_{22}=1.6. Moreover, the triplehop channel (third branch) is assumed to be consisted of Rician and Nakagamim fading channels associated with K=5 dB and m_{32}=0.9 and m_{33}=3. The validity of our asymptotic results in (21a) and (21b) are attested to in the figure, where the asymptotic curves correctly predict the diversity order and the array gain of the exact SER.
In Figure 3, we evaluated the SER performance of the cooperative wireless systems, when assuming both BPSK and 64QAM baseband modulation schemes. These figure also demonstrate the accuracy of the asymptotic SER evaluated with the aid of the MGF of the received SNR in comparison with the SER obtained by simulations. Note that the corresponding curves are plotted only for direct transmission and M=3 branches case, to avoid entanglement. The results of Figure 3 shows that the asymptotic SER is very accurate especially in high SNR region. Note that our observation of the outage performance of 8QAM and 16QAM modulations, which for the sake of clarity are not shown in Figure 3, reveals that the asymptotic SER matches the exact results in higher SNR values, as the modulation order increases. We interpret this behavior as a consequence of approximation, used in [28] to derive the closedform SER expressions.
Next, we compare the performance of the optimum and EPA, with the latter equally distributing the power among all the relay nodes. Figure 4 shows the outage performance of the proposed OPA scheme in (25) in the context of various scenarios. Specifically, in our simulations, we consider M=2 and M=3 branches cases, where channels undergo the same statistical process as that in Figure 2. Curves for 16QAM modulation are plotted for SC and MRC receivers, while corresponding curves from analysis are plotted only for M=2, to avoid entanglement. We observe that the power allocation shows significant improvement in performance compared to those of a system with EPA. Moreover, the gap in performance increases further with increase in SNR values and the number of branches. Note that, instead of minimizing the outage probability in (25), we can minimize the SER performance of the system. Figure 5 shows a comparison between the equal power and OPA schemes for multihop multibranch system using BPSK modulation. SER Curves for SC and MRC receivers are plotted and channels undergo the same statistical process as that in Figure 2.
Figure 6 shows a comparison of the outage probability of the singlerelay cooperative system for two different relay positions: relay is midway between the source and the destination, and relay is located closed to the destination. In this figure, we assume that $\mathbb{E}\left(\right{h}_{\mathit{\text{SD}}}{}^{2})={\Omega}_{\mathit{\text{SD}}}=1/{d}_{\mathit{\text{sd}}}^{\alpha}$, $\mathbb{E}\left(\right{h}_{\mathit{\text{SR}}}{}^{2})={\Omega}_{\mathit{\text{SR}}}=1/{d}_{\mathit{\text{sr}}}^{\alpha}$, and $\mathbb{E}\left(\right{h}_{\mathit{\text{RD}}}{}^{2})={\Omega}_{\mathit{\text{RD}}}=1/{d}_{\mathit{\text{rd}}}^{\alpha}$, where d_{ s d }, d_{ s r }, and d_{ r d } are, respectively, the source–destination, source–relay, and relay–destination distances and α is the path loss exponent. We also set α=3. The solid curves are the outage probability with EPA and the dotted curves are the outage probability with OPA in (28). Clearly and as expected, for these two relay placements, EPA does not give the best performance. Moreover, the OPA is more suitable to be utilized in singlerelay cooperation networks in which relay is located close to the destination.
7 Conclusion
We investigated the performance of multihop multi branch AF relay systems in generalized fading environment with MRC and SC receivers. A range of closedform results has been derived for both the statistics of the output SNR and the asymptotic performance of the system under study. We substantiated the tightness of such asymptotic expressions and the accuracy of our theoretical analysis using simulation results. Moreover, we developed two power allocation strategies for further improving the cooperation. The first strategy sought to minimize the total transmit power; the second strategy aimed at minimizing the outage probability, which was parameterized by the total power available to the relay nodes and the source node. We found that the OPA shows significant improvement in performance when relay nodes are asymmetrically placed at fixed locations when compared to a system with EPA.
Endnotes
^{a}We assume that the noise power is identical in all receiving nodes. Note that this assumption is not essential and can easily be relaxed, but at the cost of complicating the derived expressions without providing additional insight.^{b}We notice that closedform expressions for the statistics of γ_{ i } are given in [3] and [4] for the special case of an AF dualhop system in Nakagamim and Rayleigh fading channels, respectively.^{c}In [35], an accurate approximation has been presented for the SER with MPSK modulation.^{d}In this study, we assume that the receiver estimates the channel perfectly from training. A discussion of channel estimation techniques is beyond the scope of this article and the reader is referred to [36, 37] for the details.^{e}There are several highquality software downloadable from the Internet, which are widely used to solve the GP using interiorpoint methods (e.g., the MOSEK package and the CVX package). ^{f}Note that we consider the MRC combiner in the proposed power allocation schemes. However, for the SC combiner, we can follow the same procedure to get the optimized transmitted powers.
Appendix
Proof of Proposition 1
Assuming that the received SNR’s from different diversity branches are independent, the cdf of the received SNR in (6) is given by
where (a) follows by using the complementary cdf of the generalized Gamma distribution. With the help of [22], Eq. (8.356.4) the cdf can be rewritten as
Using the inequality $1+\sum _{\ell =1}^{K}{z}_{\ell}\le \prod _{\ell =1}^{K}(1+{z}_{\ell})\le exp\left(\sum _{\ell =1}^{K}{z}_{\ell}\right)$[33], Appendix II we get
By approximating the exponential term in the righthand side of the inequality with the first two terms of the well known Taylor series, for high SNRs ($1/{\stackrel{\u0304}{\gamma}}_{\mathit{\text{ij}}}\to 0$), the cdf in (31) is further simplified as
Our simulation results in Section 6 show that for k=0, a fairly tight asymptotic bound for the outage probability of the multihop system is achieved. The reason is that the outage probability is proportional to $1/{\left({\stackrel{\u0304}{\gamma}}_{\mathit{\text{ij}}}\right)}^{k}$ and thus for sufficiently high values of SNRs decays very fast with k≥1. Therefore, substituting ξ_{ i j } into (32), setting k=0, and using the fact that Γ(1+z)=z Γ(z) [34] the desired result in (8) is achieved.
Proof of Lemma 1
Let we define
where (a) follows by using [22], λ_{ k }:=λ_{k−1}+ν_{ k }+1, and Δ_{ k } is defined in (15d). To prove the proposition, we will use mathematical induction. Clearly, Lemma 1 holds for n=1 and n=2, i.e.,
where (a) follows by taking the integral of the pdf of z_{1}=x_{1} and (b) follows by using the cdf of the z_{1}. Suppose the result in (15a) is true for 2<n=(k−1)<M, then for n=k we have
where (a) follows by induction assumption. Therefore, the closedform cdf in (15a) is valid for n=k, which completes the proof.
GP
In this section, we give a brief review of the GP and refer the reader to [18], Ch. 4 for details.
Let x_{1},…,x_{ n } be n real positive variables and x denotes the vector of n variables. We define a monomial as a function of $f:{\mathcal{R}}_{++}^{n}\to \mathcal{R}$:
where c>0 and ${\alpha}_{\ell}\in \mathcal{R}$, ℓ=1,2,…,n. A sum of monomials is called a posynomial function, which has the form
where c_{ k }≥0 and ${\alpha}_{k}\in \mathcal{R}$, ℓ=1,2,…,n, k=1,2,…,K.
Minimizing a posynomial subject to posynomial upper bound inequality constraints and monomial equality constraints is called GP. Therefore, a GP in standard form in x=[x_{1},…,x_{ n }] is given as
With a logarithmic change of variables as y_{ i }= logx_{ i } (or equivalently x_{ i }= exp(y_{ i }) which enforces the positivity constraint on x_{ i }) we can turn the GP in (38) into the following equivalent problem in x
which is a convex problem, since the objective function and the inequality constraint functions are all convex and the equality constraint functions are affine (note that the logsumexp function is convex [18]).
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This work is supported in part by the Ministry of Industries and Mines of Iran.
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Mohammadi, M., Ardebilipour, M., Mobini, Z. et al. Performance analysis and power allocation for multihop multibranch amplifyandforward cooperative networks over generalized fading channels. J Wireless Com Network 2013, 160 (2013). https://doi.org/10.1186/168714992013160
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Keywords
 Power Allocation
 Outage Probability
 Relay Node
 Moment Generate Function
 Maximal Ratio Combine
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