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Power control for multipoint cooperative communication with hightolow sinr scenario
EURASIP Journal on Wireless Communications and Networking volume 2012, Article number: 276 (2012)
Abstract
In order to solve the power control problem for multipoint cooperative communication with hightolow SINR scenario, the cooperative SINR receiving model is established. Moreover, considering its nonconvex property, a novel power control algorithm is proposed, which is based on the geometric programming and a series of convex approximations are taken to achieve the global optimization in hightolow SINR scenario. The numerical results show the power of cellular users can be brought into a global optimization range whether users in highSINR area or lowSINR area compared with the existing algorithm, also the CDF of users’ SINR is optimized, and its SINR coverage distribution could be balanced in varying degrees, which improve the fairness and mitigate the intercell interference effectively.
Introduction
Recently, convex optimization theory gradually become a hot tool to solve the hard problems in communication area, for many such problems can be converted into the form of convex optimization. In brief, convex optimization mainly refers to the minimization of a convex objective function subject to convex constraints, and some modern software’s have been developed into solve convex problems, such as CVX, SeDuMi, YALMIP, et al. [1]. However, the rate maximization problem is not yet amendable to a convex formulation for the interference channel in actual systems [2]. Especially, due to the frequency reuse in cellular networks and the limitation of signal to interference plus noise ratio (SINR), it’s a long standing open problem in interferencelimted wireless networks to achieve weighted throughput maximization through power control [2, 3]. According to this open problem, some related research work have been done in single antenna scenario. For example, Chiang et al. [4] gives a singleantennabased power control algorithm by means of geometric programming (GP) method, which divides the scenarios into two aspects, directly takes GP in highSINR region, while adopts a successive convex programming and equivalent signomial programming (SP) condensation algorithm to solve it in lowSINR region. Otherwise, Qian et al. [3] proposes a MAPEL (Multiplicative linear fractional programmingbased power allocation) algorithm to achieve global optimality for a nonconvex wireless singleantennabased power control problem. The computation times of MAPEL algorithm drastically increase when accuracy increases, and MAPEL algorithm needs more convergence time with highcomplexity.
With the development of novel technologies, there are some new challenges for the traditional power control method. On the one hand, it’s still difficult to the straightforward application of single antenna power control in the multipleinput multipleoutput (MIMO) context due to coordination between receiving antennas and nonlinear dependence between interference and eigenspaces of the channel matrices. Considering this situation, Chen et al. [5] proposes an iterative channel inversion power control algorithm for the uplink of cellular MIMO spatial multiplexing systems, but not consider the power control problem with cellular cooperation scenario. On the other hand, recently cellular cooperative communication for multiple base stations and multiple users is drawing attention as a solution to achieve high system throughput in celledge for the future mobile communication systems, such as coordinated multipoint (CoMP) transmission, cooperative beam, cooperative resource control, cooperative transmission, cooperative relaying, etc [6–8]. As discussed in the IMTadvanced standardizing groups, it is expected to be essential for cooperative communication technologies in the next generation cellular networks [7]. With this background, Fodor et al. [9] gives a near optimum power control method under fairness constraints in CoMP transmission systems based on Lagrangian penalty function, which aims to improve the transmitting power allocation efficiency in multicell spatial multiplexing wireless systems. Considering the objective function is nonconvex for the PHY constraints, [9] takes use of the link capacity approximation and replaces it into the objective function, which is cited from the rate approximation inequality in [10, 11]. By means of this approximation of PHY constraints, [9] changes this nonconvex power control problem into a convex optimization problem to solve. However, the rate approximation condition needs the high SINR situation, and such approximation is hard to application in the low SINR condition. Moreover, it actually gives an analysis of singlepoint transmission scenario still without giving a theoretical derivation of CoMP scenario.
In 2005, Chiang [12] gives an explanation about why is GP useful for general communication systems, respectively from stochastic models and deterministic models. Also, Chiang et al. [4] gives an introduction that GP can be used to efficiently compute the globally optimal power control in many of these problems. As a special case, GP is also suitable for cooperative communication systems. Specifically, considering the characteristic of power control problem in CoMP systems, it needs to optimize the objective function with some other data stream from different base stations/users (such as throughput, power strength, SINR, etc.). The GP method could play an effective role in these optimization problems. For GP, its product operation can be easily converted to the summation operations of logarithm, written as logsumexp function. Based on it, the nonconvex problem can be converted into a convex optimization problem. Moreover, its Lagrange duality gap is zero under mild conditions and the global optimum can always be computed very efficiently [13].
Considering this situation, by means of GP, we propose an optimized power control algorithm with hightolow SINR scenario for cooperative communication under MIMO cellular system. Our contributions include the following aspects: Compared with [5, 9], the cooperative SINR receiving model is established and the cooperative base station sets are analyzed in our method, then the algorithm is extended into cooperative communication. In order to solve the nonconvex property, the proposed algorithm is based on the GP method and a series of convex approximations are taken to achieve the global optimization in hightolow SINR scenario, which makes our method could be applied in both high SINR and low SINR scenario. The numerical results show the power of cellular users can be brought into a global optimization range whether users in highSINR area or lowSINR area compared with the existing algorithm, also the CDF of users’ SINR is optimized, and its SINR coverage distribution could be balanced in varying degrees, which improve the fairness and mitigate the intercell interference effectively.
The rest of this paper is organized as follows: The system model is introduced in System model. The overview of power control problem in MIMO system is made in Power control problem. The power control is analyzed and solved by GP respectively in Power control analysis by GP. Then a series of convex approximations is described in A series of convex approximation. The performance analysis is given in Performance analysis. Finally, the conclusion is made in Conclusion.
System model
As shown in Figure 1, it gives the network topology of cellular cooperative communication system, where several access points (APs) are connected into eNodeBs and some celledge users are served by the cooperative AP.
Link Model
Define ${x}_{k}\in {\u2102}^{{N}_{t}\times 1}$is the data vector that is also assumed to be zeromean, normalized and uncorrelated, $E\left({x}_{k}{x}_{k}^{\u2020}\right)={\mathbf{I}}_{{N}_{t}}$. In the uplink, β_{ k,i } denotes a scalar coefficient depending on the total transmit power P_{ k } of user k, written as
where X_{ k,i } is the lognormal shadow fading and d_{ k,i } is the distance between the k^{th} user and the i^{th} base station. H_{ki} denotes the $\left({N}_{r},{N}_{t}\right)$channel transfer matrix, T_{ k,i } denotes the MSk$\left({N}_{t},{N}_{t}\right)$diagonal power loading matrix. In order to keep the total transmit power as a constant, T_{ k,i } should satisfy
Besides, assume n_{ k } is an ${N}_{r}\times 1$additive with Gaussian noise vector at the k^{th} base station with zero mean and the covariance matrix ${R}_{{n}_{k}}$ is defined as (3).
Further, we define the coordinated transmission point (CTP) sets ξ, constructed by some cooperative APs. For cell i, if it is in the CTP sets, written as $i\in \xi $. Similarly, for cell j, if it isn’t in the CTP sets, written as $j\in \overline{\xi}$. On this basis, the received signal Y_{ k } at the k^{th} AP is represented as
Particularly, for single cellular signal without considering the cooperative transmission scenario, the received signal y_{ k } can be simplified as (5). Moreover, the dimension of y_{ k } and y_{ k } is according to the sent signal (original signal), which is respectively an expression of the received signal, whether it’s in the form of the vector or not depends on the sent signal.
Linear MMSE Receiver
Considering the linear MMSE receiver, assume G_{ k } denotes an MMSE weighting matrix, the received signal y_{ k } is estimated as ${\widehat{x}}_{k}$, that is
The MMSE weighting matrix G_{ k } is equivalent to (7) with equal power allocation case. Specially, ${T}_{k}={\text{I}}_{{N}_{\text{t}}}$.
On the other hand, we denote ${\epsilon}_{\mathit{MMSE}}$ as the MMSE estimation error matrix $\left({N}_{r}\times {N}_{r}\right)$for the kth base station, that is [5, 9]
Based on the MMSE estimation error matrix ${\epsilon}_{\mathit{MMSE}}$, the postprocessing SINR by the linear MMSE receiver is denoted as [14, 15]:
Furthermore, we assume the noise power is irrelevant with the number of cells. Considering the spatial diversity gain from each cooperative cell set, the joint postprocessing SINR could be derivated from Eq.(9), which is defined as
Power control problem
Nowadays, it’s a common problem for power control in MIMO systems, which may include many different actual scenarios. For example, noncooperative power control [16], power control subjected to objective function [17], power control in cooperative communications [18], and so on. In this section, the noncooperative power control and the cooperative power control subjected to objective function are respectively introduced as two major problems.
NonCooperative Power Control in MIMO systems
In MIMO system, the power allocation for userk in multicellular system should satisfy following inequality based on [5], which is under the equal allocation assumption.
where ${\Gamma}_{k}$denotes as a given SINR target value for userk. Moreover, we define $\mathbf{\Lambda}=dig\left\{{\Gamma}_{1},{\Gamma}_{2},\cdots ,{\Gamma}_{k}\right\}$, and ${\mu}_{max}\left(\xb7\right)$is the maximum eigenvalue of a Hermitian matrix. Besides, ${\mathbf{\Omega}}_{k,j,1},{\mathbf{\Omega}}_{k,j,2}$ is respectively defined as
According to [5], in a cellular MIMO system with K cells, the postprocessing SINR for the k^{th} user with linear MMSE receiver can be defined as follows:
On this basis, the optimized power factor for power control problem in MIMO systems can be derived as [5, 19]
where N is a ${N}_{t}\times 1$dimension noise vector, and F is defined as
Cooperative Power Control in MIMO System subjected to Objective Function
In cooperative communication area, the objective of power minimum and rate maximum is contradictory in power control problem. In order to mitigate the uplink intercell cofrequency interference, it’s necessary to reduce the power in transmitter, making the power minimum ahead of actual service. Otherwise, in order to improve transmission rate, it’s necessary to raise the sum of rate in each sublink. It’s a nonconvex for such problem, and nowadays many research work aims to solve its global optimization value [3]. Considering the gap between capacity and actual transmission rate, by means of an SINRgap approximation in the Shannon’s theory, the transmission rate r_{k} is written as [4]
where W_{ k } is the bandwidth for user k, and also denotes user k’s weight, the coefficient K is the gap to capacity and always smaller than 1. Specially, K is equivalent to − 1.5/ log (5BER) for MQAM, with BER = 10^{−3}.
On this basis, the power allocation problem in cooperative communication generally aims to solve the weighted throughput extremum and mitigate the intercell interference, including two types: One needs to maximize the sum rates with constraint capacity, written as Type I. The other needs to minimize the sum power at the user side with constraint rate, written as Type II.
Over the years, the weighted throughput extremum (WTE) is one of plagued problem in interferencelimited wireless networks, for mutual interference in the transmission link constitutes a nonconvex optimization problem [1]. The existing research focus on the high SINR scenario, in whose condition the WTE problem can be transmitted into convex optimization by mathematical approximation method. However, for the low SINR scenario, the constraint conditions for such approximation method is difficult to set up, making no sense for the existing solution schemes in general scenario. In order to solve the WTE problem with hightolow SINR scenario, it’s necessary to study the nonconvex power control algorithm with global convergence.
Power control analysis by geometric programming
In MIMO system, both the capacity maximization problem subject to power constraint and the power minimization problem subject to SINR constraints can be formulated as a convex problem. However, due to the interference channel where multiple transmitters and multiple receivers interfere with each other in a shared medium, the rate maximization problem like (18) isn’t yet amendable to a convex formulation [2]. Nowadays, GP is introduced into solve nonconvex optimization problem, and the above WTE problem can also be efficiently sovled by GP ways.
Generally, GP is a type of mathematical optimization problem characterized by objective and constraint functions that have a special form, which is also a class of nonlinear optimization and its standard form is still a nonconvex optimization problem, because the polynomials are not convex functions [20, 21]. However, by means of a logarithmic transition, the variables and multiplicative constraints can respectively be turned into logarithmic variables and plus constraints, which is a convex form for the logsumexp function is convex. On this basis, it enables to take convex optimization methods to solve the above GP problem although it’s nolinear and nonconvex in the form.
Considering the objective function r_{ k } = W_{ k } log (1 + Kγ_{ k })in (18) is nonconvex, a GPbased power control algotithm is given in [12] to solve this WTE problem. In that paper, the transmitting rate is approximated by r_{ k } = W_{ k } log (Kγ_{ k }) when the SINR is in the high regime that is much larger than 0 dB in each link, or the spreading gain in CDMA systems is large [12]. With this approximation, (18) is transformed into the convex optimization problem in the form of GP after logsumexp variables change [4]. Figure 2 shows the processing flow for the power control problem solved by existing GP method [22]. First, a threshold SINR value is defined for the power control problem, and then the scenario is devided into two parts, respectively high SINR scenario and low SINR scenario. As shown in Figure 2, the original power control problem is directly changed into a GP problem under high SINR scenario, but under low SINR scenario, it needs two steps of transition, first by the SP method and then through complementary condensed. After these prepare, it becomes a traditional GP problem. But how to set the hightolow SINR threshold is not given in [22]. Due to the change speed of user location and channel variability, it’s hard to define the SINR threshold for a uncertain randomness. Moreover, this method only gives a conceptual optimization strategy but in actual scenario it inevitably lead to a certain lag and the defined complexity of SINR threshold.
However, the high SINR scenario assumption isn’t always valid in actual wireless networks, and it gives a segment solution by existing GP method, which is only a local optimization solution for the SP condensation [3]. Moreover, the algorithm complexity increases for the interior steps of SP and complementary GP. In this paper, we propose another transition way from the original nonconvex power control problem into the GP problem, which can get a global optimization solution by means of inequality iterative approximation to the objective.
Specially, the processing flow for power control problem solved by proposed GP method is shown in the Figure 3, which includes three parts, respectively logexponential variable transformation, rateSINR inequality approaching and GP problem. In logexponential variable transformation, take any positive real number a as an example, assume ã = log a, then the exponential variable transformation can be taken as a = e^{ã}. In rateSINR inequality approaching, for example, we choose the inequality ϕ log (x) + ϕ ≤ log (1 + x), where ϕ and ϕ belong to real restriction parameters. Then this inequality can be taken to approach the throughput by iteration, and the original problem can be changed into GP problem.
Due to such method does not depend on SINR threshold, it can be found that the proposed power control approach is not only specific to cooperative networks but also can be applied to more general scenarios. Under hightolow SINR scenario, take a logexponential variable transformation for the critical parameters in power control problem, turning the original problem into a convex optimization problem that the logsumexp function belongs to convex case [23]. For the objective function, search a tight iterative inequality between transmission rate and SINR, making the formula related with SINR approach to the rate. Then, solve this GP problem to get the optimized power and SINR value. Section V describe the calculation process in detail.
A series of convex approximation
It is worth mentioning that an inequality related the rate in [2] is given to solve resource allocation under highSINR approximation. Specially, it is also taken in [10] to solve the rate approximation problem by means of Lagrangian penalty function method. For this algorithm, it is constrainted with high SINR situation for such mathimatical approximation and only has optimum solution under high SINR condition. Moreover, the algorithm complexity increases for computing the tighted coefficients. What’s important, these coefficients aim to let the formula value related withlog (SINR) tightly approximate into log (1 + SINR). When directly taking an equivalent approximation from log (SINR) to SINR, the approaching condition would be changed, making the computation error under these related coefficients may be larger and not in the form of optimal approximated values, which is neglected in [9]. Considering this problem, a global optimization solution is proposed for global SINR approximation in this paper. By means of the inequality ϕ log (x) + ϕ ≤ log (1 + x), where ϕ and ϕ are approximation constants atx = x_{0}, respectively
For ϕ_{ k } log (Kγ) + ϕ_{ k } ≤ log (1 + Kγ), where it should satisfy ${\varphi}_{k}=\frac{K{\gamma}_{k}}{1+K{\gamma}_{k}}$, ϕ_{ k } = log (1 + Kγ_{ k }) − ϕ_{ k } log (Kγ_{ k }), so we can get
Based on (20), a convex approximation of objective function is presented to solve power control optimization problem with HightoLow SINR Scenario, as shown in (21).
Considering the Lagrange dual function [24], we get
where μ_{ k }, λ_{ k } are the Lagrangian multipliers. By means of Lagrange dual function, we can get the update form according to the Newton iterative method:
where [ · ]^{+} = max (0, · ), and its proof is given in the Appendix. On the other hand, we still need to prove the formula (23) is convergent, which can be proved from three aspects, respectively positivity, monotonicity and scalability. The proof of its convergence is given as follows: Since W/K is a constant value, we only need to make an analysis of μ_{ k }ϕ_{ k }/(1 + λ_{ k }). Let J(P) = μ_{ k }ϕ_{ k }/(1 + λ_{ k }), where${\mu}_{k}={\left[{\mu}_{k}+{\epsilon}_{\mu}\left({r}_{k}^{\text{target}}{\displaystyle {\sum}_{n}W\left({\varphi}_{k}\text{log}\left(K{\tilde{\gamma}}_{k}\right)+{\varphi}_{k}\right)}\right)\right]}^{+},$ and ${\lambda}_{k}={\left[{\lambda}_{k}+{\epsilon}_{\lambda}\left({\displaystyle {\sum}_{n}P}{P}_{tot.}\right)\right]}^{+}$, then we prove that (23) satisfies positivity, monotonicity and scalability.
Positivity: Each component in J(P)is nonnegativity.
Monotonicity: Assumeτ ≥ 1, letQ = τP, soP ≤ Q, we have
Scalability: Also take τ ≥ 1, so we can get
Based on the above analysis, the proposed power control algorithm with hightolow SINR scenario by GP method is shown as following:
Algorithm 1: Power Control with HightoLow SINR Scenario by GP Method

1.
Initialize for all users. When t = 0, ϕ_{ k }^{(0)} = Kγ_{ k }^{(0)}[1 + γ_{ k }^{(0)}]^{−1}, ${\varphi}_{k}^{\left(0\right)}=log\left[\left(1+K{\gamma}_{k}^{\left(0\right)}\right){\left(K{\gamma}_{k}^{\left(0\right)}\right)}^{{\varphi}_{k}}\right]$

2.
For s = 1:1:N, ϕ_{ k }^{(s)} = Kγ_{ k }^{(s)}[1 + γ_{ k }^{(s)}]^{−1}, ${\varphi}_{k}^{\left(s\right)}=log\left[\left(1+K{\gamma}_{k}^{\left(s\right)}\right){\left(K{\gamma}_{k}^{\left(s\right)}\right)}^{{\varphi}_{k}}\right]$

3.
Solve the subproblem (7) by Lagrange dual function L(γ_{ k }, P_{ k }) to give solution

4.
Compute γ_{ k }, P_{ k } according to the following formula ${P}_{k}={e}^{{\tilde{P}}_{k}}$, ${r}_{k}={e}^{{\tilde{r}}_{k}}$

5.
If $\frac{\Vert {P}_{k}^{\left(s\right)}{P}_{k}^{\left(s1\right)}\Vert}{{P}_{k}^{\left(s1\right)}}\ge \eta $, then go to step 2, update ϕ_{ k }^{(s)}, ϕ_{ k }^{(s)}and repeat iteration. Else, stop iteration, output γ_{ k }, P_{ k }
Performance analysis
In simulation, we consider 19cell system, the cell radius is 1 km, while the ISD takes 500 m. The channel model chooses rayleigh fading and the bandwidth takes 5 MHz. The thermal noise density is 174 dBm/Hz. The form of nTX_nRX antennas is 2_2. Moreover, the macroscopic pathloss is 128.1 + 37.6log10(R). The scheduler scheme takes round robin [25]. Moreover, the CVX tool is also taken in the analysis of GP problem [23]. The performance analysis of the proposed algorithm is given as follows:
Figure 4 shows the individual power levels in seven cases for Fodor algorithm and the proposed algorithm, where each case represents a single independent Monte Carlo simulation with the same parameter definition. The algorithm level1 denotes the users in cellcenter that decrease power to reach the target rate. In the initial state, the cellcenter power is distributed from 70 mW to 100 mW. After taking such two algorithms respectively, the cellcenter power for each user is distributed from 30 mW to 50 mW. Moreover, compared with Ford algorithm, the effectiveness of inhibiting cellcenter power is relatively weak by the proposed algorithm, which enables to raise the actual SINR values for cellcenter users. On the other hand, the algorithm level2 denotes the users in celledge that increase power to reach the target rate. In the initial state, the celledge power is distributed from 20 mW to 40 mW. After taking such two algorithms respectively, the celledge power for each user is distributed from 60 mW to 70 mW. It can be seen from this result that compared with Ford algorithm, the effectiveness of raising celledge power is relatively higher by the proposed algorithm, which also enables to raise the actual SINR values for celledge users. From the above comparison, powers of both cellcenter user and celledge user are raised higher by the proposed algorithm than by Fodor algorithm. Especially for celledge users, whose states are always in a lowlevel SINR, its performance is improved effectively by the proposed method, for the reason is that the global optimized power distribution is established among the whole cell. From the above analysis, it can be found that compared with the comparative analysis of unfair/fair rate allocation in [9], the fairness of the proposed algorithm is more or comparable in each cases for cellular users.
However, for the proposed method both takes Lagrange method and Newton iterative method, which belongs to the square convergence, and its algorithm complexity is O(N^{2}). But the algorithm in [9] only takes Lagrange method, whose convergence speed is faster than the proposed method. Moreover, its complexity is O(N), which needs less iteration steps than the proposed method. Although the complexity of the proposed method is a little higher than [9], its practical implementation is more adaptive to actual scenario than [9], for the uncertain change of SINR range. Considering the significant increase in complexity, which may be undesirable in cellular networks with mobile users and rapidly changing channel conditions, this analysis result motivates future research to design some measures to decrease the algorithm complexity.
On the other side, in order to verify the performance improvement of both cellcenter and celledge users brought by the proposed algorithm, we describe the SINR Cumulative Distribution Function (CDF) curve of four cellular users as an example to illustrate it. As shown in Figure 5, it gives the SINR CDF curve for such four users, respectively in cellcenter and celledge. Among them, user 1 and user 2 are in the cellcenter, while user 3 and user 4 are in the celledge. Especially, when the CDF value takes 40%, the SINR is around 4 dB. When SINR takes 10 dB, the CDF value is around 70% to 80%. It can be seen that the CDF distribution of SINR values reach to similar range levels for both cellcenter and celledge users after taking the proposed algorithm, which shows that the rate close to each other playing the same role with a minimum power, improving users’ fairness of experience.
Considering the cellcenter users are always with high SINR, while the celledge users are always with low SINR, the Figure 6 shows the SINR coverage in cell after using the proposed algorithm, where the X axis and Y axis denote the spatial location of user distribution, while the Z axis denotes the SINR value of current users. About two hundred thousand users are generated by Monte Carlo method and distributed in 19cell group. It can been seen that the range of SINR is around from 5 dB to 15 dB, where the red part denotes stronger SINR area and the blue part denotes lower SINR area. The celledge users with better channel conditions could reach to higher range SINR, colour shown as red to yellow. But the performance of celledge users with relatively poor channel conditions is improved and the number of such users decrease into a limited range, colour shown as blue, whose coverage is smaller than those red and yellow area. The result illustrates the distribution of users’ SINR is optimized. The proposed algorithm reduces the interference power effectively, and the SINR of the users is raised in varying degrees.
Conclusion
In order to solve the power control problem for multipoint cooperative communication with hightolow SINR scenario, the cooperative SINR receiving model is established. Considering the power control algorithm problem from its nonconvex property, we propose a novel algorithm that is based on geometric programming. On this basis, a series of convex approximations are taken to achieve the global optimization in hightolow SINR scenario for this algorithm. The numerical results show that the power distribution is improved in both cellcenter or celledge. Moreover, the CDF of users’ SINR is optimized, and the distribution of users’ SINR is balanced and more fairness. The SINR of cellular users can be improved in varying degrees whether users in highSINR area or lowSINR area, which means the intercell interference could be effectively mitigated.
Appendix
According to (18), this problem is not convex subject to the physical conditions, the solution is given by geometric programming. Considering the nocovexity properties in (18), we try to transit it into convex approximation problem by means of logsumexp function in geometric programming. Let ${\tilde{P}}_{k}=log{P}_{k}$, ${\tilde{r}}_{k}=log{r}_{k}$ and ñ_{ k } = log n_{ k }, then take an exponential variable transformation ${P}_{k}={e}^{{\tilde{P}}_{k}}$, ${r}_{k}={e}^{{\tilde{r}}_{k}}$ and ${n}_{k}={e}^{{\tilde{n}}_{k}}$. Moreover, because of ${\gamma}_{k}=\frac{{G}_{k}{P}_{k}}{{n}_{k}+{\displaystyle {\sum}_{j\ne k}{G}_{\mathit{kj}}{P}_{j}}}$, where G_{ kk } is the link gain matrix, P_{ k } is the transmitting power, n_{ k }is the Gaussian noise power, P_{ j } is the interfered power and G_{ kj } is the channel coupling matrix, which is not a concave function of P_{ j }[12]. Then, we get the logarithmic form of SINR ${\tilde{\gamma}}_{k}$ as
Considering the Shannon’s theory r_{ k } = W log (1 + Kγ_{ k }), which is nonconvex in the subject condition. Then we try to get an approximate value. According to the approximate formula for the rate, that is${\tilde{R}}_{k}\left(P;{\varphi}_{k},{\varphi}_{k}\right)\triangleq W\left({\varphi}_{k}\text{log}\left(K{\gamma}_{k}\right)+{\varphi}_{k}\right)$, so the constrained optimization function is equivalent to (A2)
On this basis, we try to transform the objective function in (A2) into a convex one by means of a series of convex approximations, using a tight iterative inequality between the transmission rate and SINR. By such approximation and the logtransformation, the objective in (18) becomes convex and monotonically increasing. Moreover, the feasibility set is also convex, as shown in (A2). Considering the Lagrange dual function, we get
Where μ_{ k }, λ_{ k }are the Lagrangian multipliers, and the derivation process is as follows
Similarly, $\frac{\partial L\left({\gamma}_{k},{\text{P}}_{k}\right)}{\partial {\mu}_{k}}$ and $\frac{\partial L\left({\gamma}_{k},{\text{P}}_{k}\right)}{\partial {\lambda}_{k}}$can be gotten as following:
On this basis, because of nonlinear optimization, we take the Newton iterative method to approach the exact solution gradually. The Newton iterative form is defined as
Where [ · ]^{+} = max (0, · ). By means of (A11) and (A12), we can get the update form of (A8) according to the Newton iterative method:
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Acknowledgement
This research is supported by State Key Laboratory of Networking and Switching Technology Open Project (No. SKLNST2011102), National Natural Science Foundation Project of China (No. 61101084), the Fundamental Research Funds for the Central Universities, and Specialized Research Fund for the Doctoral Program of Higher Education (No.20110031110028).
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Zhang, H., Fang, Y., Xie, Y. et al. Power control for multipoint cooperative communication with hightolow sinr scenario. J Wireless Com Network 2012, 276 (2012). https://doi.org/10.1186/168714992012276
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Keywords
 CoMP
 OFDMA
 Intercell interference
 Power control
 Cooperative communication