Power minimization for cooperative MIMOOFDM systems with individual user rate constraints
 Chihyu Hsu^{1}Email author,
 Phee Lep Yeoh^{1} and
 Brian S. Krongold^{1}
https://doi.org/10.1186/s1363801605414
© Hsu et al. 2016
Received: 14 May 2015
Accepted: 28 January 2016
Published: 9 February 2016
Abstract
We propose a continuous rate and power allocation algorithm for multiuser downlink multipleinput multipleoutput orthogonal frequencydivision multiplexing (MIMOOFDM) systems with coordinated multipoint (CoMP) transmission that guarantees to satisfy individual rate target across all users. The optimization problem is formulated as a total transmit power minimization problem subject to peruser rate targets and perantenna power constraints across multiple cooperating base stations. While the perantenna power constraint leads to a more complex optimization problem, it is a practical consideration that limits the average transmit antenna power and helps to control the resulting high peak powers in OFDM. Our proposed algorithm uses successive convex approximation (SCA) to transform the nonconvex power minimization problem and dynamically allocate power to cochannel user terminals. We prove that the transformed power minimization problem is convex and that our proposed SCA algorithm converges to a solution. The proposed algorithm is compared with two alternative approaches: (1) iterative waterfilling (IWF) and (2) zeroforcing beamforming (ZFB) with semiorthogonal user selection. Simulation results highlight that the SCA algorithm outperforms IWF and ZFB in both medium and lowinterference environments.
Keywords
1 Introduction
Intercell interference (ICI) is a limiting factor on the throughput performance of downlink multiuser multipleinput multipleoutput (MIMO) orthogonal frequencydivision multiplexing (OFDM) systems. User terminals (UTs) located at the cell edge are particularly susceptible to interference from base stations (BSs) that are operating in proximity within the same frequency. In this paper, we consider the use of coordinated multipoint (CoMP) transmission with joint processing to mitigate the effect of ICI, which is a key technology in nextgeneration networks [1–3]. Joint processing is accomplished by sharing channel state information and user data between multiple BSs via a highspeed lowdelay optical backhaul. In doing so, ICI can be mitigated by transmitting user data to a UT simultaneously from all the cooperating BSs [4, 5].
In addition to mitigating ICI using joint processing, resource allocation algorithms can be employed in conjunction with CoMP to achieve substantial improvements in multiuser MIMO system performance [6–8]. In [6], the system performance is improved by joint power allocation and linear precoding for multiuser MIMO systems with CoMP under perantenna power constraints. In [7], the received signaltointerferenceplusnoise ratio for individual user is enhanced by adaptive nonlinear precoding and power allocation for CoMP systems with multiuser MIMO under total BS and perBS power constraints. In [8], the joint linear precoding and power allocation for multiuser MIMO systems with CoMP are solved by convex optimization techniques under perBS power constraints to improve the system performance.
The resource allocation problem for downlink MIMOOFDM systems has been studied extensively for the singleuser case [9, 10]. However, the optimization problem for multiuser MIMOOFDM systems becomes mathematically challenging as the problem becomes nonconvex in the presence of interference. As a result, obtaining a globally optimal solution is difficult to achieve. Dirty paper coding (DPC) was first proposed in [11] to achieve broadcast channel capacity for singlecell MIMO systems, and it was extended to solve the nonconvex sumrate maximization problem for multicell systems [12]. The DPC employs a nonlinear precoding scheme which presubtracts interference to achieve channel capacity. However, DPC requires high computational demands in successive encodings and decodings which makes it difficult to be implemented in practice.
Suboptimal strategies, such as iterative waterfilling (IWF) [13] and zeroforcing beamforming (ZFB) [14], have been proposed to solve the nonconvex problem. The IWF approach in [13] treats interference as a channel noise component which transforms the optimization problem into a convex one. As a result, an equilibrium can be achieved by performing a competitive waterfillingbased algorithm iteratively across all UTs. The ZFB in [14] eliminates interference by employing zeroforcing beamformers. This allows powers to be allocated in interferencefree OFDM subchannels via the waterfilling strategy. However, the performance of ZFB is limited by the number of transmit antennas and the mutual orthogonality of the UT channel gains. As a result, a semiorthogonal user selection is proposed in [15] to select a subgroup of UTs that results in the lowest mutual interference.

We establish an optimization approach for minimizing total transmit power while achieving perUT rate targets. We perform eigenbeamforming on each MIMOOFDM subchannel, with the aid of singular value decomposition, to obtain precoding and postprocessing matrices for the BS and UT, respectively.

We derive an iterative algorithm, which is based on the SCA approach in [18], to solve the nonconvex power minimization problem in which a minimum rate target is achieved for each UT. A convexequivalent optimization problem is obtained using the proposed iterative algorithm. In doing so, we provide a convexity proof for the transformed problem and we show that the proposed algorithm can converge to a unique solution.

We consider the perantenna average transmit power constraint, which limits the average transmit antenna power. As a result, the high peak power of each transmit antenna can be indirectly constrained. This ensures that the peak power is limited at an acceptable level which does not exceed the dynamic range of a highpowered amplifier, thereby causing nonlinear transmission effects. The issue of high peak powers is often overlooked in resource allocation problems which only consider a total power constraint.

We compare our proposed algorithm with two other suboptimal algorithms IWF [24] and ZFB with semiorthogonal user selection [15]. We adopt an empirical path loss model, the COST231 Hata empirical model [25], to model various interference environments.
A much more complicated problem would be the joint adaptive beamforming design and power allocation with a minimum mean square error receiver used to suppress the interuser interference. While this problem tends to be intractable, our proposed algorithm could be applied on top of a coordinated beamforming method across all cooperating base stations. Furthermore, the proposed SCA algorithm is suited to fixedwireless applications in sparsely populated regions that require high UT data rates over large network areas. A prime example is the provision of wireless broadband in rural areas where the channel gains are quasistationary [26]. Our algorithm is also suitable for implementation in small cells with low user mobility.
The paper is organized as follows. The system model is introduced in Section 2. The formulation of total transmit power minimization problem is presented in Section 3. The fundamental of the SCAbased algorithm is outlined in Section 4. This section also includes the convexity proof for the convexapproximated optimization problem transformed by the proposed SCA algorithm and the convergence of the proposed algorithm. Section 5 presents the numerical results of the optimization problem. Concluding remarks are presented in Section 6.
2 System model
In this paper, we consider a downlink multiuser MIMOOFDM system with N subchannels. The system consists of M cooperating BSs each with L _{ T } transmit antennas, as shown in Fig. 1. These BSs are interconnected by a highspeed optical backhaul for exchanging CSI and user data for joint processing. The optical backhaul is then connected to a central processor for executing a centralized implementation of our proposed power allocation algorithm, which is based on the CSI of each OFDM subchannel from the cooperating BSs. There are K UT, each equipped with L _{ R } receive antennas. The spatial degree of freedom for the MIMOOFDM system is defined as L≤ min(M L _{ T },L _{ R }). We assume that perfect CSI knowledge between transmitreceive antenna pairs is known to both BSs and UTs. The CoMP configuration with joint processing operation can be envisioned as a multiuser MIMO system with distributed transmit antenna. The channel gains of these distributed transmit antennas consist various path loss profiles depending on the relative distances between the distributed transmit and receive antennas.
where \(\mathbf {U}_{n}^{k}\in \mathbb {C}^{L_{R}\times L_{R}}\) and \(\mathbf {V}_{n}^{k}\in \mathbb {C}^{{ML}_{T}\times {ML}_{T}}\) are the unitary transmit precoding and receiver shaping matrices, respectively, and \({\boldsymbol \Lambda _{n}^{k}}\in \mathbb {R}^{L\times {ML}_{T}}\) is the diagonal matrix with nonnegative singular values \(\sqrt {\gamma _{n,l}^{k}},\,l = 1,\ldots,L\) as the gain for the (n,l)th spatial subchannel [25]. The operator (·)^{H} represents the Hermitian transpose.
As such, a Nsubchannel MIMOOFDM system can be decomposed into a total of N×L spatial subchannels and with full CSI knowledge, intelligent power, and bit allocation algorithms can be employed to optimize system performance across all the spatial subchannels. The application of the eigenbeamforming does not eliminate the interuser interference for cochannel users (like ZFB). The interuser interference is caused by the mismatch between the jth UT transmit precoding matrix and all the UT receiver shaping matrices.
Before we formulate the optimization problem, we formally define the following two signal and power domains, which will be used throughout the paper.
Definition 1.
The antenna domain consists of powers that are physically transmitted by the antennas at the BSs.
Definition 2.
The spatial domain consists of effective powers and signals sent in the spatial subchannels resulting from SVD.

\(\tilde {R}_{n,l}^{k} =\) spatial rate in (n,l)th spatial subchannel for
UT k

\(\tilde {P}_{n,l}^{k} =\) spatial power in (n,l)th spatial subchannel for
UT k

P _{ n,m }= transmit power in subchannel n from antenna
m,
where a spatial subchannel pair is denoted by an accent with subscripts (n,l) and a subchannelantenna pair is denoted by subscripts (n,m).
The physical interpretation of the interuser interference gain \(\mathbf {G}_{n}^{k,j}\) can be explained as the interference function from the jth UT projecting onto the receiving direction of the kth UT. This gives in a weighted sum of the transmitted signal in all L spatial subchannels as a result of a conjugate mismatch between the transmit beamforming weights \(\mathbf {V}_{n}^{j}\) and the postprocessing of \(\mathbf {U}_{n}^{k}\). In the next section, we present the optimization problem that satisfy perUT rate targets for given perantenna transmit power constraints.
3 Power minimization problem formulation
The resource allocation problem in MIMOOFDM systems can be formulated into a power minimization (PM) problem. The PM problem aims to minimize the transmit power while satisfying rate targets for each UT and transmit power constraints. For conventional rate adaptive problems, in which the objective is to maximize the total system throughput subject to a total transmit power constraint, it is intuitive that by allocating powers to the UT who has the best channel condition will maximize the overall system throughput for a given transmit power constraint. Those UTs with less favorable channel conditions will receive very little or even no data throughput as there is no rate constraint on the individual UT. In contrast, the PM problem guarantees perUT rate targets to be satisfied while minimizing total transmit power for a given set of perantenna power constraints.
where \(R_{\mathrm {T}}^{k}\) is the desirable rate target for the kth UT. These rate targets must be feasible, which means there must exist a feasible power allocation such that the peruser rate target is satisfied and the perantenna power constraints not being violated. To enhance readability, we now write \(\tilde {R}_{n,l}^{k}\) without explicitly stating it being a function of \(\tilde {\mathbf {P}}_{n}, \, {\sigma _{n,l}^{k}}^{2}, \, \mathrm {G}_{n,l}^{k,k}\) and \(\mathbf {G}_{n}^{k,j}\).
We simplify the PM problem in (7) by converting the objective function and perantenna power constraints into the spatial domain. In doing so, we derive an important relationship between spatial average powers and antenna average powers. Assuming the data symbols sent in each spatial subchannel are uncorrelated, which is expected, with zero mean and normalized to unit variance, it can be shown that, for a given subchannel n, the relationship between spatial and antenna powers is given by the following lemma.
Lemma 1.
The relationship between antenna powers \(\mathbf {P}_{n}^{k}\) and spatial powers \(\tilde {\mathbf {P}}_{n}^{k}\) is given by \(\mathbf {P}_{n}^{k} = \mathbf {A}_{n}^{k}\tilde {\mathbf {P}}_{n}^{k}\), where \(\mathbf {A}_{n}^{k}(m,l) = \left \mathbf {V}_{n}^{k}(m,l)\right ^{2}\).
Proof.
where Tr(·) denoted as the trace of a matrix and ·^{2} denoted as the squared magnitude operation.
The term \(\mathbf {A}_{n}^{k}\) refers to the power gain transformation from spatial powers to antenna powers in the nth subchannel for the kth UT and is equal to the elementbyelement squaredmagnitude of the transmit precoding matrix, \(\mathbf {V}_{n}^{k}\).
This relationship allows us to transform antenna powers into spatial powers which result in effective rates in each spatial subchannel. Moreover, the perantenna power constraint prevents unbalanced power allocation among all the cooperating BSs. In the case of the total average transmit power constraint, the majority of the power would be allocated to BSs with better channel conditions. This makes the inherent peaktoaveragepower ratio (PAPR) problem in the OFDM more problematic as the resulting peak transmit power at the transmit antenna may exceed the dynamic range of highpowered amplifiers (HAP) during transmission. As a result, the transmitted signal will experience nonlinear transmission effects, which compromises signal quality and, consequently, affecting the overall system performance. With perantenna power constraints in place, the average transmit power of each antenna is constrained to a threshold in which the resulting high PAPR would not be problematic to cause irreversible nonlinear transmission effects.
From the expression in (9), it can be seen that it is of the form of difference of concave functions (DoCF) of \(\tilde {\mathbf {P}}_{n}\). Obtaining globally optimal solutions for optimization problems involve with DoCF is difficult and NPhard [28].
4 The proposed SCA algorithm
The lower bound is improved successively by evaluating and updating α and β according to (11a) and (11b) at each iteration, respectively, based on the new value \(\bar {x}\). A locally optimal solution will be obtained as the lower bound converges to the actual achievable rate curve [18].
where \(\boldsymbol \lambda =\left [\lambda _{1}\ldots \lambda _{{ML}_{T}}\right ]\) is the 1×M L _{ T } vector of Lagrange multipliers associated with each transmit antenna and μ=[μ _{1}…μ _{ K }] is the 1×K vector of Lagrange multipliers associated with each UT rate target. The proof for the convexity of the perUT rate target is provided in Lemma 2.
Lemma 2.
The perUT rate target in (12) is a concave function with the substitution of \(\tilde {\mathbf {P}}_{n}~=~e^{\hat {\mathbf {P}}_{n}}\).
Proof.
and since \({\sigma _{n,l}^{k}}^{2}\) is nonnegative and therefore, the CauchySchwarz inequality holds [23].
This indicates that the dual problem can be solved by optimizing Nindependent dual subproblems, each for \(\breve {g}_{n}^{k}\left (\boldsymbol \mu,\boldsymbol \lambda \right),\,\forall \,k=1,\ldots,K\). As a result, the overall implementation cost can be reduced significantly if the same procedure is executed repeatedly for solving each subproblem, or alternatively, K parallel processors can be adopted for solving N dual subproblems simultaneously to improve the convergence time of the algorithm.
where 1 is the 1×M L _{ T } vector of ones and \({\boldsymbol \alpha _{n}^{j}}={\left [\alpha _{n,1}^{j}\ldots \alpha _{n,L}^{j}\right ]}^{\mathrm {T}}\) is the L×1 convex approximation constant vector for the nth OFDM subchannel of the jth UT. We note that the term \(\mathbf {G}_{n}^{j,k}(l,:)\) quantifies the impact of allocating \(\tilde {P}_{n,l}^{k}\) to the kth UT on all other UTs, which results in an altruistic approach of allocating powers to UTs that have the minimal mutual interference. This differs from the egoistic approach of IWF by maximizing the signaltonoise ratio without regard to resulting mutual interference to all UTs.
The power allocation strategy in (26) is a standard interference function which is guaranteed to coverage to a unique solution [29]. To demonstrate this, we apply Yates’ definition of standard interference function in [29] to (26) which is introduced in the following definition.
Definition 3.

Positivity: \(\mathcal {I}(\textbf {p}) > 0\)

Monotonicity: If \(\textbf {p} \succeq \textbf {p}^{\prime }\phantom {\dot {i}\!}\), then \(\mathcal {I}(\textbf {p}) \geq \mathcal {I}(\textbf {p}^{'})\)

Scalability: For all θ>1, \(\theta \,\mathcal {I}(\textbf {p}) > \mathcal {I}(\theta \,\textbf {p})\)
Proof.

Positivity: This follows from the fact that each term in \(\mathcal {I}_{n,l}^{k}(\tilde {\mathbf {P}})\) in (27) is nonnegative.

Monotonicity: Suppose \(\tilde {\mathbf {P}} \geq \tilde {\mathbf {P}}^{'}\), the monotonicity property follows from$$\begin{array}{*{20}l} \mathcal{I}_{n,l}^{k}(\tilde{\mathbf{P}}) &= \frac{\mu_{k}\alpha_{n,l}^{k}}{\ln 2\left(\mathbf{1}+\boldsymbol\lambda\right)\mathbf{A}_{n}^{k}(:,l)+\sum\limits_{j\neq k}\frac{\mathbf{G}_{n}^{j,k}(l,:){\boldsymbol\alpha_{n}^{j}}\mu_{j}}{\mathbf{G}_{n}^{k,j}(l,:)\tilde{\mathbf{P}}^{j}_{n}+{\sigma_{n,l}^{k}}^{2}}}\\ &\geq \frac{\mu_{k}\alpha_{n,l}^{k}}{\ln 2\left(\mathbf{1}+\boldsymbol\lambda\right)\mathbf{A}_{n}^{k}(:,l)+\sum\limits_{j\neq k}\frac{\mathbf{G}_{n}^{j,k}(l,:){\boldsymbol\alpha_{n}^{j}}\mu_{j}}{\mathbf{G}_{n}^{k,j}(l,:){{}\tilde{\mathbf{P}}^{j}_{n}}^{'}+{\sigma_{n,l}^{k}}^{2}}}\\ &= \mathcal{I}_{n,l}^{k}(\tilde{\mathbf{P}}^{'}) \end{array} $$(28)

Scalability: Suppose \(\tilde {\mathbf {P}}=\theta \tilde {\mathbf {P}}^{'}\) for θ>1, the scalability property follows from$$\begin{array}{*{20}l} \theta\,\mathcal{I}_{n,l}^{k}(\tilde{\mathbf{P}}) &= \frac{\mu_{k}\alpha_{n,l}^{k}}{\frac{1}{\theta}\ln 2\left(\mathbf{1}+\boldsymbol\lambda\right)\mathbf{A}_{n}^{k}(:,l)+\frac{1}{\theta}\sum\limits_{j\neq k}\frac{\mathbf{G}_{n}^{j,k}(l,:){\boldsymbol\alpha_{n}^{j}}\mu_{j}}{\mathbf{G}_{n}^{k,j}(l,:)\tilde{\mathbf{P}}^{j}_{n}+{\sigma_{n,l}^{k}}^{2}}}\\ &> \frac{\mu_{k}\alpha_{n,l}^{k}}{\ln 2\left(\mathbf{1}+\boldsymbol\lambda\right)\mathbf{A}_{n}^{k}(:,l)+\sum\limits_{j\neq k}\frac{\mathbf{G}_{n}^{j,k}(l,:){\boldsymbol\alpha_{n}^{j}}\mu_{j}}{\mathbf{G}_{n}^{k,j}(l,:)\theta{{}\tilde{\mathbf{P}}^{j}_{n}}^{'}+{\sigma_{n,l}^{k}}^{2}}}\\ &= \mathcal{I}_{n,l}^{k}(\theta\,\tilde{\mathbf{P}}^{'}) \end{array} $$(29)
respectively, for some fixed \({\tilde {\mathbf {P}}_{n}}^{k}\), where ε and ν are step sizes for each iteration, and s is the iteration number. The updated Lagrange multipliers μ ^{[s+1]} and λ ^{[s+1]} are then substituted back into (26) to obtain the new power allocation \({{}{\tilde {\mathbf {P}}_{n}}^{k}}^{[s+1]}\), and the resulting rate allocation \({{}\tilde {R}_{n,l}^{k}}^{[s+1]}\) is obtained from \({{}{\tilde {\mathbf {P}}_{n}}^{k}}^{[s+1]}\) using (4). The iterative procedure terminates when the duality gap between the primal and dual objective function approaches to zero. The PMSCA algorithm is outlined in Algorithm 1. We initialize the algorithm with a highSINR approximation with α=1 and β=0 [18]. Before we present numerical results in next section, we introduce IWF and ZFB with semiorthogonal user selection, which we used to compare the performance of our proposed algorithm.
4.1 IWF
In IWF, the power allocation for each MIMOOFDM subchannel is performed by assuming that the interuser interference is constant and treating it as a part of channel noise. As a result, the original nonconvex optimization problem is transformed into a convex one. An equilibrium is achieved by performing the waterfilling solution iteratively across all the UTs in the system. In numerical simulations, we first perform a SVD on individual MIMOOFDM subchannel to obtain the individual subchannel gains. These subchannel gains are then used to perform the power allocation, which is based on the iterative waterfilling algorithm across all the UTs.
4.2 ZFB with semiorthogonal user selection
In ZFB, orthogonal beamformers are used to eliminate the interuser interference for cochannel UTs. This transforms the original nonconvex optimization problem into a convex one, and the waterfilling algorithm is performed across MIMOOFDM subchannels to obtain a suboptimal solution. However, an efficient user selection algorithm is needed for finding cochannel UTs with less mutual interference in order to maximize the system performance, in particular, when the number of UTs is large. Therefore, a semiorthogonal user selection is introduced for effectively finding nearorthogonal cochannel UTs to occupy the limited number of zeroforcing beamformers, which is governed by the number of transmit antennas.
5 Simulation results and discussion
Simulation parameters  

Bandwidth (MHz)  10 
Carrier frequency (GHz)  2 
BS height (m)  30 
Maximum Tx power (dBm)  46 
RF feeder cable/connector loss (dB)  2 
Antenna gain (dBi)  18 
Receiver height (m)  5 
Antenna gain (dBi)  8 
Noise figure (dB)  7 
Thermal noise (dBm/Hz)  −174 
Receiver noise floor (dBm)  −97 
Slow fading margin (dB)  8 
where \(\Lambda _{n,l}^{k}\) is the effective channel gain after precoding and postprocessing. The noise power is assumed to be equal across all OFDM subchannels. We average the simulation results over a total of 16,000 channel realizations, which is obtained from 100 simulation iterations for each UT subchannel.
Next, we investigate the relationship between the minimum achievable rates perUT and the coverage radius, r. To ensure the distribution of UT changes with the coverage radius, we place the UTs uniformly distributed on the circumference of the coverage circle as the circle expands to simulate UTs scatter between cooperating BSs. The minimum achievable rates refer to the minimum rate between perUT rate targets when the perantenna powers are close to be fully utilized. The results are obtained from the average of 100 simulation iterations.
5.1 Complexity analysis

For the proposed algorithm: The eigenbeamforming of each MIMOOFDM subchannel is obtained by the SVD of MIMOOFDM subchannel \(\mathbf {H}_{n}^{k}\). The channel matrix \(\mathbf {H}_{n}^{k}\) is a L _{ R }×M L _{ T } complex matrix. To obtain the SVD of each \(\mathbf {H}_{n}^{k}\) requires 8(4L _{ R } ^{2} M L _{ T }+8L _{ R } (M L _{ T })^{2}+9(M L _{ T })^{3}) complex floating point operations [32]. The total number of complex floating point operations across all MIMOOFDM subchannel and UTs is approximately$$ \sum\limits_{k=1}^{K}\sum\limits_{n=1}^{N}\,8kn\left[4{L_{R}}^{2}{ML}_{T} + 8L_{R}\left({ML}_{T}\right)^{2} + 9\left({ML}_{T}\right)^{3}\right]. $$(35)Therefore, the overall computational complexity of the proposed algorithm is$$ \mathcal{O}\left\{8KN\left[4{L_{R}}^{2}{ML}_{T} + 8L_{R}\left({ML}_{T}\right)^{2} + 9\left({ML}_{T}\right)^{3}\right]\right\}. $$(36)

The overall computational complexity of IWF is approximately the same as the proposed algorithm since the eigenbeamforming is performed across all MIMOOFDM subchannels, which is given by$$ {\mathcal{O}}\left\{8KN\left[4{L_{R}}^{2}{ML}_{T} + 8L_{R}\left({ML}_{T}\right)^{2} + 9\left({ML}_{T}\right)^{3}\right]\right\}. $$(37)

For ZFB with semiorthogonal user selection: This algorithm consists of two stages: (1) semiorthogonal user selection and (2) obtaining zeroforcing beamformers. The computational complexity of semiorthogonal user selection is given by \(\mathcal {O}[KN(L_{T})^{3}]\) [33]. Finding zeroforcing beamformers involves block diagonalization across all the MIMOOFDM subchannels, which can be obtained by performing SVD. The total number of complex floating point operations is approximately$$ \sum\limits_{k=1}^{K}\sum\limits_{n=1}^{N}\,8nk\left[8L_{R}\left({ML}_{T}\right)^{2} + 9\left({ML}_{T}\right)^{3}\right]. $$(38)Therefore, the overall computational complexity of ZFB with semiorthogonal user selection is given by$$ \mathcal{O}\left\{KN\left[64L_{R}\left({ML}_{T}\right)^{2}+{L_{T}^{3}}+72\left({ML}_{T}\right)^{3}\right]\right\}. $$(39)
6 Conclusions
In this paper, the individual UT rate target is achieved by transforming a nonconvex optimization problem into a tractable set of successive convex approximations. A convex lower bound is updated at each iteration to improve the approximation of the achievable rate region, where a dual Lagrange decomposition and a subgradient method is efficient in obtaining the locally optimal solution. Average power constraints are enforced on each antenna for all BSs, which helps manage the resulting peak power effects (via OFDM’s inherently high PAPR) for all transmission highpowered amplifiers. We envision this work to be more suited for small cells with low user mobility, but more importantly, for fixedwireless applications in sparsely populated regions that require high data rates to UTs over very large network areas.
The effectiveness of our proposed SCAbased algorithm was demonstrated through a performance comparison of SCA and the alternative approaches of IWF in [24] and ZFB in [15]. Comparing SCA and IWF, we see that SCA provides a lower total transmit power and higher minimum perUT rate target relative to IWF in a range of interference environments. In general, we find that the higher the interference between UTs, the larger difference in terms of total transmit power and minimum perUT target rate between SCA and IWF. As expected, ZFB performs well in highinterference environments as it provides interferencefree subchannels for the scheduled UTs. However, the performance of ZFB is limited by the number of transmit antennas and the mutual orthogonality of the scheduled UTs’ channel conditions. As such, we find that ZFB results in a higher total transmit power and lower minimum achievable rate solution than SCA and IWF in both medium and lowinterference environments.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 A Ghosh, R Ratasuk, B Mondal, N Mangalvedhe, T Thomas, LTEAdvanced: nextgeneration wireless broadband technology. IEEE Wirel. Commun. 17(3), 10–22 (2010).View ArticleGoogle Scholar
 M Sawahashi, Y Kishiyama, A Morimoto, D Nishikawa, M Tanno, Coordinated multipoint transmission/reception techniques for LTEAdvanced. IEEE Wirel. Commun. Mag. 17(3), 26–34 (2010).View ArticleGoogle Scholar
 D Lee, H Seo, B Clerckx, E Hardouin, D Mazzarese, SNK Sayana, Coordinated multiple transmission and reception in LTEAvanced: deployment scenarios and operational challenges. IEEE Commun. Mag. 50(2), 148–155 (2012).View ArticleGoogle Scholar
 R Irmer, H Droste, P Marsch, M Grieger, G Fettweis, S Brueck, HP Mayer, L Thiele, V Jungnickel, Coordinated multipoint: concepts, performance, and field trial results. IEEE Commun. Mag. 49(2), 102–111 (2011).View ArticleGoogle Scholar
 J Lee, Y Kim, H Lee, BL Ng, D Mazzarese, J Liu, W Xiao, Y Zhou, Coordinated multipoint transmission and reception in LTEAvanced systems. IEEE Commun. Mag. 50(11), 44–50 (2012).View ArticleGoogle Scholar
 S Kaviani, WA Krzymień, in Proc. IEEE Wireless Communications and Networking Conference. Sum rate maximization of MIMO broadcast channels with coordinated of base stations (Las Vegas, 2008), pp. 1079–1084.Google Scholar
 W Hardjawana, B Vucetic, Y Li, Multiuser cooperative base station systems with joint processing and beamforming. IEEE J. Sel. Topics Signal Process. 3(6), 1079–1093 (2009).View ArticleGoogle Scholar
 R Zhang, Cooperative multicell block diagonalization with perbasestation power constraints. IEEE J. Sel. Areas Commun. 28(9), 1435–1445 (2010).View ArticleGoogle Scholar
 CY Hsu, BS Krongold, in Proc. IEEE Global Communications Conference. Coordinated multipoint transmission of MIMOOFDM system with perantenna power constraints (Anaheim, 2012).Google Scholar
 BS Krongold, K Ramchandran, DL Jones, Computationally efficient optimal power allocation algorithm for multicarrier communication systems. IEEE Trans. Commun. 48(1), 23–27 (2000).View ArticleGoogle Scholar
 MHM Costa, Writing on dirty paper. IEEE Trans. Inf. Theory. 29(3), 439–441 (1983).View ArticleMATHGoogle Scholar
 DHN Nguyen, T LeNgoc, Sumrate maximization in the multicell MIMO broadcast channel with interference coordination. IEEE Trans. Signal Process. 62(6), 1501–1513 (2014).View ArticleMathSciNetGoogle Scholar
 W Yu, W Rhee, S Boyd, JM Cioffi, Iterative waterfilling for Gaussian vector multipleaccess channel. IEEE Trans. Inf. Theory. 50(1), 145–152 (2004).View ArticleMathSciNetMATHGoogle Scholar
 QH Spencer, AL Swindlehurst, M Haardt, Zeroforcing methods for downlink spatial multiplexing in multiuser MIMO channels. IEEE Trans. Signal Process. 52(2), 461–471 (2004).View ArticleMathSciNetGoogle Scholar
 S Kaviani, WA Krzymień, in Proc. IEEE Global Communications Conference. User selection for multipleantenna broadcast channel with zeroforcing beamforming (New Orleans, 2008).Google Scholar
 M Pischella, JC Belfiore, Distributed resource allocation for rateconstrained users in multicell OFDMA networks. IEEE Commun. Lett. 12(4), 250–252 (2008).View ArticleGoogle Scholar
 C Hellings, M Joham, W Utschick, Gradientbased power minimization in MIMO broadcast channels with linear precoding. IEEE Trans. Signal Process. 60(2), 877–890 (2012).View ArticleMathSciNetGoogle Scholar
 J Papandriopoulos, JS Evans, SCALE: A lowcomplexity distributed protocol for spectrum balance in multiuser DSL networks. IEEE Trans. Inf. Theory. 8(8), 3711–3724 (2009).View ArticleMathSciNetGoogle Scholar
 NU Hassan, M Assaad, in Proc. IEEE International Workshop on Signal Processing Advances in Wireless Communications. Optimal downlink beamforming and resource allocation in MIMOOFDMA systems (Marrakech, 2011).Google Scholar
 L Venturino, N Prasad, X Wang, Coordinated scheduling and power allocation in downlink multicell OFDMA networks. IEEE Trans. Veh. Technol. 6(58), 2835–2848 (2009).View ArticleGoogle Scholar
 H Zhu, J Wang, Chunkbased resource allocation in OFDMA systems—part i: chunk allocation. IEEE Trans. Commun. 57(9), 2734–2744 (2009).View ArticleGoogle Scholar
 H Zhu, J Wang, Chunkbased resource allocation in OFDMA systems—part ii: joint chunk, power and bit allocation. IEEE Trans. Commun. 60(2), 499–509 (2012).View ArticleGoogle Scholar
 S Boyd, L Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004).View ArticleMATHGoogle Scholar
 M Kobayashi, G Caire, in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing. Iterative waterfilling for weighted rate sum maximization in MIMOOFDM broadcast channels, (2007).Google Scholar
 A Goldsmith, Wireless Communications (Cambridge University Press, New York, 2005).View ArticleGoogle Scholar
 HR Anderson, Fixed Broadband Wireless System Design (Wiley, UK, 2003).View ArticleGoogle Scholar
 GL Stüber, J Barry, SW McLaughlin, YG Li, MA Ingram, TG Pratt, Broadband MIMOOFDM wireless communication. Proc. IEEE. 92:, 271–294 (2004).View ArticleGoogle Scholar
 R Horst, H Tuy, Global Optimization: Deterministic Approaches, 2nd edn (Springer, Berlin, 1993).View ArticleGoogle Scholar
 RD Yates, A framework for uplink power control in cellular radio system. IEEE J. Sel. Areas Commun. 13(7), 1341–1347 (1995).View ArticleMathSciNetGoogle Scholar
 H Holma, A Toskala (eds.), WCDMA for UMTS  HSPA Evolution And LTE, 4th edition (Wiley, UK, 2007).Google Scholar
 H Holma, A Toskala (eds.), LTE for UMTS: OFDMA and SCFDMA Based Radio access (Wiley, UK, 2009).Google Scholar
 GH Golub, CFV Loan, Matrix Computations (John Hopkins University Press, Baltimore, 1996).MATHGoogle Scholar
 J Mao, J Gao, Y Liu, G Xie, Simplified semiorthogonal user selection for MUMIMO systems with ZFBF. IEEE Wirel. Commun. Lett. 1(1), 42–45 (2012).View ArticleGoogle Scholar