A distributed power control and mode selection algorithm for D2D communications
 Norbert Reider^{1}Email author and
 Gabor Fodor^{2, 3}
https://doi.org/10.1186/168714992012266
© Reider and Fodor; licensee Springer. 2012
Received: 2 September 2011
Accepted: 5 July 2012
Published: 21 August 2012
Abstract
Devicetodevice (D2D) communications underlaying a cellular infrastructure has recently been proposed as a means of increasing the resource utilization, improving the user throughput and extending the battery lifetime of user equipments. In this article we propose a new distributed power control algorithm that iteratively determines the signaltonoiseandinterferenceratio (SINR) targets in a mixed cellular and D2D environment and allocates transmit powers such that the overall power consumption is minimized subject to a sumrate constraint. The performance of the distributed power control algorithm is benchmarked with respect to the optimal SINR target setting that we obtain using the Augmented Lagrangian Penalty Function method. The proposed scheme shows consistently near optimum performance both in a singleinputmultipleoutput and a multipleinputmultipleoutput setting. We also propose a joint power control and mode selection algorithm that requires single cell information only and clearly outperforms the classical cellular mode operation.
Keywords
Introduction
Devicetodevice (D2D) communications in cellular spectrum supported by a cellular infrastructure holds the promise of three types of gains. The reuse gain implies that radio resources may be simultaneously used by cellular as well as D2D links thereby tightening the reuse factor even of a reuse1 system [1–3]. Secondly, the proximity of user equipments (UE) may allow for extreme high bit rates, low delays and low power consumption [4]. Finally, the hop gain refers to using a single link in the D2D mode rather than using an uplink and a downlink resource when communicating via the access point in the cellular mode. Additionally, D2D communications may increase the reliability of cellular communications [5] and also facilitate new types of wireless peertopeer [3, 6, 7] and multicast services [8].
Although the idea of enabling D2D communications as a means of relaying in cellular networks was proposed by some early works on ad hoc networks [9, 10], the concept of allowing local D2D communications to (re)use cellular spectrum resources simultaneously with ongoing cellular traffic is relatively new [2, 3, 11, 12]. Because the nonorthogonal resource sharing between the cellular and the D2D layers has the potential of the reuse gain, proximity gain and hop gain and at the same time increasing the resource utilization [13–15], D2D communications underlaying cellular networks has received considerable interest in the recent years.
A series of articles analyzes and evaluates the single (isolated) cell scenario in a singleinputsingleoutput (SISO) system to provide some basic insight into the impact of power control and resource (e.g. OFDM resource block) allocation [16–19]. The multicell problem scenario is considered in, for example [20], that assumes that the base station (BS) has all the involved channel state information (CSI) to select the optimal resource sharing mode (D2D mode reusing cellular resources, D2D mode using orthogonal resources and cellular mode in which the D2D pair communicates through the cellular BS). The heuristic mode selection (MS) algorithm proposed in [20] uses probing signals between the D2D transmitter and receiver to estimate the interference plus noise power and the BS has the task to estimate the transmit power, SINR and throughput in each possible communication modes on a small time scale matching with that of the transmission time interval. As stated by Doppler et al. [20], their proposed method has significant signaling load though it is expected to be feasible in low mobility scenarios. In other articles dealing with MS [15–17], the problem is addressed as finding the optimal mode for communication in terms of highest achieved rate, which requires the evaluation of the rate in all of the considered communication modes. Xiao et al. [18] proposed heuristics for joint subcarrier allocation, power control and MS to minimize the total downlink transmission power in a singlecell SISO system.
Gu et al. [21] studied a multicell system focusing on a SISO power control scheme that helps minimize the interference from the D2D layer to the cellular users and assuming that D2D users operate in D2D mode reusing cellular resources. D2D communication in MIMO systems is considered in [22], where interferenceavoiding precoding schemes are proposed for downlink MIMO transmissions in the presence of intracell D2D links. In [23], a new interference management strategy is proposed to enhance the overall capacity of cellular networks and D2D systems when the BS equipped with multiple antennas enables multiple cellular UEs to communicate simultaneously with the help of MIMO spatial multiplexing techniques.
Since the main motivation and justification of allowing D2D communications in cellular spectrum is ultimately to harvest some capacity, sumrate or sumpower gain, many articles apply optimization techniques to explore the potential of cellular D2D communications [14–16, 19]. These works provide important reference cases when the assumption can be made that the BS is aware of the CSI not only between transmitterreceiver pairs, but also of the interference links, such as, for example the state of the link between the D2D transmitter and the cellular receiver (BS) and/or the cellular transmitter (e.g. cellular UE) and the D2D receiver.
Typically, state of the art works give priority to the cellular users or avoids or constraints the interference caused by the D2D users to the cellular layer, see for example [15–17, 22–26]. However, it can be argued that D2D traffic should be treated near equally to the cellular traffic as long as fairness between all cellular spectrum users (i.e. cellular and D2D users) are handled [27, 28], since they all use cellular spectrum under operator controlled charging conditions.
In this article, our purpose is to propose and study the joint performance of a practically viable power control and MS algorithm applicable in multicell cellular systems supporting D2D communications, such that the algorithms use only limited CSI. To this end, we only require that the receiver nodes can estimate (measure) the covariance of the total received interference and feed it back to their respective transmitters. This piece of information is then used by the transmitters in a distributed fashion to adjust their respective transmit powers such that some predefined SINR targets are reached. Next, this basic algorithm can be optionally combined with an SINR target setting algorithm that allows to minimize the overall used power subject to some sum rate target such that a minimum link quality is also guaranteed for both the cellular and the D2D transmission links. Finally, we also propose a practical MS algorithm that only requires the CSI (specifically the large scale fading) information of the useful and interfering links in the own cell.
The current article is a substantially revised and extended version of [30]^{a}. First, we revised the distributed power control algorithm (Algorithm 1) such that it is based on the measured covariance of the total received interference and noise and investigate the impact of the measurement error. Second, the description of the optimum power allocation method using the augmented Lagrangian penalty function (ALPF) scheme has been revised and illustrated through a specific numerical example. Also, in this article, we provide the detailed derivations of the steps needed in the SINR target setting scheme (Algorithm 2). Third, we introduce a practical MS algorithm that requires only average CSI information from the own cell. Furthermore, new numerical results are presented to evaluate the potential gains of D2D communications under strong and weak intercell interference situations. Finally, the performance of the distributed power control scheme with and without adaptive SINR target adjustment is evaluated jointly with the proposed MS algorithm in various parameter configurations of a 7cell system.
Our scheme does not consider the scheduling or pairing problem that is concerned with selecting the specific cellular users and D2D pairs and allocating OFDM resource blocks or subcarriers to them [13, 15, 27, 31–33]. Therefore, we believe that our work can be an efficient complement to these resource allocation and pairing schemes.
We structure the article as follows. The next section describes our system model and formulates the D2D power control problem as an optimization task. Next, in Section “An iterative D2D power control scheme”, we propose an iterative power control scheme to meet predefined SINR targets. A second algorithm is presented in Section “Determining the optimum SINR target” that aims to set the SINR targets that help to minimize the overall used power in the system. In Section “Mode selection”, the proposed MS algorithm is presented that relies on single cell information and dynamically selects between cellular and D2D communication modes. Section “Numerical results” discusses numerical results and Section “Conclusions” highlights our findings.
Throughout the article, we use the following notations. A^{−1}, A^{T} and A^{H}denote the pseudoinverse, the transpose and the conjugate transpose of matrix A, respectively. {A}^{(i,j)} is the (i,j)^{th} element of matrix A, while diag(a_{1},…,a_{ N }) denotes a N × N diagonal matrix whose diagonal elements are the scalars a_{1},…,a_{ N }. The absolute value of a real or complex number z is denoted by z. Furthermore, trace(·) and $\mathbb{E}(\xb7)$ represent the trace and the expectation operations of matrix A, respectively.
System model
Modeling the received signal

N_{ t }is the number of transmit antennas and N_{ r }is the number of receive antennas;

${\alpha}_{k,j}=\sqrt{{P}_{j}{d}_{k,j}^{{\rho}_{k,j}}{\chi}_{k,j}/{N}_{t}}$ is a scalar coefficient depending on the total transmit power P_{ j }for user j, the lognormal shadow fading χ_{k,j}and distance d_{k,j}between the k th receiver and the j th transmitter with path loss exponent ρ_{k,j}. The values of ρ_{k,j}and χ_{k,j}depend on the transmitter and receiver being a transmitter UE, a receiver UE or a cellular access point respectively, the specific environment (e.g. indoor or outdoor deployment, femto or macro type of access point), etc.

${\mathbf{x}}_{k}\in {\mathbb{C}}^{{N}_{t}\times 1}$ is the data vector that is assumed to be zeromean, normalized and uncorrelated, $\mathbb{E}\left({\mathbf{x}}_{k}{\mathbf{x}}_{k}^{\u2020}\right)={\mathbf{I}}_{{N}_{t}}$;

H_{k,j}denotes the (N_{ r } × N_{ t })channel transfer matrix; and

T_{ k } is the UEk(N_{ t } × N_{ t })diagonal power loading matrix. To keep the total transmit power constant, T_{ k }must satisfy$\text{trace}\left({\mathbf{T}}_{k}{\mathbf{T}}_{k}^{\u2020}\right)=\sum _{i=1}^{{N}_{t}}{\left\{{\mathbf{T}}_{k}\right\}}^{(i,i)}{}^{2}={N}_{t}\phantom{\rule{1em}{0ex}}\forall k;$
n_{ k } is a N_{ r } × 1 additive white Gaussian noise vector at the k th receiver with zero mean and covariance matrix ${\mathbf{R}}_{{n}_{k}}=\mathbb{E}\left({\mathbf{n}}_{k}{\mathbf{n}}_{k}^{\u2020}\right)={\sigma}_{n}^{2}{\mathbf{I}}_{{N}_{r}}\forall k$.
It is easy to show that v_{ k } is zeromean with covariance ${\mathbf{R}}_{{v}_{k}}={\mathbf{R}}_{{z}_{k}}+{\mathbf{R}}_{{n}_{k}}$.
MMSE receiver error matrix and the effective SINR
where ${\mathbf{R}}_{{H}_{k}}={\alpha}_{k,k}^{2}{\mathbf{H}}_{k,k}^{\u2020}{\mathbf{R}}_{{v}_{k}}^{1}{\mathbf{H}}_{k,k}$, see e.g. ([37], Chapter 12).
Summary
In this section we defined the multicell MIMO received signal model (2) and, assuming a linear MMSE receiver, derived the associated effective SINR (γ_{k,s}) for each stream of the received signal. Equations (5) and (6) are important because they capture the dependence of the SINRs on the transmission powers of the own UE and the interfering UEs (both at an access point and at a receiving UE of a D2D pair) through the ${\mathbf{R}}_{{H}_{k}}$’s and the ${\mathbf{R}}_{{v}_{k}}$’s. Thus, these relations serve as the basis for the optimization problems of the next section.
An iterative D2D power control scheme
denotes the effective interference after MMSE processing and {·}^{(i,j)} denotes the operation of acquiring the matrix element of the i th row of the j th column. In [36], a heuristic algorithm for distributing the transmit power over different streams was presented. By inverting Equation (7) for fixed SINR targets, the algorithm finds a near optimal (sum power minimizing) power loading matrix for these given SINR targets assuming perfect knowledge of the own and cross channel matrices H_{k,j}.
Unfortunately, in the mixed cellular and D2D communications scenario, the availability of the cross channel matrices at the transmitters cannot be assumed, because that would require extensive reference signal processing and channel quality information reporting. Therefore, in this article, we relax the assumption on the knowledge of all the H_{k,j} channel matrices at all transmitters. Our assumption instead is that Receiverk estimates the covariance of the total received signal and noise (Φ_{ k }) and feeds it back to its transmitter. We further assume that Transmitterk knows its channel to its receiver (H_{k,k}), which is reasonable considering that in practice a D2D pair typically communicates over a bidirectional channel and that the D2D link can be expected to operate in a time division duplex (TDD) mode [3, 24].
Transmitterk can then calculate the effective interference ζ after the MMSE processing based on (8).
The covariance estimation based iterative power control algorithm is summarized by the pseudo code of Algorithm 1. (In practice, the receiver can estimate the covariance matrix of the received interferenceplusnoise and feed back this reduced covariance matrix ${\mathbf{\Phi}}_{k}^{\mathrm{red}}$ as defined in Algorithm 1.) Algorithm 1 iteratively adjusts the power loading matrix T_{ k }such that the MIMO streams that suffer from higher effective interference ζ are allocated higher transmit power, since the given fixed SINR target $\mathit{\Gamma}\triangleq \text{diag}\left({\gamma}_{1}^{\mathrm{tgt}},\dots ,{\gamma}_{K}^{\mathrm{tgt}}\right)$ where ${\gamma}_{k}^{\mathrm{tgt}}$ is the assumed given SINR target at Receiverk is set equal to all streams of Transmitterk. Without unequal power loading, when the “weakest” stream’s SINR is raised to the target, the stronger streams tend to overshoot the SINR target and thereby to waist transmit power. The transmit power itself (P_{ k }) is determined by the MIMO stream that requires the highest transmit power (proportional to the effective interference and target SINR (${\gamma}_{k}^{\mathrm{tgt}}$)).
Algorithm 1 Iterative transmit power and power loading optimization
Given t = 0(iteration number), P_{tot}, ϵ_{gap} and ${\mathbf{T}}_{k}^{\left(0\right)}={\mathbf{I}}_{{N}_{t}}\phantom{\rule{2.77695pt}{0ex}}\forall \phantom{\rule{2.77695pt}{0ex}}k$. {·}^{(i,j)} denotes the operation of acquiring the matrix element of the i^{ th }row of the j^{ th }column.
Initialize SINR targets ${\mathit{\Gamma}}^{\left(0\right)}=\text{diag}\left({\gamma}_{k}^{\mathrm{tgt}}\right)$, where ${\gamma}_{k}^{\mathrm{tgt}}$ is the assumed given SINR target at Receiverk, and initial transmit powers p^{(0)}.
 1.
t = t + 1.
 2.
for k = 1 to K do
Receiverk feeds the estimated (measured) Φ_{ k }back to Transmitterk;
end
until $\phantom{\rule{1em}{0ex}}\mid {P}_{k}^{\left(t\right)}{P}_{k}^{(t1)}\mid \le {\u03f5}_{\mathrm{gap}},\phantom{\rule{1em}{0ex}}\forall \phantom{\rule{2.77695pt}{0ex}}k;$
Determining the optimum SINR target
Determining the optimum SINR target is useful for benchmarking purposes. For smaller systems, in which the number of interfering transmitters is limited, it is possible to determine the optimum SINR targets by the method we apply in this section. For larger systems, the distributed algorithm of the next section is more practical. We note that, in this section, we assume full and perfect channel knowledge at each transmitter.
Notation and assumptions for optimum SINR target setting
with each MSk. In what follows, we search for SINR targets ${\gamma}_{k}^{\text{tgt}}$ which are feasible for the lowerbound (and hence for each individual stream) and $\mathit{\Gamma}\triangleq \text{diag}({\gamma}_{1}^{\mathrm{tgt}}\dots {\gamma}_{K}^{\mathrm{tgt}})$.
Minimizing the sum power under predetermined fixed SINR targets
This observation is the basis for determining the SINR targets such that the sum transmit power is minimized, as shown in the next section.
The problem of optimal SINR target selection
in the optimization variables $\mathit{\Gamma}\triangleq \text{diag}({\gamma}_{1}^{\mathrm{tgt}}\dots {\gamma}_{K}^{\mathrm{tgt}})$ (SINR targets) and p (transmit power).
Solution approach: employing the ALPF
where μ is the Lagrange multiplier and ϵ is the so called penalty parameter.
It can be shown that if the optimum Lagrange multipliers are known, the solution to this unconstrained problem corresponds to the solution of the original problem (24) regardless of the value of the penalty parameter ϵ, see e.g. ([38], Chapter 9). Since we obviously do not know the value of the Lagrange multiplier, we start with an arbitrary value (e.g. zero) and develop a procedure that moves the multiplier closer to its optimum value. This procedure is detailed in the following subsection.
Updating the lagrange multipliers
A numerical example
Using this relationship, the M_{ ij } parameters are expressed as the functions of ${\gamma}_{1}^{\mathrm{tgt}}$ and ${\gamma}_{2}^{\mathrm{tgt}}$ (see Appendix 3). That is, for a specific capacity target c_{m}, p and the sum of its components are expressed as a twovariable function of ${\gamma}_{1}^{\mathrm{tgt}}$ and ${\gamma}_{2}^{\mathrm{tgt}}$. Using (25), it is straightforward to find the stationary points of the unconstrained problem and, ${\gamma}_{1}^{\mathrm{tgt}}$ and ${\gamma}_{2}^{\mathrm{tgt}}$. Using to find the local optimum solutions (that is, the local minimum points) of (24). In our Mathematica^{ ® }implementation, we found that in all considered practically relevant examples, a simple heuristic can then easily identify the near optimum solution (see also the numerical section).
In the following, we describe the steps of the complete optimization process implemented in Mathematica^{ ® }and detailed in Algorithm 4 (see Appendix 4: the process of optimal SINR target selection). In Steps 1 and 2, we drop the cellular UE (UE1) and the D2D pair according to a surface uniform distribution. Then, the signal model is recalculated (see Steps 3–7) and the sum power vector is expressed in the function of the SINR targets (Steps 8 and 9). In Step 10, the ALPF optimization is executed using the inits = 0 vector as initial points. The variables maxIter and convTolerance denote the maximum number of iterations performed by ALPF and the convergence tolerance specifying the maximum value by which the constraints can be violated.
Step 11 executes an other optimization using the NMinimize builtin Mathematica^{ ® }method which applies the NelderMead (also called as the downhill simplex) heuristic approach [39] (i.e., it is not a true global optimization algorithm). As opposed to the gradient based ALPF, the NelderMead technique is a direct search method which does not use derivative information and has the advantage to better tolerate the presence of noise in the function and constraints at the cost of slow convergence time [39]. We use the output of Step 11 as the starting points of another ALPF execution in Step 13. Then, we compare the solutions of Step 10 and 13, and accept the results if both ALPF optimizations converged within maxIter iterations and returned the same solutions (see Steps 14 and 15) otherwise the Monte Carlo drop is discarded and a new one is drawn.
We note that the optimization process of Algorithm 4 does not ensure true global optimum in all cases, though it turned out to be practically useful in finding reference points in all of the examined cases.
Iterations of the ALPF optimization method in an example scenario
Iteration  Points$\left(\begin{array}{c}{\gamma}_{1}^{\text{tgt}}\\ {\gamma}_{2}^{\text{tgt}}\end{array}\right)$  Objective function  Lagrange multipliers  Max. violation 

0  $\left(\begin{array}{c}0\\ 0\end{array}\right)$  357.133  $\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right)$  0 
1  $\left(\begin{array}{c}6.48572\\ 1.47024\end{array}\right)$  25.4479  $\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right)$  0.0659845 
2  $\left(\begin{array}{c}6.48727\\ 1.47633\end{array}\right)$  25.4411  $\left(\begin{array}{c}\phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}0\\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}0\\ 1.31969\end{array}\right)$  0.0684702 
3  $\left(\begin{array}{c}6.47661\\ 1.44415\end{array}\right)$  25.4818  $\left(\begin{array}{c}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}0\\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}0\\ 1.31969\end{array}\right)$  0.0548636 
4  $\left(\begin{array}{c}6.46081\\ 1.39942\end{array}\right)$  25.555  $\left(\begin{array}{c}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}0\\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}0\\ 1.31969\end{array}\right)$  0.035204 
5  $\left(\begin{array}{c}6.4493\\ 1.36862\end{array}\right)$  25.6172  $\left(\begin{array}{c}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}0\\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}0\\ 1.31969\end{array}\right)$  0.0211488 
6  $\left(\begin{array}{c}6.44168\\ 1.34844\end{array}\right)$  25.6633  $\left(\begin{array}{c}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}0\\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}0\\ 1.31969\end{array}\right)$  0.0117258 
7  $\left(\begin{array}{c}6.43088\\ 1.3186\end{array}\right)$  25.7391  $\left(\begin{array}{c}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}0\\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}0\\ 6.39165\end{array}\right)$  <0.01 
A distributed algorithm to set the SINR targets
The insight of the previous (and as we will see the numerical) section is that setting the SINR targets to a uniform value that is suitable for both cellular and D2D links is nonoptimal due to several reasons. First, due to the presence of D2D transmitters and receivers, the distances between any transmitter and receiver can vary between a close proximity and the cell diameter resulting in extremely large SINR fluctuations. Note that this observation holds for both the D2D and the cellular traffic, since a D2D transmitter may get close to the cellular BS. Specifically, to minimize the sum power with respect to a sum capacity target, strong (low path loss) links must be granted high SINR targets, while weak links must be set to low values. Second, different services (e.g., voice or video streaming) have different quality of service (QoS) requirements and therefore maintaining a minimum (link specific) SINR target for any link is desirable.

It should rely only on large scale fading information;

It should allow for setting a minimum link quality (SINR target) value;

It should reward the transmitters whose transmit power increase yields high capacity increase. This requirement is justified by the intuition (confirmed and illustrated in the numerical section) that higher SINR targets should be granted to links with low pass loss, while “weak” links should be set to their respective minimum SINR target.

It should not require a central entity, but it can assume the availability of large scale fading information to surrounding receivers.
When D2D communications is enabled in cellular spectrum, it is expected that new types of reference signals and associated measurement reporting schemes will be designed to facilitate various RRM algorithms. Therefore, the last assumption is reasonable, since it assumes large scale fading information only.
We propose an algorithm (Algorithm 2) that meets the above requirements by starting from a minimum SINR target and iteratively adjusting them for all links to reach a near optimal power allocation subject to a sum capacity constraint. Algorithm 2 tries to successively increase the SINR targets until a predefined C^{sum}capacity target is reached. In each iteration it increases the SINR target of the one user that contributes the most to the sum capacity increase by calculating a benefit value b_{ k }. More specifically, in Step 1, it estimates a power value ΔP_{ k } that is needed to increase the SINR by a Δvalue for link k, and then calculates the capacity increase corresponding to this increased SINR. The calculation of the power increase is detailed in Appendix 2: derivation of Δ P in Algorithm 2. Next, it computes a benefit value b_{ k }that indicates how beneficial it is to increase the power for link k in terms of bit/sec/Hz/mW, i.e., what is the gain of the increased SINR in capacity for that link. In Step 2, the transmitter can compose a vector b containing the benefit values for all links and then select the link to increase its SINR target which has the highest benefit value. These steps are repeated until the desired sum capacity target C^{sum} is reached.
Algorithm 2 Adaptive SINR target setting
Input: C^{sum},SINR^{min}>0,Δ>1,ρ path loss exponent, ε>0 and ${g}_{k,j}={d}_{k,j}^{\rho}{\chi}_{k,j}$, k = 1,…,K,j = 1,…,J, as in Equation (1) where K and J are the number of receivers and transmitters, respectively.
Output: Γ=diag(γ_{ k }). Given t=0(iteration number), ${\mathbf{b}}^{\left(0\right)}=\phantom{\rule{0.3em}{0ex}}[{b}_{1}^{\left(0\right)},\dots ,{b}_{K}^{\left(0\right)}]=\mathbf{0},$ and ${\gamma}_{k}^{\left(0\right)}={\text{SINR}}_{\mathrm{min}}$, ${p}_{k}^{\left(0\right)}={\gamma}_{k}^{\left(0\right)}\xb7{\sigma}_{n}^{2}/{g}_{k,k}$, k=1,…,K.
 1.
for k=1 to K do
 2.Select user with the highest benefit value as:if $\left({b}_{i}^{\left(t\right)}{b}_{j}^{\left(t\right)}<\epsilon ,\forall i,\forall j,i\ne j\right)$then${\mathrm{bestUE}}^{\left(t\right)}=\text{argmax}\{{g}_{1,1},\dots ,{g}_{k,k}\}$
 3.Update SINR target for the user with the highest benefit as:${\gamma}_{{\mathrm{bestUE}}^{\left(t\right)}}^{(t+1)}={\gamma}_{{\mathrm{bestUE}}^{\left(t\right)}}^{\left(t\right)}\xb7\mathrm{\Delta .}$
 4.Calculate current sum capacity as:${C}^{(t+1)}=\sum _{s=1}^{{N}_{t}}\underset{2}{log}\left(1+{\gamma}_{k}^{(t+1)}\right).$
 5.
t = t+1;
until C^{sum}≤C^{(t)};
An important feature of this algorithm is that if the slow fading information (including path loss and shadowing) is available for all links at all transmitters (g_{k,j},∀k,j), i.e., if the k th cell is aware of the slow fading channel state between its receiver and all the transmitters of the network (g_{k,j},∀j), and all cells exchange this information using slow scale BSBS communications, then each transmitter can execute this algorithm in a distributed fashion, since then each transmitter can calculate the benefit vector by itself. This algorithm is a networkwise optimization in the sense that it uses multicell channel knowledge (slow fading information) to determine the SINR target for a user.
An additional feature of this algorithm is that a minimum SINR can be set for all links (SINR^{ min }), which guarantees a minimum link quality. Setting this parameter to a higher value for all users prevents boosting the best channel only. Later, in Section “7cell system results”, we will use this parameter to ensure that all UEs experience a certain QoS.
The convergence of this algorithm is not analyzed in this article. In practice, the maximum number of iterations would be limited and the target capacity could be adjusted. In the evaluated scenarios, the numerical results show that the proposed method converges.
Summary
While Section “An iterative D2D power control scheme” proposed a heuristic algorithm that allocates transmit powers and tunes the power loading matrix at the transmitter such that a predefined SINR target vector is reached, in Sections “Determining the optimum SINR target” and “A distributed algorithm to set the SINR targets” we considered the problem of setting the SINR targets that minimize the sum power subject to a target capacity constraint. To this end, we proposed a heuristic algorithm that requires the slow changing path loss and shadowing matrix knowledge at each transmitter. The availability of this information can be assumed in systems with an interBS backhaul network or with a central node such as a radio network controller.
Mode selection
In the development of the MS algorithm, we assume that exactly one cellular UE is allocated on an OFDM resource block, that is without D2D communications, intracell orthogonality is maintained. We also assume that at most one D2D link is allocated to a resource block that is used by a cellular UE, meaning that on any one OFDM resource block, there are at most two links (one cellular and one D2D) multiplexed.
It is intuitively clear that for a given D2D candidate the benefit of direct mode communication (as compared to communicating through the BS) depends on the geometry of the D2D pair and the UEs in the own cell and neighbor cells using the same resource blocks. MS is a D2D specific function that allows the BS to dynamically adjust the characteristics of the D2D link and to change the communication mode (cellular mode: via the BS or D2D mode: via the direct link) of two communicating UEs. MS plays a similar role for D2D communications as handover does for traditional cellular communications in the sense that the D2D transmitter can switch its transmission between the D2D receiver and its serving BS.

It should rely only on large scale fading information;

It should rely on information available in the own cell only rather than trying to coordinate MS decisions among multiple cells. We justify this requirement by noting that intercell interference can be addressed by proper resource allocation (scheduling) and power control and by arguing that multicell MS would lead to unacceptable complexity in real systems.

It should take into account the geometry of the D2D link and the cellular UE that are multiplexed onto the same resources (physical resource blocks), in terms of the large scale fading of the useful as well as interfering links.

It should preferably be executable independently of the transmit power setting to mitigate the complexity of joint power control and MS.

${g}_{{\mathrm{BS}}_{l},{\mathrm{CellUE}}_{l}}={d}_{{\mathrm{BS}}_{l},{\mathrm{CellUE}}_{l}}^{\rho}\xb7{\chi}_{{\mathrm{BS}}_{l},{\mathrm{CellUE}}_{l}}$: Large scale fading between the cellular UE and its serving BS of Celll (see g1 link in Figure 5);

${g}_{{\mathrm{RxD}}_{l},{\mathrm{TxD}}_{l}}={d}_{{\mathrm{RxD}}_{l},{\mathrm{TxD}}_{l}}^{\rho}\xb7{\chi}_{{\mathrm{RxD}}_{l},{\mathrm{TxD}}_{l}}$: Large scale fading between the D2D transmitter and receiver of Celll (see g2 link in Figure 5);

${g}_{{\mathrm{BS}}_{l},{\mathrm{TxD}}_{l}}={d}_{{\mathrm{BS}}_{l},{\mathrm{TxD}}_{l}}^{\rho}\xb7{\chi}_{{\mathrm{BS}}_{l},{\mathrm{TxD}}_{l}}$: Large scale fading between the D2D transmitter and the BS of Celll (see g3 link in Figure 5);

${g}_{{\mathrm{RxD}}_{l},{\mathrm{CellUE}}_{l}}={d}_{{\mathrm{RxD}}_{l},{\mathrm{CellUE}}_{l}}^{\rho}\xb7{\chi}_{{\mathrm{RxD}}_{l},{\mathrm{CellUE}}_{l}}$: Large scale fading between the cellular UE and the D2D receiver of Celll (see g4 link in Figure 5).
The fourth requirement implies that the MS algorithm should rely on SNR rather than SINR metrics, since the measured SINR at the receivers (D2D receiver or cellular BS) depend on the transmit powers of the interferers (see also our proposed SNR metric in Algorithm 3). Finally we note that the proposed MS algorithm does not consider the hop gain that is described in the Introduction of this article. That is, the MS algorithm is somewhat biased towards favoring the cellular mode, since it disregards the potential hop gain of the D2D mode. Based on these requirements, in this article we propose a simple MS algorithm described by Algorithm 3.
Algorithm 3 Simple MS algorithm based on singlecell knowledge
Input: $\Delta ,\rho ,{\sigma}_{n}^{2},p={p}_{\mathrm{max}}$, number of cells (L), and ${g}_{k,j}={d}_{k,j}^{\rho}{\chi}_{k,j}$, k = 1,…,K,j = 1,…,J, as in Equation (1) where K and J are the number of receivers and transmitters, respectively.
Output: Decision on which mode is preferred (D2D or Cellular) for all cells:
useD2D_{ l }∈{True,False}, l=1,…,L.
Notations:
BS_{ l } the cellular BS of cell l
CellUE_{ l } the cellular UE in cell l
RxD_{ l } the D2D receiver in cell l
TxD_{ l } the D2D transmitter in cell l
 1.The useful (u) signal path loss in Cellular (C) mode is ${g}_{{\mathrm{BS}}_{l},{\mathrm{CellUE}}_{l}}$, hypothetical SNR${\gamma}_{l}^{u,\mathrm{C}}=\frac{p\xb7{g}_{{\mathrm{BS}}_{l},{\mathrm{CellUE}}_{l}}}{{\sigma}_{n}^{2}};$
 2.The useful signal path loss in D2D mode is ${g}_{{\mathrm{RxD}}_{l},{\mathrm{TxD}}_{l}}$, hypothetical SNR${\gamma}_{l}^{u,\mathrm{D}2D}=\frac{p\xb7{g}_{{\mathrm{RxD}}_{l},{\mathrm{TxD}}_{l}}}{{\sigma}_{n}^{2}};$
 3.The interfering (i) signal path loss in Cellular mode is ${g}_{{\mathrm{BS}}_{l},{\mathrm{TxD}}_{l}}$, hypothetical SNR${\gamma}_{l}^{i,\mathrm{C}}=\frac{p\xb7{g}_{{\mathrm{BS}}_{l},{\mathrm{TxD}}_{l}}}{{\sigma}_{n}^{2}};$
 4.The interfering signal path loss in D2D mode is ${g}_{{\mathrm{RxD}}_{l},{\mathrm{CellUE}}_{l}}$, hypothetical SNR$\begin{array}{lcr}{\gamma}_{l}^{i,\mathrm{D}2D}& =& \frac{p\xb7{g}_{{\mathrm{RxD}}_{l},{\mathrm{CellUE}}_{l}}}{{\sigma}_{n}^{2}};\end{array}$
 5.
Select whether Cellular or D2D mode is beneficial to use as:
if $\left(\underset{2}{log}\right(1+{\gamma}_{l}^{u,\mathrm{D}2D})+\underset{2}{log}(1+{\gamma}_{l}^{u,\mathrm{C}})\underset{2}{log}(1+{\gamma}_{l}^{i,\mathrm{D}2D})\underset{2}{log}(1+{\gamma}_{l}^{i,\mathrm{C}})>\Delta )$ then
useD2D_{ l }=True
else useD2D_{ l }=False;
end
The proposed algorithm is based on the geometry of the UEs in the own cell, i.e., the geometry situations in the neighbor cells are not considered. Figure 5 illustrates the idea of Algorithm 3, where the useful and the interference path loss links are shown for a particular D2D candidate pair and a cellular UE in a specific Monte Carlo drop. The useful path loss links are denoted with bold black arrows (g1 and g2), while the interference path loss links (g3 and g4) are marked with dashed blue arrows. The algorithm first calculates hypothetical SNR values for each link according to Step 1–4. The proposed algorithm selects D2D mode for the D2D candidate if the useful links (g1 and g2) are stronger than the interfering links (g3 and g4). More specifically, D2D mode is selected if the hypothetical capacity values corresponding to the useful links are higher than the hypothetical capacity values corresponding to the interfering links plus a Δ value (see Step 5 of Algorithm 3), which is a tunable system parameter measured in bit/sec/Hz. The transmit power value p in Step 1)– 4) is set to an arbitrary positive value. By increasing Δ, the MS algorithm becomes more conservative and selects D2D communication more cautiously. Selecting a negative Δ implies a more frequent D2D MS. This algorithm is not a networkwise optimization in the sense that it uses only single cell slow fading (distance dependent path loss and shadowing) information to determine the communication mode of a cell. An important feature of this algorithm is that it meets Requirement 4 by relying on SNR rather than SINR metrics.
Numerical results
The input parameters used in the simulations
Input parameters  

Inter site distance (ISD)  500 m 
Number of access points (base stations)  2 or 7 
Path loss exponent  3.07 
Shadow fading  Lognormal; st. dev: 5 dB 
Fast fading model  Rayleigh flat 
AWGN noise power  −60 dBm 
Max. per user transmit power  250 mW 
Antenna configurations  1 × 2 SIMO and 2 × 4 MIMO 
Nr. of Monte Carlo experiments  40000 
System operation
 1.
Run the MS algorithm (Algorithm 3) in each cell to select between the cellular and D2D communication modes (i.e, to select the links for transmission) on the time scale of few hundred milliseconds based on large scale fading (distance dependent path loss and shadowing) information of the own cell.
 2.
Execute the adaptive SINR target setting algorithm (Algorithm 2) on the transmission links (selected by MS) to minimize the sum transmit power. The time scale is the same as that of the MS.
 3.
Run the distributed power control scheme (Algorithm 1) to set the transmit power for each link in each transmission slot taking into account fast fading information as well.
The above operation of the system is reasonable, since MS should decide first which links are going to transmit in the next few transmission slots. As discussed, for instance, in [28], the time scale of MS should match that of handover and should rely on large scale fading information only (see also the requirements in Section “Mode selection”). The execution of the SINR target setting algorithm is optional, though significant power can be saved by tuning the SINR target according to the large scale channel conditions while maintaining some fairness criterion as well (see the numerical results of Sections “2cell system results” and “7cell system results”). Finally, the proposed distributed power control scheme combats against fast fading by measuring the covariance of the total received interference and noise in each transmission slot.
2cell system results
We consider two sets of numerical results. The first set focuses on the performance of Algorithm 1 given a fixed set of SINR targets. The second set shows the gains when setting the SINR targets in an optimal or heuristic fashion.
Simulation scenarios
 1.
D2D mode: The two UEs of the D2D pair communicate via a direct link. In this mode, the D2D link uses the same OFDM resource blocks as the UE1 uses to communicate with its serving AP.
 2.
Cellular mode: The two UEs of the D2D pair communicate via the serving AP. In this case the UE1 and UE2 use orthogonal uplink resources (either in the time or in the frequency domain). For example, assuming a time domain separation, during the first period only UE1 transmits to AP1 followed by a period when only UE2 transmits to AP1. (The resources are split equally between UE1 and UE2).
The two performance measures of interest are the sum power for a given sum capacity target (UE1 + UE 2 + UE 3) and the probability that the (fixed or set) SINR targets are infeasible. Some of the simulation parameters are listed in Table 2. Recall that for the SINR target optimization, fast fading is taken into account in the reference (centralized) case, whereas only distance dependent path loss and shadowing are considered in the distributed approach.
Results for predefined SINR targets
The upper graph of Figure 7 shows the sum power results for the 1 × 2 SIMO case. As UE1 moves from its cell center position towards the cell edge, the average sum power (on the three links) required to reach their respective SINR targets gradually increases both when the D2D pair communicates in D2D mode and when they communicate in cellular mode. Recall that in cellular mode, we first assume that only UE1 transmits and then only UE2 transmits to the AP (when only UE2 transmits, the required power is obviously independent from the UE1 position, since UE1 does not transmit). What is important to notice here is that the sum power is always lower (roughly 30% of the average power used in cellular mode) in the D2D mode than in cellular mode due to the reuse and proximity gains in D2D mode.
The lower graph of Figure 7 shows the probability that in a Monte Carlo experiment the SINR targets are infeasible. As expected, the probability of infeasibility increases as UE1 moves towards the cell edge, but this probability is significantly lower (typically half or less) in D2D mode.
Figure 8 shows the sum power and the probability of infeasibility in Scenario 1 for the 2 × 4 MIMO case and setting the SINR target per stream to 4 dB (that is setting the sum capacity target to twice of that required in Figure 7). This high SINR per stream target is basically only feasible when UE1 is in the cell center. Similarly to the 1 × 2 case, the D2D mode between UE2 and its D2D pair is clearly superior to the cellular mode both in terms of sum power and feasibility.
Results for optimal and heuristic SINR targets
We discuss the results when the SINR targets are not fixed, but set optimally or by means of the proposed heuristic SINR target setting algorithm such that the sum rate capacity is the same as in the fixed SINR target case of the previous section (that is 5.44 bps/Hz in the 1 × 2 SIMO case and 2 × 5.44 bps/Hz in the 2 × 4 MIMO case).
Recall from Figure 8 that in this case the fixed SINR targets were typically infeasible. With optimal and heuristic SINR targets, the same sum rate becomes feasible except when UE1 is close to the cell edge. Also the sum power in the feasible drops becomes only a fraction of what is required in the fixed SINR case.
In both the 1 × 2 SIMO and the 2 × 4 MIMO case we also notice that D2D mode provides better performance than cellular mode.
7cell system results
In this network, when all D2D candidates transmit directly, i.e., in D2D mode there are 14 simultaneous transmissions. In this case, we set the fixed SINR target for all links to 2 dB resulting in 19.18 b/s/Hz spectral efficiency. When each cell communicates in cellular mode, we have seven simultaneous transmissions in the whole system and the fix SINR target is set to 7.54 dB in order to achieve exactly the same sum capacity as with pure D2D mode.
Recall from Section “Mode selection”, that we assume that there are at most two links (one cellular and one D2D) multiplexed on a single OFDM resource block. Therefore, just like e.g. [36], we focus on a single resource block (used by at most 3 users), since each resource block of the system bandwidth can be studied in isolation.
Potential of D2D communication
Numerical results with MS
The performance result of the MS algorithm together with 1 dB minimum SINR is shown by the light blue color (“Mode selection− adaptive SINR + F”). As it can be seen, the employment of MS gives some additional gains to D2D mode with minimum SINR. This gain comes from that that the MS algorithm avoids using D2D mode in such cases when, for example, a cellular UE is placed very close to a D2D receiver and would suppress the transmission of the D2D transmitter. In Figure 16, it is clearly visible that MS combined with adaptive SINR target setting can provide superior performance, even when a minimum SINR target is required on all links.
Looking at the lower plot of Figure 16, we can observe that the infeasibility probability is in line with the result of the average sum power results of the upper plot. These results highlight the importance of MS combined with adaptive SINR target setting.
The gain of the MS algorithm comes from the fact that it avoids using D2D mode in cases when the transmission of one layer (D2D or cellular) would be suppressed due to the proximity of the receiver of the other layer. This algorithm can be thought of as an additional sanity check to adapt to realistic situations and avoid using simultaneous transmissions within a cell, i.e., D2D mode when high intracell interference can be expected.
Figure 17, presents empirical cumulative distribution functions (CDF) of the received SINR in different cases in Cell1, which is the cell in the middle among the seven cells. In this figure the D2D candidate operates in D2D mode. This cell is in the worst situation, since it receives the most interference from the neighbors. We use a 1 2 SIMO system where the maximum D2D distance is also limited to 100 m. We focus on the cellular UE position 5, i.e., when the cellular UE is around the same distance from the BS as from the cell edge. We compare four different cases, where the black curve shows the CDF of the received SINR at the receiver device of the D2D pair when fixed SINR targets are set and D2D mode is used in the cell (Cell1). The reason why it is hard to distinguish the black curve (“RxDfixed SINRD2D”) is that all points are at exactly 2 dB as expected, verifying that the power setting algorithm (Algorithm 1) works well. The result is similar to the SINR at the cellular BS (red curve), since 2 dB target SINR is set for the cellular UE as well. The next two curves (green and blue) show the same results when employing adaptive SINR targets (“RxDAdaptive SINRD2D”, “Cellular BSAdaptive SINR D2D”). In this case, we set the minimum SINR to −10 dB. The SINR of the receiver device can be in very wide range from −10 to 30 dB as shown by the green curve, which also confirms that it is hard to set one single SINR target that is optimal or “good enough” for both D2D and cellular modes. The problem of adaptive SINR target is illustrated by the blue curve of Figure 17 (“Cellular BSAdaptive SINR D2D”) where in the 90% of the cases, the SINR at the cellular BS is around or below −10 dB. This means that the algorithm puts this link into outage. There is a need to introduce the concept of the minimum SINR to avoid situations in which one of the transmission links is practically muted.
Computational complexity of the distributed SINR target setting algorithm
Conclusions
In this article we developed a distributed power control and MS algorithm for cellular network assisted D2D communications. The power control algorithm consists of an SINR target setting part that aims to set the individual SINR targets such that the required sum power is minimized with respect to a sum rate target and a power allocation part that sets the power levels and power loading matrices over multiple MIMO streams. The MS algorithm considers the geometry of the D2D candidate and the cellular UE communicating with the cellular access point and determines if the D2D candidate should use the direct D2D link or should communicate via the cellular access point.
Numerical results clearly indicate that in order to take advantage of the proximity and reuse gains of D2D communications, adaptively setting the SINR targets for both the cellular and D2D links and adaptively determining the communication mode for the D2D candidate are necessary. To this end, we proposed low complexity power control and MS algorithms that rely on slow scale CSI. When the proposed power control and MS algorithms are employed, D2D communication is clearly superior both in terms of the required sum power and the feasibility of a predefined sum rate target to the classical cellular mode of operation.
The numerical examples also suggest that due to the combination of the intra and intercell interference, it becomes important that the power control algorithm ensures some level of fairness between the D2D and the cellular links. The proposed power control algorithm is therefore capable of guaranteeing a predefined minimum SINR target to each link. This feature of the power control algorithm along with the low complexity of the MS algorithm make them interesting candidates for future networks supporting D2D communications.
Endnotes
^{a}This article is a substantially revised and extended version of the article “A Distributed Power Control Scheme for Cellular Network Assisted D2D Communications” presented at the IEEE Global Communication Conference (Globecom), in Houston, TX, USA, December 2011 [30].^{b}It is advantageous to use uplink resources for the D2D link, because in some countries regulatory requirements may not allow to use downlink resources by UEs in the future. Therefore, in this article we only deal with the case when the D2D links use UL cellular resources, such as the uplink OFDM resource blocks in a cellular Frequency Division Duplexing system or the uplink time slots in a TDD system [28, 40, 41].
Appendices
Appendix 1: derivation of the MMSE estimation error matrix
where ${\mathbf{R}}_{{H}_{k}}={\alpha}_{k,k}^{2}{\mathbf{H}}_{k,k}^{\u2020}{\mathbf{R}}_{{v}_{k}}^{1}{\mathbf{H}}_{k,k}$.
Appendix 2: derivation of Δ P in Algorithm 2
The approximated transmission power needed to increase the SINR by Δ can be calculated from (27) and (28) as ${\mathrm{\Delta P}}_{k}^{\left(t\right)}={p}_{k}^{\left(t\right)\prime}{p}_{k}^{\left(t\right)}$.
Appendix 3: components of the sum power vector
where $\kappa ={2}^{{c}_{\mathrm{m}}\underset{2}{log}\left(1+{\gamma}_{1}^{\mathrm{tgt}}\right)\underset{2}{log}\left(1+{\gamma}_{2}^{\mathrm{tgt}}\right)}1$, and ${D}_{p}=1{\gamma}_{1}^{\mathrm{tgt}}{\gamma}_{2}^{\mathrm{tgt}}{F}_{1,2}{F}_{2,1}\kappa \left({\gamma}_{1}^{\mathrm{tgt}}{F}_{1,3}{F}_{3,1}{\gamma}_{2}^{\mathrm{tgt}}{F}_{2,3}{F}_{3,2}{\gamma}_{1}^{\mathrm{tgt}}{\gamma}_{2}^{\mathrm{tgt}}\left({F}_{1,2}{F}_{2,3}{F}_{3,1}+{F}_{1,3}{F}_{2,1}{F}_{3,2}\right)\phantom{\rule{15.0pt}{0ex}}\right)$.
Appendix 4: the process of optimal SINR target selection
Algorithm 4 Optimization process
for pos = 1 to number of UE1 Positions do
 1.
Drop UE1 in the interval of ((pos−1)·r,pos·r], where r=R/10;
 2.
Drop UE2 (Tx Device) and Rx Device according to a surface uniform distribution within Cell1;
 3.
Calculate distances between the k ^{ th }receiver and the j ^{ th }transmitter d _{k,j},∀k,j;
 4.
Draw fast fading H _{k,j},∀k,j;
 5.
Calculate Ω _{k,j},∀k,j according to (18);
 6.
Draw shadow fading χ _{k,j},∀k,j;
 7.
Calculate F _{k,j}according to (23);
 8.
Express the sum power vector p as defined in (25);
 9.
Substitute ${\gamma}_{3}^{\mathrm{tgt}}$ in (25) with the right side of (26);
 10.Run ALPF optimization method {minValue^{ALPF1}, minPoints ^{ALPF1}} =ALPF (obj, vars, inits, cons, maxIter, convTolerance), where$\phantom{\rule{14.0pt}{0ex}}\begin{array}{l}\phantom{\rule{5.5em}{0ex}}\mathrm{obj}=\sum _{k=1}^{K}{\mathbf{p}}_{i},\\ \phantom{\rule{4.5em}{0ex}}\mathbf{vars}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}[{\gamma}_{1}^{\mathrm{tgt}},{\gamma}_{2}^{\mathrm{tgt}}],\\ \phantom{\rule{4.5em}{0ex}}\mathbf{inits}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}[0.0,0.0],\\ \phantom{\rule{4.5em}{0ex}}\mathbf{cons}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}[{\gamma}_{1}^{\mathrm{tgt}}\ge 0,{\gamma}_{2}^{\mathrm{tgt}}\\ \phantom{\rule{7.5em}{0ex}}\ge 0,{2}^{{c}_{\mathrm{m}}\underset{2}{log}\left(1+{\gamma}_{1}^{\mathrm{tgt}}\right)\underset{2}{log}\left(1+{\gamma}_{2}^{\mathrm{tgt}}\right)}\\ \phantom{\rule{8.5em}{0ex}}1\ge 0],\\ \phantom{\rule{3.3em}{0ex}}\mathrm{maxIter}=20,\\ \mathrm{convTolerance}=0.01;\end{array}$
 11.
Run NMinimize Mathematica ^{ ® } builtin numerical optimization method {minValue^{NMin},minPoints ^{NMin}}=NMinimize(obj,vars,cons);
 12.
Set new initial points to ALPF as inits=minPoints ^{NMin};
 13.
Run ALPF optimization method {minValue^{ALPF2}, minPoints^{ALPF2}} =ALPF (obj, vars, inits, cons, maxIter, convTolerance) ;
 14.
if ALPF converged in Steps 10 and 13, and minValue^{ALPF1} =minValue^{ALPF2} ±10^{3}then Potential global optimum is found: {minValue, minPoints} ={minValue^{ALPF2}, minPoints ^{ALPF2}}else Discard MC drop (i.e., decrease i by one) and go to Step 1;
 15.
Save the optimization results optResults {pos, i} ={minValue, minPoints};
end
Calculate the average sum power and infeasibility ratio measures in UE1 position pos for all MC drops;
end
Declarations
Acknowledgements
The authors gratefully acknowledge the private discussions with Dr. Claes Tidestav and Dr. Szabolcs Malomsoky at Ericsson Research, whose comments and suggestions helped a lot to improve the article. We also thank the anonymous reviewers for the valuable comments that improved the contents and the presentation of the article.
Authors’ Affiliations
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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.