# Power control for multipoint cooperative communication with high-to-low sinr scenario

- Hui Zhang
^{1, 2}Email author, - Ying Fang
^{2}, - Yifeng Xie
^{2}, - Hong Wu
^{2}and - Yanyan Guo
^{3}

**2012**:276

https://doi.org/10.1186/1687-1499-2012-276

© Zhang et al.; licensee Springer. 2012

**Received: **15 December 2011

**Accepted: **10 August 2012

**Published: **30 August 2012

## Abstract

In order to solve the power control problem for multipoint cooperative communication with high-to-low SINR scenario, the cooperative SINR receiving model is established. Moreover, considering its non-convex 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 high-to-low SINR scenario. The numerical results show the power of cellular users can be brought into a global optimization range whether users in high-SINR area or low-SINR 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 inter-cell interference effectively.

## Keywords

## 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 interference-limted 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 single-antenna-based power control algorithm by means of geometric programming (GP) method, which divides the scenarios into two aspects, directly takes GP in high-SINR region, while adopts a successive convex programming and equivalent signomial programming (SP) condensation algorithm to solve it in low-SINR region. Otherwise, Qian et al. [3] proposes a MAPEL (Multiplicative linear fractional programming-based power allocation) algorithm to achieve global optimality for a non-convex wireless single-antenna-based power control problem. The computation times of MAPEL algorithm drastically increase when accuracy increases, and MAPEL algorithm needs more convergence time with high-complexity.

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 multiple-input multiple-output (MIMO) context due to coordination between receiving antennas and nonlinear dependence between interference and eigen-spaces 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 cell-edge 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 IMT-advanced 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 non-convex 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 non-convex 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 single-point 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 log-sum-exp function. Based on it, the non-convex 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 high-to-low 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 non-convex property, the proposed algorithm is based on the GP method and a series of convex approximations are taken to achieve the global optimization in high-to-low 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 high-SINR area or low-SINR 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 inter-cell 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

#### Link Model

*β*

_{ k,i }denotes a scalar coefficient depending on the total transmit power

*P*

_{ k }of user

*k*, written as

*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 MS-

*k*$\left({N}_{t},{N}_{t}\right)$diagonal power loading matrix. In order to keep the total transmit power as a constant,

*T*

_{ k,i }should satisfy

*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).

*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

*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

*G*

_{ k }denotes an MMSE weighting matrix, the received signal y

_{ k }is estimated as ${\widehat{x}}_{k}$, that is

_{ k }is equivalent to (7) with equal power allocation case. Specially, ${T}_{k}={\text{I}}_{{N}_{\text{t}}}$.

*k*-th base station, that is [5, 9]

### Power control problem

Nowadays, it’s a common problem for power control in MIMO systems, which may include many different actual scenarios. For example, non-cooperative power control [16], power control subjected to objective function [17], power control in cooperative communications [18], and so on. In this section, the non-cooperative power control and the cooperative power control subjected to objective function are respectively introduced as two major problems.

#### Non-Cooperative Power Control in MIMO systems

*k*in multi-cellular system should satisfy following inequality based on [5], which is under the equal allocation assumption.

*k*. 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

*k*

^{th}user with linear MMSE receiver can be defined as follows:

#### Cooperative Power Control in MIMO System subjected to Objective Function

_{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 (5*BER*) for M-QAM, with *BER* = 10^{−3}.

Over the years, the weighted throughput extremum (WTE) is one of plagued problem in interference-limited wireless networks, for mutual interference in the transmission link constitutes a non-convex 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 high-to-low SINR scenario, it’s necessary to study the non-convex 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 non-convex 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 non-convex 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 log-sum-exp function is convex. On this basis, it enables to take convex optimization methods to solve the above GP problem although it’s nolinear and non-convex in the form.

*r*

_{ k }=

*W*

_{ k }log (1 +

*Kγ*

_{ k })in (18) is non-convex, a GP-based 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 log-sum-exp 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 high-to-low 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 non-convex power control problem into the GP problem, which can get a global optimization solution by means of inequality iterative approximation to the objective.

*a*as an example, assume

*ã*= log

*a*, then the exponential variable transformation can be taken as

*a*=

*e*

^{ ã }. In rate-SINR 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 high-to-low SINR scenario, take a log-exponential variable transformation for the critical parameters in power control problem, turning the original problem into a convex optimization problem that the log-sum-exp 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

*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 at

*x*=

*x*

_{0}, respectively

*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

*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 non-negativity.

*τ*≥ 1, let

*Q*=

*τP*, so

*P*≤

*Q*, we have

*τ*≥ 1, so we can get

Based on the above analysis, the proposed power control algorithm with high-to-low SINR scenario by GP method is shown as following:

**Algorithm 1: Power Control with High-to-Low 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(s-1\right)}\Vert}{{P}_{k}^{\left(s-1\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 19-cell 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:

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.

## Conclusion

In order to solve the power control problem for multipoint cooperative communication with high-to-low SINR scenario, the cooperative SINR receiving model is established. Considering the power control algorithm problem from its non-convex 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 high-to-low SINR scenario for this algorithm. The numerical results show that the power distribution is improved in both cell-center or cell-edge. 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 high-SINR area or low-SINR area, which means the inter-cell interference could be effectively mitigated.

## Appendix

*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

*r*

_{ k }=

*W*log (1 +

*Kγ*

_{ k }), which is non-convex 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)

*μ*

_{ k },

*λ*

_{ k }are the Lagrangian multipliers, and the derivation process is as follows

^{+}= max (0, · ). By means of (A11) and (A12), we can get the update form of (A8) according to the Newton iterative method:

## Declarations

### Acknowledgement

This research is supported by State Key Laboratory of Networking and Switching Technology Open Project (No. SKLNST-2011-1-02), 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).

## Authors’ Affiliations

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## Copyright

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