 Research
 Open Access
Distributed energyaware resource allocation in multiantenna multicarrier interference networks with statistical CSI
 Alessio Zappone^{1}Email author,
 Giuseppa Alfano^{2},
 Stefano Buzzi^{3} and
 Michela Meo^{2}
https://doi.org/10.1186/168714992013205
© Zappone et al.; licensee Springer. 2013
 Received: 19 December 2012
 Accepted: 22 July 2013
 Published: 9 August 2013
Abstract
Resource allocation for energy efficiency optimization in multicarrier interference networks with multiple receive antennas is tackled. First, a onehop network is considered, and then, the results are extended to the case of a twohop network in which amplifyandforward relaying is employed to enable communication. A distributed algorithm which optimizes a systemwide energyefficient performance function, and which is guaranteed to converge to a stable equilibrium point, is provided. Unlike most previous works, in the definition of the energy efficiency, not only the users’ transmit power but also the circuit power that is required to operate the devices is taken into account. All of the proposed procedures are guaranteed to converge and only require statistical channel state information, thus lending themselves to a distributed implementation. The asymptotic regime of a saturated network in which both the active users and the number of receive antennas deployed in each receiver grow large is also analyzed. Numerical results are provided to confirm the merits of the proposed algorithms.
Keywords
 Power Allocation
 Energy Efficiency
 Channel State Information
 Resource Allocation Algorithm
 Potential Game
1 Introduction
In the recent past, the growing concerns on sustainable growth due to the exponential increase in the use of mobile communication devices, as well as the need to maximize battery life in mobile handsets, garnered a great interest for resource allocation techniques aimed at the maximization of the energy efficiency (EE), measured in bits per Joule, at the physical layer of wireless networks. While more traditional resource allocation techniques pursue the optimization of the network’s performance in terms of throughput, which usually comes at the expense of battery life, an energyefficient resource allocation aims at the maximization of performance measures that take into account the tradeoff between achieving a high throughput and saving as much battery power as possible.
Pioneering works in the area of distributed resource allocation for EE are [1–3]. There, the EE is defined as the ratio between the users’ throughput and the transmit power, and energyefficient power control algorithms for the uplink of wireless networks are provided.
The results of [1–3] have been extended in several directions in many following studies. In [4], joint linear receiver design and power control is pursued for CDMA networks, while in [5], spreading code optimization is also plugged into the resource allocation process. In [6], the impact of widely linear filtering on the EE of wireless networks is investigated. In [7], a hierarchical approach based on Stackelberg games is used to tackle the energyefficient power control problem, while in [8], a power control algorithm based on a repeatedgame approach is proposed. In [9], the impact of relaying on the EE of wireless networks is studied.
Most of the above works deal with multipleaccesschannel systems. In [10], instead, a multicarrier interference network is considered, and the problem of distributed EE maximization through individual power allocation over the available subcarriers is investigated. However, the proposed algorithm is not guaranteed to converge to a stable equilibrium point. In [11], the uplink of multicell orthogonal frequencydivision multiple access (OFDMA) network is considered, and a resource allocator aiming at minimizing the total transmitted power subject to individual rate constraints is studied. The results indicate that the proposed algorithm converges to a stable resource allocation policy only when the interference load is below a certain threshold. In [12], a gametheoretic approach to subcarriers, modulation format, and power allocation for transmit power minimization with rate constraints is considered. Again, the proposed game is not guaranteed to converge to an equilibrium, and for this reason, a virtual referee is introduced to dictate the resource allocation and force it to a stable and efficient equilibrium point. In [13], an auction approach to subcarrier, modulation, and coding scheme allocation in singlecell and multicell OFDMA networks is proposed, and the simplification is made to divide the users in each cell between interior and edge users, assuming that the inner users in each cell do not interfere with adjacent cells. In [14], energyefficient subcarrier and transmit power allocation in the uplink of an OFDMA interference network is considered. However, in order to ensure the convergence of the resulting algorithm, subcarrier assignment is performed in a centralized fashion. In [15], a novel systemwide energyefficient performance metric is proposed, which allows for a completely distributed and convergent power and subcarrier allocation in multicarrier interference networks. There, the resource allocation problem is formulated as a potential game [16], with potential function given by the systemwide performance function to be maximized. Potential games for resource allocation in wireless multiuser networks have also been considered in [17–19].
In [15], as in all previously cited works, as far as the definition of the EE is concerned, only the transmit power is considered, whereas a more meaningful definition of the EE is obtained when not only the transmit power but also the power that is dissipated in the electronic circuitry of each terminal in order to operate the device is accounted for [20–25]. Moreover, previous works on distributed EE in interference networks deal with singleantenna systems, whereas it is confirmed that a crucial key ingredient of today’s and future wireless networks will be the use of multiple antennas, which may be very helpful in providing diversity gains and dramatically increasing the system performance in terms of effective data rates and network coverage [26–28]. First, results on EE in multipleantenna systems are [22, 29], where singleuser multipleinput multipleoutput (MIMO) systems are considered and the optimal energyefficient covariance matrix is determined.
Motivated by this background, this paper extends the results of [15] in the following directions:

The transmitter circuit powers are taken into account in the definition of the users’ energy efficiency, which complicates the analysis.

Multipleantenna receivers are considered, and unlike [15], only statistical channel state information (CSI) is assumed at the transmitter side, which reduces the amount of feedback required to implement the proposed algorithms.

A twohop interference network in which communication takes place by means of an amplifyandforward relay is also considered.
In this challenging scenario, distributed algorithms for energyefficient joint subcarrier and power allocation are devised. The proposed algorithms require only the solution of convex problems, which makes their implementation feasible in real world systems, and will be shown to always converge to a stable equilibrium point. EE maximization will also be tackled in the asymptotic regime of a saturated interference network in which both the number of active users and of receive antennas grow large with a fixed ratio.
The remainder of this paper is organized as follows. Section 2 describes the considered system model. Section 3 investigates the energyefficient resource allocation problem, providing the tools to devise suitable resource allocation algorithms. Section 4 contains the asymptotic analysis for saturated networks. In Section 5, the results of 3 are extended to the relayassisted scenario, while Section 6 is devoted to the numerical results. Concluding remarks are given in Section 7.
2 System model
Consider the uplink of a K×J multicarrier interference network. Each receiver is equipped with M receive antennas, and for all k=1,…,K, a_{ k }∈{1,2,…,J} denotes the intended receiver for user k. It is assumed here that receiver assignment has been performed in a previous phase, and we focus on the resource allocation problem only. Let $\mathcal{\mathcal{L}}=\{1,2,\dots ,L\}$ be the set of available subcarriers in the system, and denote by p_{ k }(ℓ) the kth user’s transmit power on subcarrier ℓ, and by h_{k,j,m}(ℓ) the complex channel gain between the kth user and the mth antenna of the jth receiver on the ℓth subcarrier, modeled as a realization of a zeromean Gaussian random variable with variance $\frac{{d}_{k,j}^{\eta}}{M}$, with d_{k,j} and η denoting the distance between the kth mobile user and the jth BS, and the path loss exponent, respectively. Let us also define the channel vector h_{k,j}(ℓ)= [ h_{k,j,1}(ℓ),…,h_{k,j,M}(ℓ)]^{ T }, with (·)^{ T } denoting transpose, modeled as a zeromean Gaussian random vector with covariance matrix R_{k,j}(ℓ).
In the following, only statistical CSI is assumed at the transmitters’ side, in terms of secondorder statistics of the channel coefficients and vectors.
In this scenario, the aim is to devise a transmit power and subcarrier allocation algorithm that optimizes the network’s EE in a distributed way. Here, by distributed, we mean that the resource allocation process is not jointly carried out by a computational center which computes the optimum resource allocation policy, feeding back the results to the transmitters. Instead, each transmitter should allocate its own resources in a selforganizing way. This is a very important feature especially in networks that, by their own nature, lack a central control unit that dictates the resource allocation policy, which is the case for example in relevant communication systems like ad hoc networks and interference networks.
3 Distributed energyefficient resource allocation
The aim of this section is to derive a distributed energyefficient resource allocation algorithm. The main challenge in deriving a distributed algorithm in interference networks is that typically global CSI is needed. Otherwise stated, each user also needs to know other users’ channels, which requires a too great amount of overhead information to feedback. In order to circumvent this problem, the proposed algorithm will be designed so as to require only statistical CSI at the transmitter side. Since channel statistics vary at a very slow rate compared to the actual channel realizations, feeding back only statistical CSI significantly reduces the amount of required overhead. Moreover, channel statistics can be estimated more easily than channel coefficients at the receivers. It is also to be mentioned that present multicell networks are typically endowed with a highspeed backhaul link which allows the receivers to exchange information with one another. Therefore, each receiver can easily learn the channel statistics also of users that are not associated to it and then feedback this information to its associated transmitters. Again, the overhead information to be exchanged on the backhaul link due to the resource allocation algorithm is quite limited since only channel statistics need to be shared. Thanks to these features, the algorithm to be developed lends itself to a distributed implementation at the transmitter side.
To begin with, we remark that since the resource allocation takes place at the transmitters, the instantaneous SINR expression (2) cannot be used for resource allocation purposes, because each transmitter only has statistical CSI. Thus, before turning to the analysis of the resource allocation algorithm, an average SINR expression is needed.
3.1 Users’ average SINR
3.2 Proposed distributed algorithm
with $G\left({\left\{{p}_{k}\right(\ell ),{\gamma}_{k}(\ell \left)\right\}}_{k=1,\ell =1}^{K,L}\right)$ being the energyefficient performance metric to optimize, which will be specified shortly. For all k=1,…,K, the solution to the kth problem in (6) yields the kth user’s power allocation for a fixed configuration of the other users’ powers, and any fixed point of iteration (6) represents a stable resource allocation policy. We remark that for all k=1,…,K, only the transmit powers ${\left\{{p}_{k}\right(\ell \left)\right\}}_{\ell =1}^{L}$ have been indicated as the optimization variables of the generic kth problem, because by choosing the transmit powers, each user automatically chooses also the transmit subcarriers. Indeed, a subcarrier can be discarded by simply transmitting zero power over it.
wherein R is the transmit data rate, Q≥1 is the packet length, D≤Q is the number of information symbols contained in each packet, γ is the achieved SINR, p the transmit power, and (1−e^{−γ})^{ Q } is the socalled efficiency function which approximates the probability of correct reception for a datapacket of length Q[4, 10, 30] and references therein. We stress that the case of bitoriented communications, i.e., Q=1 is included as a special case in our definition of the energy efficiency and all results to follow will hold true also for Q=1. Moreover, it should be mentioned that also the case Q>1 is of practical interest in modern OFDMA systems, such as LTE [31].
Another widely used efficiency function is the achievable rate log(1+γ) [21, 23, 32]. However, such a choice applies to strictly static channels but has no informationtheoretic meaning in the considered scenario where the channels are rapidly varying. In our context, an informationtheoretic meaningful function would be the ergodic achievable rate E[log(1+γ)]. Such an approach, which has been considered in [22] for the simpler scenario of singleuser MIMO systems, appears more challenging in interference networks and is left as future work.
Due to its multiplicative nature, it is unlikely that a maximizer of (10) results in one of the users’ throughputs to be very low, since each user’s throughputs is a factor of the product in the numerator of (10). Moreover, (10) is also a systemwide performance function, since it is an increasing function of the players’ energy efficiencies. We stress that the maximization of products of utility functions in order to obtain fair resource allocation policies is also considered in contexts other than EE maximization [33, 34].
i.e., the nth user’s SINR with interferencefree transmission. The following proposition holds.
Proposition 1
with Ei(·) denoting the exponential integral function.
Proof
Hence, the thesis. □
It should be stressed that the computation of the coefficients ${\left\{{\beta}_{n}\right\}}_{k=1}^{K}$ needs to be performed just once and can be carried out offline because each β_{ n } only depends on the constant networks parameters Q, P_{m a x,n}, σ^{2}, and $E[\parallel {\mathit{h}}_{n,{a}_{n}}(\ell ){\parallel}^{2}$.
Accordingly, the resource allocation algorithm can be expressed as follows:
Algorithm 1 Distributed resource allocation
Convergence in Algorithm 1 is declared when the difference between the values of the objective function (16) achieved at the end of two successive outer loops is below a predetermined tolerance. The following proposition guarantees the convergence of Algorithm 1.
Proposition 2
For any feasible initialization point${\left\{{p}_{k}^{\left(0\right)}\right(\ell \left)\right\}}_{k=1,\ell =1}^{K,L}$, Algorithm 1 is guaranteed to converge.
Proof
The objective (16) depends on all of the users’ KL transmit powers. After the initialization, we have${\hat{\mathrm{GEE}}}^{\left(0\right)}({\left\{{p}_{1}^{\left(0\right)}\right(\ell \left)\right\}}_{\ell =1}^{L},\dots ,{\left\{{p}_{K}^{\left(0\right)}\right(\ell \left)\right\}}_{\ell =1}^{L})$. After the first iteration of the for cycle in Algorithm 1, (16) is maximized with respect to${\left\{{p}_{1}\right(\ell \left)\right\}}_{\ell =1}^{L}$ while keeping the other (K−1)L powers fixed. Let us denote by${\left\{{p}_{1}^{\left(1\right)}\right(\ell \left)\right\}}_{\ell =1}^{L}$ the L powers resulting from such optimization. Then, after this first optimization, the new value of the objective is${\hat{\mathrm{GEE}}}^{\left(1\right)}({\left\{{p}_{1}^{1}\right(\ell \left)\right\}}_{\ell =1}^{L},{\left\{{p}_{2}^{\left(0\right)}\right(\ell \left)\right\}}_{\ell =1}^{L},\dots ,{\left\{{p}_{K}^{\left(0\right)}\right(\ell \left)\right\}}_{\ell =1}^{L})$, and clearly, we have${\hat{\mathrm{GEE}}}^{\left(1\right)}\ge {\hat{\mathrm{GEE}}}^{\left(0\right)}$. In the second iteration of the cycle, the powers${\left\{{p}_{2}\right(\ell \left)\right\}}_{\ell =1}^{L}$ are optimized. Thus, after the optimization, we have${\hat{\mathrm{GEE}}}^{\left(2\right)}({\left\{{p}_{1}^{1}\right(\ell \left)\right\}}_{\ell =1}^{L},{\left\{{p}_{2}^{\left(1\right)}\right(\ell \left)\right\}}_{\ell =1}^{L},{\left\{{p}_{3}^{\left(0\right)}\right(\ell \left)\right\}}_{\ell =1}^{L},\dots ,{\left\{{p}_{K}^{\left(0\right)}\right(\ell \left)\right\}}_{\ell =1}^{L})$, and it holds${\hat{\mathrm{GEE}}}^{\left(2\right)}\ge {\hat{\mathrm{GEE}}}^{\left(1\right)}$. It is seen that cyclically iterating this procedure originates a sequence of values of${\hat{\mathrm{GEE}}}^{\left(n\right)}$ which is nondecreasing. As a consequence, since$\hat{\mathrm{GEE}}$is upperbounded with respect to the transmit powers, Algorithm 1 will eventually converge. □
Now, in order to complete the resource allocation design, the solution to problem (17) remains to be tackled. Such a problem is not convex, because the objective is not concave, but it is possible to recast it as a convex problem without loss of optimality, by exploiting the result in the following proposition. Next, we will also provide an algorithm to solve problem (17) that employs the alternating maximization technique [35], rather than using the convex reformulation.
Proposition 3
Moreover, the objective of (18) has a unique maximizer, which lies in its concave region.
Proof
Hence, the thesis. □
Thus, Proposition 3 allows to reformulate the nonconvex problem (18) as a convex one by restricting the problem domain to the concave region of the objective function, which can be done by simply imposing the additional constraints${p}_{k}\left(\ell \right)\le {\stackrel{\u0304}{p}}_{k}\left(\ell \right)$for all ℓ=1,…,L in (18). This causes no loss of optimality, since the global maximum has been proved to lie in the concave region of the objective function.
Equipped with this result, the formal alternating maximization algorithm to solve problem (18) can be stated as follows
Algorithm 2 Solution of problem (18)
Similarly as for Algorithm 2, convergence in Algorithm 2 is declared when the value of the objective of (18) after two successive outer loops is below a given threshold. Convergence of Algorithm 2 can be proved with similar algorithms as for Algorithm 1.
From (31), it also follows that for large P_{max,k}, Algorithm 2 is guaranteed to converge to the global solution of (18) in L iterations. Indeed, for large P_{max,k}, after L iterations of the alternating maximization, the resulting transmit powers will be${p}_{k}\left(\ell \right)={p}_{k}^{\ast}\left(\ell \right)$, which is the global maximizer of solution of (18). In Section 6, the performance obtained using Algorithm 2 to solve (18) will be contrasted to that achieved by solving (18) through its convex reformulation.
3.3 The proposed algorithm as a potential game
In this section, we will briefly provide a different look on the proposed algorithm, showing how it fits into the framework of game theory and in particular of potential games. Let us first give some details on noncooperative games and potential games.
with s_{ k }and s_{−k}being the kth player’s strategy and set of the other player’s strategies, respectively. The coupled problems (32) are usually referred to as bestresponse dynamics (BRD), because for all$k\in \mathcal{K}$, in the kth iteration, given the strategies of the other players s_{−k}, player k responds by choosing his own strategy s_{ k }in order to maximize his own utility function. Each fixed point of (32), if any, is termed Nash equilibrium (NE). At an NE, no user can unilaterally improve its own utility by taking a different strategy, thus implying that each user, provided that the other users’ strategies do not change, is not interested in changing his own strategy. In general, given a generic strategicform game, convergence of the BRD to an NE is not guaranteed, even if one or more NEs exist.
The function V is called the potential function of the game. A very attractive property of potential games is that at least one NE is guaranteed to exist and that the BRD always converges to an NE, provided the potential function is upperbounded. In our scenario, it can be seen that the distributed resource allocation algorithm can be seen as a potential game${\mathcal{G}}_{\mathrm{pot}}$, with the mobile users as players, with potential function V given by (21) and utility functions given by u_{ k }=V−C_{ k }, for all k=1,…,K, with C_{ k }being the additive constant that appears in (56). Thus, the resource allocation policy obtained at the fixed point of Algorithm 1 can be regarded as an NE of${\mathcal{G}}_{\mathrm{pot}}$.
4 Energyefficient resource allocation in saturated networks with fairness constraint
and we notice that, as already anticipated, (35) does not contain the users’ channels realizations and therefore can be employed at the transmitters’ side for resource allocation purposes.
and the following proposition holds.
Proposition 4
Iterations (39) are guaranteed to converge to an equilibrium in K iterations.
Proof
Since multiuser interference is averaged out in the asymptotic SINR, we infer that the K problems in (39) can be decoupled and independently solved. Therefore, after all of the K users have solved their corresponding maximization problem in (39), a fixed point is reached. □
We remark that the solution to the generic kth problem in (39) can be obtained by means of the alternating maximization algorithm, in a similar fashion as for Algorithm 2.
5 Distributed energyefficient resource allocation in relayassisted systems
This introduces a coupling between the relay and the mobile users which is not always easy to manage in distributed systems like an interference network. Instead, the normalization approach has the advantage of making the relay completely independent of the mobile users, ensuring that the relay amplifier never goes into saturation without having to enforce additional constraints on the transmitters.
wherein g_{ j }(ℓ) denotes the vector channel between the relay and receiver j, modeled as a realization of a zeromean Gaussian random vector with covariance matrix${\mathit{R}}_{{g}_{j}}\left(\ell \right)$. The main difference of the considered scenario with respect to onehop systems is that the useful power p_{ k }(ℓ) also appears at the denominator of the SINR, whereas in the onehop case the SINR was simply proportional to the useful power.
Finally, defining for notational convenience the quantities${R}_{{g}_{{a}_{k}}}\left(\ell \right)=\left({\mathrm{tr}}^{2}\left({\mathit{R}}_{{g}_{{a}_{k}}}\right(\ell \left)\right)+2\mathrm{tr}\left({\mathit{R}}_{{g}_{{a}_{k}}}\right(\ell \left){\mathit{R}}_{{g}_{{a}_{k}}}^{H}\right(\ell \left)\right)\right)$and${z}_{k}\left(\ell \right)={\sigma}_{r}^{2}{\sigma}_{{h}_{k}}^{2}\left(\ell \right)\left({a}_{R}^{2}\left(\ell \right){R}_{{g}_{{a}_{k}}}\left(\ell \right)+{\sigma}^{2}\mathrm{tr}\left({\mathit{R}}_{{g}_{{a}_{k}}}\right(\ell \left)\right)\right)$, for all k=1,…,K, the average SINR enjoyed by the kth user in its assigned receiver on subcarrier ℓ is expressed as
Next, we follow the approach of Section 3 and consider the optimization problem (6) with (16) as objective function and the new SINR expression (48). The variables to optimize are not only the users’ powers but also the relay amplification factors${\left\{{a}_{R}^{2}\right(\ell \left)\right\}}_{\ell =1}^{L}$. With respect to the choice of (16) as objective in relayassisted systems, one remark is in order. The main reason to optimize the EE in wireless networks is to limit the energy consumption of batterypowered terminals, in order to prolong their lifetime. However, fixed relays are usually linked to the electrical supply network and therefore have a virtually indefinite lifetime, which makes the investigation of energyefficient relaying protocols of little practical interest. For this reason, the power consumed by the relay will not be included in the denominator of (48) and the scenario to be tackled is that in which the relay always employs its full power to maximize (16) in order to help the mobile nodes improve their own energysaving capabilities. Then, a similar resource allocation algorithm as Algorithm 1 can be stated as follows.
Algorithm 3 Distributed resource allocation in relayassisted systems
5.1 Relay optimization
The following proposition holds.
Proposition 5
The objective of problem (49) is a logconcave function of${\left\{{a}_{R}^{2}\right(\ell \left)\right\}}_{\ell =1}^{L}$.
Proof
□
As a consequence, the relay amplification factors on each subcarrier can be found by solving a convex problem equivalent to (49), which is obtained by simply considering the logarithm of the objective.
5.2 Transmitters optimization
In order to solve problem (17) for all k=1,…,K, the following proposition is provided, which extends the results of Proposition 3 to the relayassisted scenario.
Proposition 6
Moreover, the objective of (52) has a unique maximizer, which lies in its concave region.
Comparing (58) with (23), it follows that, up to the constant term$\frac{{\mu}_{k}\left(\ell \right)}{{\alpha}_{k}\left(\ell \right)}$, for all ℓ=1,…,L, f_{ k }(p_{ k }(ℓ)) is formally equivalent to the function g_{ k }(p_{ k }(ℓ)) given by (23), which was encountered in the proof of Proposition 3. Consequently, since the constant term$\frac{{\mu}_{k}\left(\ell \right)}{{\alpha}_{k}\left(\ell \right)}$vanishes when computing the derivatives of (58), and since the properties that were shown to hold for g_{ k }(p_{ k }(ℓ)) in the proof of Proposition 3 were derived assuming no particular expression for the coefficients a_{ k }(ℓ) and c_{ k }(ℓ), the two functions f_{ k }(p_{ k }(ℓ)) and g_{ k }(p_{ k }(ℓ)) will enjoy similar properties and the line of reasoning employed in Proposition 3 can be replicated here to obtain the thesis. □
Therefore, similarly to the onehop case, also in the relayassisted scenario, it is possible to recast the generic kth problem in (6) in convex form without loss of generality by simply restricting the feasible set to the concave region of the objective. Moreover, for applications in which computational complexity is a major issue, a similar algorithm as Algorithm 2 can be devised resorting to the alternating maximization technique, in order to convert problem (17) into a sequence of scalar problems.
6 Numerical results
ρ = 0, average number of outer loops needed for Algorithm 1 to converge versus the number of active users
Instantaneous CSI  Statistical CSI  

K=2  4.90  4.25 
K=4  7.59  6.25 
K=6  9.17  7.39 
K=8  10.49  8.39 
K=10  11.54  9.29 
K=12  12.63  10.05 
K=14  13.85  10.80 
evaluates the performance of Algorithm 1 in terms of the minimum EE, over the users, versus the number of active users. Otherwise stated, the shown performance metric is mink EE_{ k }, with EE_{ k }being the standard EE (8), achieved at the fixed point of Algorithm 1. It is to be stressed that the instantaneous EE has been plotted, meaning that once Algorithm 1 has converged, the resulting transmit powers have been used to evaluate the instantaneous achieved SINR (2) and then to compute the EE (8). The following scenarios have been contrasted:

Algorithm 1 implemented by solving the generic kth problem of (17) according to Algorithm 2

Algorithm 1 implemented by solving the generic kth problem of (17) by means of its convex reformulation.

As a benchmark, the minimum EE obtained when Algorithm 1 is run with perfect, instantaneous CSI is reported.

Initial minimum EE resulting from a random power allocation over the available subcarriers. This scenario is shown for comparison purposes, since it represents the minimum EE before the resource allocation scheme comes into play.
Results clearly show that the proposed games with statistical CSI largely improve the initial EE while suffering a limited gap with respect to the instantaneous CSI benchmark, even for very high network loads. It is also interesting to note how the performance of Algorithm 1 are virtually identical when problem (17) is solved by means of Algorithm ?? and by its convex reformulation, thus indicating that the lowcomplexity Algorithm 2 can effectively substitute standard numerical algorithms.
Figure 3 considers a similar scenario, with the difference that the shown performance metric is the social welfare function (21). Similar remarks as for Figure 2 hold.
Table 1 contrasts the average number of outer loops needed for Algorithm 1 to reach convergence when statistical and instantaneous CSI is available. As expected, the number of iterations increases with the network load, but it is still satisfactory even for high network loads. Also, it is seen that the number of required iterations with statistical CSI is only slightly lower than that with perfect CSI.
7 Conclusions
This paper has dealt with the problem of distributed resource allocation in the uplink of a multicarrier interference network with multiple receive antennas. A performance metric that trades off between the need for improved overall EE and the need for a fair resource allocation has been proposed, and distributed resource allocation algorithms that are guaranteed to converge to a stable equilibrium have been devised, relying only on statistical CSI at the transmitters’ side. Both onehop and twohop networks have been considered. The proposed algorithms have an affordable computational load, since they only require the solution of convex problems. Individual EE maximization in onehop saturated network in which the number of users and of receive antennas in each receiver grow large with a fixed ratio has also been tackled. Also in this case, a distributed resource allocation algorithm that converges to a stable equilibrium has been designed. Finally, numerical results have been provided to assess the performance of the proposed procedures.
Endnotes
^{a} A function is said to be Sshaped if there exists a point below which it is convex and above which it is concave.
^{b} The forwarded signal is transmitted on an orthogonal channel, usually by means of frequency or time division duplex, to avoid interference with the incoming signal from the users.
Declarations
Acknowledgements
The work of Alessio Zappone has received funding from the German Research Foundation (DFG) project CEMRIN, under grant ZA 747/11. The work of Giuseppa Alfano, Stefano Buzzi, and Michela Meo has received funding from the European Union Seventh Framework Programme (FP7/20072013) under grant agreement no. 257740 (Network of Excellence "TREND"), from the Regional Project MASP, Regione Piemonte, Torinowireless 2009, and by the Network of Excellence NEWCOM ♯, grant agreement no. 318306.
Authors’ Affiliations
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