Performance analysis and power allocation for multi-hop multi-branch amplify-and-forward cooperative networks over generalized fading channels
© Mohammadi et al.; licensee Springer. 2013
Received: 10 June 2012
Accepted: 12 February 2013
Published: 12 June 2013
In this article, efficient power allocation strategies for multi-hop multi-branch amplify-and-forward 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 signal-to-noise 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 interior-point methods. Numerical results are provided to substantiate the analytical results and to demonstrate the considerable performance improvement achieved by the power allocation.
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.  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.  exploit the Moment Generating Function (MGF)-based approach for performance analysis of multi-hop multi-branch networks over fading channels. Amarasuriya et al.  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 , SER , or numerical methods , 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 . 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.
2 System model
especially for sufficiently large values of SNR.
where ξ i j =Γ(m i j +1/υ i j )/Γ(m i j ) and , with being the expectation operator.
3 Asymmetric multi-hop system
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.
3.1 Statistics of the end-to-end SNR
To analyze the performance of the multi-hop system, we need statistical characterization of its end-to-end SNR. In this section, we derive the cumulative distribution function (cdf), pdf and MGF of the received SNR.
The following proposition summarizes the results on statistics of γ i in the high-SNR regime.
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 multi-hop system. We focus in what follows on outage probability and average SER performance measures.
3.2.1 Outage probability
which is consistent with the result obtained in .
where g q a m =3/2(M−1).
Closed-form 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 multi-hop multi-branch system
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 . 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.
To obtain the statistics of , i.e., sum of several independent variables, we need the following lemma.
where λ0=0 and D0=1.
See Proof of Lemma 1 in Appendix. □
The following propositions summarize the results for the cdf and MGF of for high-SNR regime.
where denotes the Laplace transform. Therefore, substituting the cdf given in (16a) into (17) and using , the MGF given in (16b) is achieved. □
The following proposition gives the cdf and the MGF of .
The MGF of 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
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 M-PSK and M-QAM. However, seeking a closed-form solution to (12) and (13) is intractable due to the integration over θ. To avoid this integration, we invoke the accurate approximations in , Eq. (34) and , Eq. (36) to get the asymptotic SER for M-PSK and M-QAM, respectively.
4.2.3 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 multi-hop multi-branch 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.
GP is well-investigated class of nonlinear, non-convex optimization problems, which can be turned into a convex optimization problem . Hence, a local optimum of a GP problem is also a global optimum, which can always be calculated efficiently using interior-point methods . The polynomial time complexity of the interior-point methods, their high speed in practice, and availability of large-scale 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
where . Each of the terms is a posynomial in P i j and the product of posynomials is also a posynomial . 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 interior-point methods for GP we can solve the power allocation problem in (24).
5.3 Minimizing the outage probability
where 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 interior-point methods.
5.3.1 Analytical results for a single-relay cooperative network
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.
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.
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.
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/codebook-based limited-rate 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 first-order and second-order 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 low-rate feedback link . 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.
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  and  for the special case of an AF dual-hop system in Nakagami-m and Rayleigh fading channels, respectively.cIn , 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.
Proof of Proposition 1
Our simulation results in Section 6 show that for k=0, a fairly tight asymptotic bound for the outage probability of the multi-hop system is achieved. The reason is that the outage probability is proportional to 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)  the desired result in (8) is achieved.
Proof of Lemma 1
where (a) follows by induction assumption. Therefore, the closed-form cdf in (15a) is valid for n=k, which completes the proof.
In this section, we give a brief review of the GP and refer the reader to , Ch. 4 for details.
where c k ≥0 and , ℓ=1,2,…,n, k=1,2,…,K.
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 ).
This work is supported in part by the Ministry of Industries and Mines of Iran.
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