Performance analysis and power allocation for multi-hop multi-branch amplify-and-forward cooperative networks over generalized fading channels

Recently, it has been shown that the throughput, coverage, and battery life of resource-constrained wireless ad hoc networks can be increased through the use of multi-hop 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 multi-hop multi-branch transmission system, a source communicates with the destination through several multi-hop 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 dual-hop amplify and forward (AF) networks has widely been analyzed in [3‐5] and [6‐10] for single and multiple relay scenarios, respectively. Multi-hop multi-branch 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 non-zero values at the origin, including Rayleigh and Rician fading channels, when the average signal-to-noise ratio (SNR) is sufficiently large. Renzo et al. [13] exploit the Moment Generating Function (MGF)-based approach for performance analysis of multi-hop multi-branch networks over fading channels. Amarasuriya et al. [14] proposed a new class of upper bounds on the end-to-end SNR of a multi-hop system and then derived an asymptotic expression for the symbol error rate (SER) of the multi-hop multi-branch set-up in independent and identically distributed (i.i.d.) Nakagami-m 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 set-up 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 dual-hop relaying scheme is typically considered in the aforementioned studies and power-optimized multi-hop relaying is only studied in [17]. To the best of the authors’ knowledge, the ultimate benefit of power control in multi-hop multi-branch 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 multi-hop multi-branch 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 interior-point 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 multi-hop system. Section 4 presents the asymptotic analysis of the multi-hop multi-branch 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.

Consider a generalized cooperative system with M diversity branches and { N i } 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 σ 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 γ ij = | h ij | 2 P i ( j − 1 ) / σ 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 power-scaling parameter. In what follows, we will use the shorthand notation X ∼ 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), Nakagami-m (υ=1), Weibull (m=1), log-normal (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 ≈ ( K + 1 ) 2 2 K + 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 γ ̄ ij = E { | h ij | 2 } P i ( j − 1 ) / σ n 2, with E { · } being the expectation operator.

In this section, we study the performance of the asymmetric multi-hop systems. We first derive the asymptotic statistics of the received SNR at the destination. Then, we obtain closed-form expressions for the outage probability and the average SER of the system under the high SNR assumption.

In this section, we study the performance of the multi-hop multi-branch 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.

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.

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 m₁₁=1. By contrast, the dual-hop link (second branch) is assumed to be consisted of Nakagami-m fading channels associated with m₂₁=1.1 and m₂₂=1.6. Moreover, the triple-hop channel (third branch) is assumed to be consisted of Rician and Nakagami-m fading channels associated with K=5 dB and m₃₂=0.9 and m₃₃=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 64-QAM 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 8-QAM and 16-QAM 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 closed-form 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 16-QAM 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 multi-hop multi-branch 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 single-relay 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 E ( | h SD | 2 ) = Ω SD = 1 / d sd α, E ( | h SR | 2 ) = Ω SR = 1 / d sr α, and E ( | h RD | 2 ) = Ω RD = 1 / d rd α, 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 single-relay cooperation networks in which relay is located close to the destination.

We investigated the performance of multi-hop multi- branch AF relay systems in generalized fading environment with MRC and SC receivers. A range of closed-form 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.

^aWe 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.^bWe notice that closed-form expressions for the statistics of γ _i are given in [3] and [4] for the special case of an AF dual-hop system in Nakagami-m and Rayleigh fading channels, respectively.^cIn [35], an accurate approximation has been presented for the SER with M-PSK modulation.^dIn 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.^eThere are several high-quality software downloadable from the Internet, which are widely used to solve the GP using interior-point methods (e.g., the MOSEK package and the CVX package). ^fNote 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.

In this section, we give a brief review of the GP and refer the reader to [18], Ch. 4 for details.

Let x₁,…,x _n be n real positive variables and x denotes the vector of n variables. We define a monomial as a function of f : R + + n → R:

where c>0 and α ℓ ∈ R, ℓ=1,2,…,n. A sum of monomials is called a posynomial function, which has the form

where c _k ≥0 and α k ∈ 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=[x₁,…,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 log-sum-exp function is convex [18]).