Per-antenna Power Constraints Research Articles

Zero-Forcing Beamforming in Multiuser MISO Downlink Systems Under Per-Antenna Power Constraint and Equal-Rate Metric

In this paper, we analyze the average sum rate of downlink multi-antenna systems with zero-forcing beamforming (ZFBF). In practical implementations, each antenna is equipped with its own power amplifier and is limited individually by linearity of the amplifier. Thus, this paper adopts a more realistic per-antenna power constraint instead of conventional sum-power constraint on transmit antennas. To this end, we first show that a distribution of the received signal-to-noise ratio (SNR) of the ZFBF scheme with per-antenna power constraint and equal-rate metric can be approximated as a minimum of chi-square random variables. Based on this result, we present an accurate formula of the average sum rate in a closed form. Furthermore, employing extreme value theory, an expression of the asymptotic average sum rate with large numbers of transmit antennas and users is derived from the limiting distribution of the received SNR. Simulation results verify the validity of our analysis even with not so large numbers of transmit antennas and users.

Beamformer Designs for MISO Broadcast Channels with Zero-Forcing Dirty Paper Coding

We consider the beamformer design for multiple-input multiple-output (MISO) broadcast channels (MISO BCs) using zero-forcing dirty paper coding (ZF-DPC). Assuming a sum power constraint (SPC), most previously proposed beamformer designs are based on the QR decomposition (QRD), which is a natural choice to satisfy the ZF constraints. However, the optimality of the QRD-based design for ZF-DPC has remained unknown. In this paper, first, we analytically establish that the QRD-based design is indeed optimal for any performance measure under a SPC. Then, we propose an optimal beamformer design method for ZF-DPC with per-antenna power constraints (PAPCs), using a convex optimization framework. The beamformer design is first formulated as a rank-1-constrained optimization problem. Exploiting the special structure of the ZF-DPC scheme, we prove that the rank constraint can be relaxed and still provide the same solution. In addition, we propose a fast converging algorithm to the beamformer design problem, under the duality framework between the BCs and multiple access channels (MACs). More specifically, we show that a BC with ZF-DPC has the dual MAC with ZF-based successive interference cancellation (ZF-SIC). In this way, the beamformer design for ZF-DPC is transformed into a power allocation problem for ZF-SIC, which can be solved more efficiently.

Weighted Sum Rate Maximization for MIMO Broadcast Channels Using Dirty Paper Coding and Zero-forcing Methods

We consider precoder design for maximizing the weighted sum rate (WSR) of successive zero-forcing dirty paper coding (SZF-DPC). For this problem, the existing precoder designs often assume a sum power constraint (SPC) and rely on the singular value decomposition (SVD). The SVD-based designs are known to be optimal but require high complexity. We first propose a low-complexity optimal precoder design for SZF-DPC under SPC, using the QR decomposition. Then, we propose an efficient numerical algorithm to find the optimal precoders subject to per-antenna power constraints (PAPCs). To this end, the precoder design for PAPCs is formulated as an optimization problem with a rank constraint on the covariance matrices. A well-known approach to solve this problem is to relax the rank constraints and solve the relaxed problem. Interestingly, for SZF-DPC, we are able to prove that the rank relaxation is tight. Consequently, the optimal precoder design for PAPCs is computed by solving the relaxed problem, for which we propose a customized interior-point method that exhibits a superlinear convergence rate. Two suboptimal precoder designs are also presented and compared to the optimal ones. We also show that the proposed numerical method is applicable for finding the optimal precoders for block diagonalization scheme.

Power Allocation in Multi-Antenna Wireless Systems Subject to Simultaneous Power Constraints

We address the problem of power allocation to maximize ergodic capacity subject to multiple power constraints assuming that perfect causal channel state information (CSI) is available at both the transmitter and the receiver. We characterize the optimal power allocation subject to both long-term and short-term power constraints, which depends upon the ratio of the two power levels. Additionally, we find a suboptimal power allocation if the input power is subject to long-term and per-antenna power constraints. We characterize the conditions for which one power constraint dominates and the other can be ignored. Numerical results suggest that, for the Rayleigh fading case, a short-term power constraint that is larger than a long-term power constraint does not significantly impact the ergodic capacity of the channel. The effect of per-antenna power constraints is also explored for the case of Rayleigh fading through our numerical results.

Iterative Mode-Dropping for the Sum Capacity of MIMO-MAC with Per-Antenna Power Constraint

We propose an iterative mode-dropping algorithm that optimizes input signals to achieve the sum capacity of the MIMO-MAC with per-antenna power constraint. The algorithm successively optimizes each user's input covariance matrix by applying mode-dropping to the equivalent single-user MIMO rate maximization problem. Both analysis and simulation show fast convergence. We then use the algorithm to briefly highlight the difference in MIMO-MAC capacities under sum and per-antenna power constraints.

Optimum Linear Block Precoding for Multi-Point Cooperative Transmission with Per-Antenna Power Constraints

Using cyclic prefix (CP), the transmission schemes, orthogonal frequency-division multiplexing (OFDM), single carrier block transmission, and time reversal, can be unified as linear block precoding. Considering frequency-selective channels, this paper studies linear block precoding for CP-based multi-point transmission with per-antenna power constraints (PAPCs) under capacity maximization and mean-square-error (MSE) minimization criteria. We show that, the optimal precoders for both criteria could be, but not necessarily, in the form of OFDM transmission (i.e., an inverse discrete fourier transformation (IDFT) matrix multiplying a complex diagonal matrix). Based on the optimal precoder structure, the two problems are simplified to two matrix-free optimization problems for which we prove strong duality holds. Moreover, it is shown that the dual problems can be equivalent to two unconstrained convex optimization problems. Efficient optimum precoding algorithms are proposed for both problems. Simulation results show that the maximum capacity (or minimum MSE) in the PAPC case almost coincides with that in the sum power constraint (SPC) case when the power budgets for each antenna are equal, but a capacity (or MSE) gap exists between the two power constraint cases when the power budgets for each antenna are different.

Achievable Rates of Multi-Antenna Downlink Channels with Peak Power Constraints

This paper considers a Gaussian multiple-input single-output (MISO) broadcast channel with a per-antenna peak power constraint (or simply peak power constraint). It is more realistic to consider the peak power constraint on each transmit antenna because in many practical implementations each antenna is equipped with its own power amplifier. Assuming the perfect knowledge of the channel state information (CSI) at the transmitter, we propose an achievable scheme using dirty-tape coding (DTC). The ideal dirty-paper coding (DPC), which is a capacity-achieving scheme for the Gaussian multiple-input multiple-output (MIMO) broadcast channel, cannot be used for our model because its optimal input distribution is Gaussian and thus the peak power constraint is violated. On the other hand, the channel input of DTC is uniformly distributed in a fixed range, which helps to control the peak power of the transmit signal easily. We also present an algorithm that finds capacity-achieving beamforming vectors and power allocation factors under a per-antenna average power constraint and use the optimized parameters in the proposed scheme. Simulation results show that the proposed scheme provides gains of 2.7dB over a non-DTC scheme based on minimum mean square error (MMSE) beamforming at high signal-to-noise ratio (SNR) when there are three receivers and the transmitter has three antennas.

Antenna Placement Optimization for Distributed Antenna Systems

In this paper, we propose new algorithms to determine the antenna location for downlink distributed antenna systems (DASs) in single-cell and two-cell environments. We consider the composite fading channel which includes small and large scale fadings. First, for the single-cell DAS, we formulate the optimization problem of distributed antenna (DA) port locations by maximizing the lower bound of the expected signal to noise ratio (SNR). In comparison to the conventional algorithm based on the squared distance criterion which requires an iterative method, our problem generates a closed form solution. Next, for the two-cell DAS, we propose a gradient ascent algorithm which determines the optimum DA locations by maximizing the lower bound of the expected signal to leakage ratio (SLR). In our work, we consider selection transmission, maximal ratio transmission and zero-forcing beamforming (ZFBF) under sum power constraint and study equal gain transmission and scaled ZFBF under per-antenna power constraint. Simulation results show that our proposed algorithms based on both the SNR and the SLR criteria offer a capacity gain over the conventional centralized antenna systems.

Sum-Rate Maximization in Distributed-Antenna Heterogeneous MIMO Downlinks: Application to Measured Channels

In a distributed antenna multiple-input multiple- output (MIMO) multiuser channel, multiple nodes simultaneously access the wireless channel by cooperating to achieve virtual MIMO links, enabling improved exploitation of the channel spatial dimensions. However, practical considerations dictate that in such a system, the transmit power from each cooperating node must be constrained in a per-antenna power constraint, which differs from the sum-power constraint typical of multiuser MIMO signaling. Furthermore, existing multiuser MIMO communication algorithms are appropriate for specific assumptions regarding the signaling strategy allowed. This paper provides a general framework able to accommodate a variety of multiuser MIMO channel topologies, precoding techniques (linear or nonlinear), and power constraints including the per-antenna constraint appropriate for the cooperative MIMO channel involving heterogeneous nodes. In contrast to prior work incorporating per-antenna power constraints, the algorithm enables reduced implementation and computational complexity by avoiding the requirement of numerical optimization. Simulation results using both modeled and measured cooperative channel responses demonstrate that the sum rate of a cooperative broadcast channel with a per-antenna power constraint is very close to the optimal bound achieved using a sum-power constraint, even in practical scenarios. Additional results demonstrate the straightforward application of the algorithm to different practical multiuser topologies.

On Gaussian MIMO BC-MAC Duality With Multiple Transmit Covariance Constraints

Owing to the special structure of the Gaussian multiple-input multiple-output (MIMO) broadcast channel (BC), the associated capacity region computation and beamforming optimization problems are typically non-convex, and thus cannot be solved directly. One feasible approach is to consider the respective dual multiple-access channel (MAC) problems, which are easier to deal with due to their convexity properties. The conventional BC-MAC duality has been established via BC-MAC signal transformation, and is applicable only for the case in which the MIMO BC is subject to a single transmit sum-power constraint. An alternative approach is based on minimax duality, which can be applied to the case of the sum-power constraint or per-antenna power constraint. In this paper, the conventional BC-MAC duality is extended to the general linear transmit covariance constraint (LTCC) case, which includes sum-power and per-antenna power constraints as special cases. The obtained general BC-MAC duality is applied to solve the capacity region computation for the MIMO BC and beamforming optimization for the multiple-input single-output (MISO) BC, respectively, with multiple LTCCs. The relationship between this new general BC-MAC duality and the minimax duality is also discussed, and it is shown that the general BC-MAC duality leads to simpler problem formulations. Moreover, the general BC-MAC duality is extended to deal with the case of nonlinear transmit covariance constraints in the MIMO BC.

Polite Water-Filling for Weighted Sum-Rate Maximization in MIMO B-MAC Networks Under Multiple Linear Constraints

The algorithms in this paper exploit optimal input structure in interference networks and is a major advance from the state-of-the-art. Optimization under multiple linear constraints is important for interference networks with individual power constraints, per-antenna power constraints, and/or interference constraints as in cognitive radios. While for single-user MIMO channel transmitter optimization, no one uses general purpose optimization algorithms such as steepest ascent because water-filling is optimal and much simpler, this is not true for MIMO multiaccess channels (MAC), broadcast channels (BC), and the non-convex optimization of interference networks because the traditional water-filling is far from optimal for networks. We recently found the right form of water-filling, polite water-filling, for some capacity/achievable regions of the general MIMO interference networks, named B-MAC networks, which include BC, MAC, interference channels, X networks, and most practical wireless networks as special cases. In this paper, we use weighted sum-rate maximization under multiple linear constraints in interference tree networks, a natural extension of MAC and BC, as an example to show how to design highly efficiency and low complexity algorithms. Several times faster convergence speed and orders of magnitude higher accuracy than the state-of-the-art are demonstrated by numerical examples.

Transmit Beamforming for MISO Frequency-Selective Channels With Per-Antenna Power Constraint and Limited-Rate Feedback

In this paper, we consider transmit beamforming (BF) design under per-antenna power constraint (PPC) for multiple-input-single-output (MISO) frequency-selective channels. Both cyclic-prefixed (CP) single carriers and orthogonal frequency-division-multiplexing (OFDM) systems are investigated. Since there is no closed-form expression for the optimal PPC BF coefficients, and the optimization tools employed to find the PPC solution are usually complicated, here, we propose three computationally more effective suboptimal solutions to minimize the arithmetic mean of the effective error probabilities. Moreover, we address the issue of limited-rate feedback, and a simple codebook design method for frequency-selective channels is proposed. Unlike conventional methods where the codebook is directly designed for BF vectors, here, we construct codebooks for the amplitude and phase of the time-domain channel state information (CSI) to reduce the rate of feedback. Further, the nonuniform feedback strategy is investigated. Simulation results show that the proposed suboptimal solutions have much lower complexities than the convex optimization-based PPC solutions. The effectiveness of the proposed uniform and nonuniform limited-rate feedback methods is also demonstrated via simulations.

MISO Capacity with Per-Antenna Power Constraint

We establish in closed-form the capacity and the optimal signaling scheme for a MISO channel with per-antenna power constraint. Two cases of channel state information are considered: constant channel known at both the transmitter and receiver, and Rayleigh fading channel known only at the receiver. For the first case, the optimal signaling scheme is beamforming with the phases of the beam weights matched to the phases of the channel coefficients, but the amplitudes independent of the channel coefficients and dependent only on the constrained powers. For the second case, the optimal scheme is to send independent signals from the antennas with the constrained powers. In both cases, the capacity with per-antenna power constraint is usually less than that with sum power constraint.

Optimal Multiuser Zero Forcing with Per-Antenna Power Constraints for Network MIMO Coordination

We consider a multi-cell multiple-input multiple-output (MIMO) coordinated downlink transmission, also known as network MIMO, under per-antenna power constraints. We investigate a simple multiuser zero-forcing (ZF) linear precoding technique known as block diagonalization (BD) for network MIMO. The optimal form of BD with per-antenna power constraints is proposed. It involves a novel approach of optimizing the precoding matrices over the entire null space of other users' transmissions. An iterative gradient descent method is derived by solving the dual of the throughput maximization problem, which finds the optimal precoding matrices globally and efficiently. The comprehensive simulations illustrate several network MIMO coordination advantages when the optimal BD scheme is used. Its achievable throughput is compared with the capacity region obtained through the recently established duality concept under per-antenna power constraints.

Cooperative Multi-Cell Block Diagonalization with Per-Base-Station Power Constraints

Block diagonalization (BD) is a practical linear precoding technique that eliminates the inter-user interference in downlink multiuser multiple-input multiple-output (MIMO) systems. In this paper, we apply BD to the downlink transmission in a cooperative multi-cell MIMO system, where the signals from different base stations (BSs) to all the mobile stations (MSs) are jointly designed with the perfect knowledge of the downlink channels and transmit messages. Specifically, we study the optimal BD precoder design to maximize the weighted sum-rate of all the MSs subject to a set of per-BS power constraints. This design problem is formulated in an auxiliary MIMO broadcast channel (BC) with a set of transmit power constraints corresponding to those for individual BSs in the multi-cell system. By applying convex optimization techniques, this paper develops an efficient algorithm to solve this problem, and derives the closed-form expression for the optimal BD precoding matrix. It is revealed that the optimal BD precoding vectors for each MS in the per-BS power constraint case are in general non-orthogonal, which differs from the conventional orthogonal BD precoder design for the MIMO-BC under one single sum-power constraint. Moreover, for the special case of single-antenna BSs and MSs, the proposed solution reduces to the optimal zero-forcing beamforming (ZF-BF) precoder design for the weighted sum-rate maximization in the multiple-input single-output (MISO) BC with per-antenna power constraints. Suboptimal and low-complexity BD/ZF-BF precoding schemes are also presented, and their achievable rates are compared against those with the optimal schemes.

Precoder and Capacity Expressions for Optimal Two-User MIMO Transmission with Per-Antenna Power Constraints

In this letter, multiuser multiple-input multiple-output (MIMO) precoding which achieves the Gaussian broadcast channel (GBC) capacity under the per-antenna power constraint (PAPC) is investigated. In particular, closed-form expressions for both the MIMO precoder and the corresponding sum capacity are presented for the two-user case. We show that the result can also be utilized to provide the two-user capacity expression under the total power constraint (TPC) of transmit antennas.

Robust MMSE Transceiver Designs for Downlink MIMO Systems with Multicell Cooperation

Therobust-generalized iterative approach(Robust-GIA),robust-fast iterative approach(Robust-FIA), androbust-decoder covariance optimization approach(Robust-DCOA) are proposed for designing MMSE transceivers of downlink multicell multiuser MIMO systems with per-cell and per-antenna power constraints and possibly imperfect channel state information. TheRobust-DCOAis the most restrictive but is always optimum, theRobust-GIAis the most general, and theRobust-FIAis the most efficient. When theRobust-DCOAis applicable and the decoder covariance matrices are full rank, the three proposed approaches are equivalent and all provide the optimum solution. Numerical results show that the proposed robust approaches outperform their non-robust counterparts in various single-cell and multicell examples with different system configurations, channel correlations, power constraints, and cooperation scenarios. Moreover, performances of the robust approaches are insensitive to estimation errors of channel statistics (correlations and path loss). With cell-cooperation, cell edge interference problems can be remedied without reducing the number of data streams by using the proposed robust approaches.

Fast Interior Point Method for MIMO Transmit Power Optimization with Per-Antenna Power Constraints

For multiple-input multiple-output (MIMO) precoded transmission that has individual constraints on the maximum power of each transmit antenna or a subset of transmit antennas, the transmit power optimization problem is a non-linear convex optimization problem with a high level of computational complexity. In this paper, assuming the use of the interior point method (IPM) to solve this problem, we propose two efficient techniques that reduce the computational complexity of the IPM by appropriately setting its parameters. Based on computer simulation, the achieved reductions in the level of the computational complexity are evaluated using the proposed techniques for both the fairness and the sum-rate maximization criteria assuming i.i.d Rayleigh fading MIMO channels and block diagonalization zero-forcing as a multi-user MIMO (MU-MIMO) precoder.

Robust THP Transceiver Designs for Multiuser MIMO Downlink with Imperfect CSIT

In this paper, we present robust joint non-linear transceiver designs for multiuser multiple-input multiple-output (MIMO) downlink in the presence of imperfections in the channel state information at the transmitter (CSIT). The base station (BS) is equipped with multiple transmit antennas, and each user terminal is equipped with one or more receive antennas. The BS employs Tomlinson-Harashima precoding (THP) for inter-user interference pre-cancellation at the transmitter. We consider robust transceiver designs that jointly optimize the transmit THP filters and receive filter for two models of CSIT errors. The first model is a stochastic error (SE) model, where the CSIT error is Gaussian-distributed. This model is applicable when the CSIT error is dominated by channel estimation error. In this case, the proposed robust transceiver design seeks to minimize a stochastic function of the sum mean square error (SMSE) under a constraint on the total BS transmit power. We propose an iterative algorithm to solve this problem. The other model we consider is a norm-bounded error (NBE) model, where the CSIT error can be specified by an uncertainty set. This model is applicable when the CSIT error is dominated by quantization errors. In this case, we consider a worst-case design. For this model, we consider robust i) minimum SMSE, ii) MSE-constrained, and iii) MSE-balancing transceiver designs. We propose iterative algorithms to solve these problems, wherein each iteration involves a pair of semi-definite programs (SDP). Further, we consider an extension of the proposed algorithm to the case with per-antenna power constraints.

Generalized co-phasing for multiple transmit and receive antennas

We propose and study a class of transmit beamforming techniques for systems with multiple transmit and multiple receive antennas with a per-antenna transmit power constraint. The per-antenna transmit power constraint is more realistic than the widely used total (across all transmit antennas) power constraint, since in practice each transmit antenna is driven by a separate power amplifier with a maximum power rating. Under the per-antenna power constraint, from an implementation perspective, it becomes desirable to vary only the phases (as opposed to both power and phase variation) of the signals departing from the transmit antennas. We name this class of techniques generalized co-phasing and formulate an optimization problem to calculate the transmit antenna phases. Furthermore, we propose five heuristic algorithms to solve the optimization problem. All the proposed algorithms except one are optimal for the case of two transmit antennas and an arbitrary number of receive antennas. For an arbitrary number of transmit and receive antennas, simulations indicate that the proposed algorithms perform very close to the optimal solution calculated through an exhaustive search of all possible transmit phases.