Optimum transmit beamforming design for multiple input single output frequency-selective channels with per-antenna power constraint

Many MIMO capacity analyses and precoding algorithms are obtained based on the assumption of total power constraint while assuming power can be allocated freely across antennas. In practical MIMO systems, however, per-antenna power constraints are often more predominant design consideration than the total power constraints. In this paper, we study the transmission schemes for MIMO systems with per-antenna power constraints. We derive the necessary and sufficient conditions for the optimal transmission scheme. We also develop an iterative algorithm and prove its convergence to the optimal solution.

Joint Beamforming and power allocation schemes are considered for cognitive radio (CR) multiple input single output (MISO) system in this paper. We considered the robust downlink beamforming and power allocation problem with per-antenna power constraints which is more realistic in practice, instead of sum-power constraint, under the imperfect channel state information (CSI). Since the given optimization problem is a non-convex problem which is difficult to handle, we transformed it into second order cone program (SOCP) problem, which can be solved efficiently. Simulation shows that the proposed scheme indeed has a better performance in CR system.

In this paper, three novel efficient power control algorithms are proposed for V-BLAST system with both total power constraint (TPC) and per-antenna power constraints (PAPC). Through analysis of the Karush-Kuhn-Tucker (KKT) conditions of the optimization problem, we find out that the antennas with the optimal power control can be divided into two sets. The optimal allocated power (OAP) of each antenna in the first set is its per-antenna power constraint, and the OAPs of the antennas in the second set can be obtained easily through a new optimization problem with TPC only. Based on the above analysis, three novel efficient power allocation algorithms are developed, which have much lower computational complexities compared with the existing dual update method. Simulation results are presented to demonstrate the efficiency of the proposed power control schemes.

This paper considers a two user 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 each antenna is equipped with its own power amplifier in many practical implementations. Assuming the perfect channel state information (CSI) at the transmitter, we propose an achievable scheme using a dirty-tape coding (DTC). The uniform input in a fixed range of the DTC scheme helps to control the peak power of the transmit signal easily. We also present an optimization algorithm that finds the capacity achieving beamforming vectors and power allocations under a per-antenna average power constraint used in our achievable scheme. Simulation results show that as the transmit power increases, the achievable rate region under the peak power constraint is getting close to the capacity region under the reduced per-antenna average power constraint by 1/3. Compared to a non-DTC scheme based on minimum mean square error (MMSE) beamforming, the proposed scheme performs better.

We propose a semi-closed-form solution to the problem of computing the capacity and optimal signaling for a Gaussian MIMO channel under a joint sum power constraint (SPC) and per-antenna power constraint (PAPC). Existing efficient solutions to this fundamental problem are only applicable to some special cases: multiple-input single-output (MISO) systems, or full column rank MIMO channels with sufficiently high transmit power, or full-rank optimal signaling. For the general case, we present an efficient numerical method to solve the considered problem which does not make any assumptions on the rank of the channel matrix or the maximum transmit power. To achieve this, the considered problem is transformed into an equivalent minimax problem. We then exploit the special structure of the minimax problem to derive a closed-form solution based on a concave-convex procedure (CCP)-like algorithm. Extensive simulation results show that our proposed algorithm outperforms the existing solutions in terms of complexity and generality.

This paper considers the design of hybrid precoders and combiners for mmWave MIMO systems with per-antenna power constraints and the additional limitations introduced by the phase-shifting network in the analog processing stage. Previous hybrid designs were obtained using a total power constraint, but in practical implementations per-antenna constraints are more realistic, specially at mmWave, given the large number of power amplifiers used in the transmit array. Assuming perfect channel knowledge, we obtain first an approximation to the alldigital solution for the precoder and the combiner given the per-antenna constraints. Then, we develop a new method for the design of the hybrid precoder and combiner which attempts to match such all- digital approximation. Simulation results show the effectiveness of the proposed approach, which performs close to the all-digital solution.

We determine the capacity-optimal transmission strategy for a multiple-input-multiple-output (MIMO) Gaussian channel under multiple power constraints, namely joint sum power constraint (SPC), per group power constraints (PGPC), and per antenna power constraints (PAPC). First, we focus on cases where we can analytically determine the optimal transmit strategy under joint SPC-PGPC-PAPC. We obtain results for the following cases: 1) $n_{t} \times 1$ multiple-input-single-output (MISO); 2) MIMO channel with full column rank and full rank optimal covariance matrix; and 3) $2 \times n_{r}$ MIMO channel. These results generalize some recent results for the special cases of PAPC only and joint SPC-PAPC. Then, we propose a projected factored gradient descent (PFGD) algorithm for the general MIMO Gaussian channel under joint SPC-PGPC-PAPC including the possibility of additional rank constraints. This algorithm matches the solution of standard convex optimization tools with lower complexity. The algorithm also overcomes the limitations of existing algorithms, in terms of accuracy and applicability to low rank channels.

MISO NOMA downlink beamforming optimization with per-antenna power constraints

Large antenna arrays at the base station can facilitate power efficient single user downlink communication due to the inherent array power gain, i.e., under an average only total transmit power constraint, for a fixed desired information rate, the required total transmit power can be reduced by increasing the number of base station antennas (e.g. with i.i.d. fading, the required total transmit power can be reduced by roughly 3 dB with every doubling in the number of base station antennas, i.e., an O(N) array power gain can be achieved with N antennas). However, in practice, building power efficient large antenna arrays would require power efficient amplifiers/analog RF components. With current technology, highly linear power amplifiers generally have low power efficiency, and therefore linearity constraints on power amplifiers must be relaxed. Under such relaxed linearity constraints, the transmit signal that suffers the least distortion is a signal with constant envelope (CE). In this paper, we consider a single user Gaussian multiple-input single-output (MISO) downlink channel where the signal transmitted from each antenna is constrained to have a constant envelope (i.e., for every channel-use the amplitude of the signal transmitted from each antenna is constant, irrespective of the channel realization). We show that under such a per-antenna CE constraint, the complex noise-free received signal lies in the interior of a “doughnut” shaped region in the complex plane. The per-antenna CE constrained MISO channel is therefore equivalent to a doughnut channel, i.e., a single-input single-output (SISO) AWGN channel where the channel input is constrained to lie inside a “doughnut” shaped region. Using this equivalence, we analytically compute a closed-form expression for an achievable information rate under the per-antenna CE constraint. We then show that, for a broad class of fading channels (i.i.d. and direct-line-of-sight (DLOS)), even under the more stringent per-antenna CE constraint (compared to the average only total power constraint), an O(N) array power gain can still be achieved with N base station antennas. We also show that with N ≫1, compared to the average only total transmit power constrained channel, the extra total transmit power required under the per-antenna CE constraint, to achieve a desired information rate is small and bounded for a broad class of fading channels (i.i.d. and DLOS). We also propose novel CE precoding algorithms. The analysis and algorithms presented are general and therefore applicable to conventional systems with a small number of antennas. Analytical results are supported with numerical results for the i.i.d. Rayleigh fading channel.

Block diagonalization (BD) is a practically favorable precoding technique that eliminates the interuser 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 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, this paper studies the BD precoder design to maximize the weighted sum-rate achievable for all the MSs. The associated optimization problem can be formulated in an auxiliary MIMO broadcast channel (BC) with a set of transmit power constraints equivalent to those for different BSs in the multi-cell system. Based on convex optimization techniques, this paper designs an efficient algorithm to solve this problem, and derives the structure of the corresponding optimal BD precoding matrix. Moreover, for the special case of single-antenna BSs and MSs, it is shown that the proposed solution leads to the optimal zero-forcing beamforming (ZF-BF) precoder design for the multiple-input single-output (MISO) BC with the per-antenna power constraints.

We consider the downlink transmission of a wireless communication system where M antennas transmit independent information to a subset of K users, each equipped with a single antenna. The Shannon capacity of this MIMO broadcast channel (MIMO-BC) can be achieved using a non-linear preceding technique known as dirty paper coding (DPC) which is difficult to implement in practice. Motivated to study simpler transmission techniques, we focus on a linear precoding technique based on the zero-forcing (ZF) algorithm. In contrast to the typical sum power constraint (SPC), we consider a per-antenna power constraint (PAPC) motivated both by current antenna array designs where each antenna is powered by a separate amplifier and by future wireless networks where spatially separated antennas transmit cooperatively to users. We show that the problem of power allocation for maximizing the weighted sum rate under ZF with PAPC is a constrained convex optimization problem that can be solved using conventional numerical optimization techniques. For the special case of two users, we find an analytic solution based on waterfilling techniques. For the case where the number of users increases without bound, we show that ZF with PAPC is asymptotically optimal in the sense that the ratio of the expected sum-rate capacities between ZF with PAPC and DPC with SPC approaches one. We also show how the results can be generalized for multiple frequency bands and for a hybrid power constraint. Finally, we provide numerical results that show ZF with PAPC achieves a significant fraction of the optimum DPC sum-rate capacity in practical cases where K is bounded

We study the design of linear precoders for secure transmission in the two-user multiple-input single-output (MISO) broadcast channel with confidential messages (BC-CM). The transmitter has multiple antennas, and each user has a single receive antenna. Two independent messages are simultaneously transmitted, one intended for each user, and each message should be kept confidential from the other user. Assuming real-valued transmitted signals, we design the linear precoders subject to total and per-antenna average power constraints, and also subject to amplitude constraints. In both cases, we tackle the design problem via weighted secrecy sum rate maximization. The resulting problem, however, involves a fractional objective, making it nonconvex and difficult to solve. Nevertheless, we show that this difficult problem can be transformed into a more tractable problem, for which a solution can be obtained by an iterative search algorithm. In addition, we characterize a condition under which the obtained solution is guaranteed to be optimal. Furthermore, we show that the problem formulation and solution approach can be easily extended to handle the robust version of the design problem with uncertain channel information. We provide numerical examples to demonstrate the performance of the proposed precoder in terms of the achievable secrecy rate regions subject to the aforementioned constraints. We also demonstrate the performance of the robust precoder under different channel uncertainty levels.

This paper studies fast downlink beamforming algorithms using deep learning in multiuser multiple-input-single-output systems where each transmit antenna at the base station has its own power constraint. We focus on the signal-to-interference-plus-noise ratio (SINR) balancing problem which is quasi-convex but there is no efficient solution available. We first design a fast subgradient algorithm that can achieve near-optimal solution with reduced complexity. We then propose a deep neural network structure to learn the optimal beamforming based on convolutional networks and exploitation of the duality of the original problem. Two strategies of learning various dual variables are investigated with different accuracies, and the corresponding recovery of the original solution is facilitated by the subgradient algorithm. We also develop a generalization method of the proposed algorithms so that they can adapt to the varying number of users and antennas without re-training. We carry out intensive numerical simulations and testbed experiments to evaluate the performance of the proposed algorithms. Results show that the proposed algorithms achieve close to optimal solution in simulations with perfect channel information and outperform the alleged theoretically optimal solution in experiments, illustrating a better performance-complexity tradeoff than existing schemes.

This paper investigates a new analog beamforming architecture for massive multiple-input multiple-output (MIMO) systems, where each of the multiple transmit antennas is switched to be on or off to form a beam according to the channel state information at transmitters. This on–off analog beamforming (OABF) scheme has the advantages in terms of both hardware complexities and algorithmic complexities. OABF can completely remove the high-cost, power-consuming, and bulky analog phase shifters that are extensively employed by traditional analog beamforming schemes; it only requires the deployment of low-cost analog switches that are easy to implement. Moreover, we show that the beams formed by such simple antenna on–off switch operations can achieve rather good performances with low-complexity beamforming algorithms. Specifically, we first propose two optimal signal-to-noise ratio maximization algorithms to determine the on–off state of each switch under the per-antenna power constraint and the total power constraint, respectively. After that, we theoretically prove that OABF can achieve the full diversity gain and the full array gain with complexities up to a polynomial order. Numerical results are consistent with our theoretical analysis. We believe that the simple structure of OABF makes massive MIMO much easier to implement in real systems.

A novel unified approach to deal with arbitrary transmitted power constraints for designing jointly the MMSE precoders and decoders in downlink multiuser MIMO systems is developed. This work is an extension of the iterative approach developed in (Jinfan Zhang et al., 2005) for the total power constraint. An explicit expression relating Lagrange multipliers with the per-user constraint can be obtained using the proposed approach to mitigate the time consuming root searching for the Lagrange multipliers in (Jinfan Zhang et al., 2005). Moreover, the explicit expression allows the MMSE solution subject to arbitrary transmit power constraints. In this paper, total, per-user, per-antenna, and peak power constraints are formulated in under the same format and solved using the proposed approach. It is shown in some two-user numerical examples that the raw BER performance obtained from the practical per-antenna power constraint is similar to those of the per-user and total power constraints and is 1-1.5 dB better than that of the peak power constraint.