Energy efficiency optimization in MIMO broadcast channels with fairness constraints

We study the multiuser diversity in both multiple-input multiple output (MIMO) multiple-access channel (MAC) and MIMO broadcast channel (BC). We consider multiuser scheduling with linear receivers for uplink and linear precoders for downlink. In each time slot, a group of users are selected in either transmission or reception. Based on the performance analysis, a criterion of selecting a number of active users to optimize error performance is presented. We show that the linear processing for downlink and uplink achieves the same SNR. This makes it convenient to use a single scheduling algorithm in both downlink and uplink. Furthermore, we present an analysis of the performance of the proposed scheduling algorithm. It is shown that the analysis agrees with the simulation.

In this paper, Multi-Stream Receive-Spatial Modulation (MSR-SM) for application to the Multiple-Input Multiple-Output (MIMO) broadcast channel is introduced and studied. MSR-SM is a closed-loop transmission scheme, which applies the concept of multistream space modulation at the receiver side. An accurate mathematical framework for the evaluation of the Bit Error Rate (BER) is proposed. In addition, the diversity order and coding gain of the new architecture are derived. Note that the proposed analytical framework takes into account both the small-scale fading and the system topology, and is directly applicable to the conventional MIMO broadcast channel. Compared with the state-of-the-art MIMO transmission in the broadcast channel, it is mathematically shown that MSR-SM achieves the same diversity order and a better coding gain, in the high Signal-to-Noise Ratio (SNR) regime. Finally, the proposed mathematical framework and the new findings are validated via Monte Carlo simulation results.

This paper considers the joint optimization of the transmitter beamforming filters and receiver combining filters for the multiple-input multiple-output (MIMO) broadcast channel. A low-complexity iterative algorithm to compute the filters is proposed and then a closed-form expression of the filters for the downlink with two transmit antennas is derived, avoiding the need for iterative computation in this case. To enable practical implementation, a new limited feedback algorithm is proposed that exploits the channel structure in the closed form solution, and is independent of the number of receive antennas. Performance is evaluated by Monte Carlo simulations as a function of the amount of feedback. With two receive antennas, the proposed method performs close to the sum capacity of the MIMO broadcast channel, without the need for multiuser diversity and with only limited feedback.

In Multiple-input multiple-output (MIMO) broadcast channels (BCs), the transmitter simultaneously broadcasts signals to multiple receivers at same frequency band, resulting in that the communication capacity is affected by both interference, channel fading and random noise. Although channel fading can be mitigated by transceivers, the signal-to-noise-ratio (SNR) of the practical communication system is still dynamically changing due to the randomly changing noise. Traditional MIMO transceiver optimization algorithms can not flexibly adapt to the dynamic changes of SNR, resulting in large performance degradation. In this paper, we comprehensively consider signal power and signal space dimensions of the received signal in MIMO BCs, and propose two transceiver optimization algorithms which can dynamically adapt to the variance of SNRs. In the proposed algorithms, SNR is adopted to be an adjustment factor to cope with its variance. When SNR is low, i.e, large noise, the algorithm parameters are automatically adjusted so that the signal power is preserved as much as possible to combat the loss of communication capacity caused by large random noise. Correspondingly, under the condition of high SNR environment, the algorithm parameters are adjusted automatically to effectively compress the inter-user-interference (IUI) and intra-user-inter-stream-interference (ISI) by optimizing signal space dimensions. Simulation results show that in different SNR environments, the proposed algorithms can automatically adjust the focus of optimization, so that the optimization of signal power and signal space dimensions can automatically adapt to different SNRs. Compared with traditional transceiver optimization algorithms, the proposed algorithms can improve the communication capacity within a large dynamic range of SNR.

We study the Gaussian multiple-input multiple-output (MIMO) broadcast channel with common and private messages. We first obtain an outer bound for the capacity region of the two-user discrete memoryless broadcast channel with common and private messages. We next show that if jointly Gaussian random variables are sufficient to evaluate this outer bound for the Gaussian MIMO broadcast channel, the dirty-paper coding (DPC) region is the capacity region of the Gaussian MIMO broadcast channel with common and private messages. However, we can evaluate only a loosened version of this outer bound, which yields the result that extending the DPC region in the common message rate direction by a fixed amount is an outer bound for the capacity region of the Gaussian MIMO broadcast channel with common and private messages. However, this fixed amount, i.e., the gap, is not finite for all channels. We derive the necessary and sufficient conditions for this gap to be finite.

The capacity-achieving coding scheme for the multiple-input multiple-output (MIMO) broadcast channel is dirty-paper coding. With this type of transmission scheme the optimal number of active users that receive data and the optimal power allocation strategy are highly dependent on the structure of the channel matrix and on the total transmit power available. In the context of packet-data access with adaptive transmission where mobile users are equipped with a single receive antenna and the base station has multiple transmit antennas, we study the optimal number of active users and the optimal power allocation. In the particular case of two transmit antennas, we prove that the optimal number of active users can be a non-monotonic function of the total transmit power. Thus not only the number of users that should optimally be served simultaneously depends on the user channel vectors but also on the power available at the base station transmitter. The expected complexity of optimal scheduling algorithms is thus very high. Yet we then prove that at most as many users as the number of transmit antennas are allocated a large amount of power asymptotically in the high-power region in order to achieve the sum-capacity. Simulations confirm that constraining the number of active users to be no more than the number of transmit antennas incurs only a marginal loss in spectral efficiency. Based on these observations, we propose low-complexity scheduling algorithms with sub-optimal transmission schemes that can approach the sum-capacity of the MIMO broadcast channel by taking advantage of multiuser diversity. The suitability of known antenna selection algorithms is also demonstrated. We consider the cases of complete and partial channel knowledge at the transmitter. We provide simulation results to illustrate our conclusions.

Recent information-theoretic results show the optimality of dirty-paper coding (DPC) in achieving the full capacity region of the Gaussian multiple-input multiple-output (MIMO) broadcast channel (BC).This paper presents a DPC based code design for BCs.We consider the case in which there is an individual rate/signal-to-interference-plus-noise ratio (SINR) constraint for each user.For a fixed transmitter power, we choose the linear transmit precoding matrix such that the SINRs at users are uniformly maximized, thus ensuring the best bit-error rate performance.We start with Cover's simplest two-user Gaussian BC and present a coding scheme that operates 1.44 dB from the boundary of the capacity region at the rate of one bit per real sample (b/s) for each user.We then extend the coding strategy to a two-user MIMO Gaussian BC with two transmit antennas at the base-station and develop the first limit-approaching code design using nested turbo codes for DPC.At the rate of 1 b/s for each user, our design operates 1.48 dB from the capacity region boundary.We also consider the performance of our scheme over a slow fading BC.For two transmit antennas, simulation results indicate a performance loss of only 1.4 dB, 1.64 dB and 1.99 dB from the theoretical limit in terms of the total transmission power for the two, three and four user case, respectively.

Zero-forcing (ZF) precoding scheme can achieve the asymptotic sum capacity as dirty-paper coding (DPC) in multiple-input multiple-output broadcast (MIMO-BC) channel when the number of users, K, approaches infinity. However, the gap between ZF and DPC is not negligible in a practical range of K, that is, K ≤ 100. The capacity loss is partly due to the excessive transmission power penalty incurred by ZF when the channel matrix of the selected user subset is poorly conditioned. To avoid this power penalty, we propose to use a variation of ZF, channel inversion regularization (CIR), as a precoding scheme in MIMO-BC channels. But, unlike the interference-free ZF, the problem of maximizing sum-rate capacity using CIR precoding becomes nonconvex, which cannot be solved by water-filling strategy. Thus, we propose an efficient algorithm based on gradient projection (GP) as the optimal power allocation strategy for selected users, and show that the proposed CIR precoding scheme can achieve asymptotically the optimum sum-rate of the DPC strategy. Moreover, simulation results show that the CIR precoding scheme with the proposed optimal power allocation scheme achieves better sum-rate performance than ZF for a wide range of K.

In this paper, we consider a general K-user Gaussian multiple-input multiple-output (MIMO) broadcast channel (BC). We assume that the channel state is deterministic and known to all the nodes. While the private-message capacity region is well known to be achievable with dirty paper coding (DPC), we are interested in the simpler linearly precoded transmission schemes. In particular, we focus on linear precoding schemes combined with rate-splitting (RS). First, we derive an achievable rate region with minimum mean square error (MMSE) precoding at the transmitter and joint decoding of the sub-messages at the receivers. Then, we study the achievable sum rate of this scheme and obtain two findings: 1) an analytically tractable upper bound on the sum rate that is shown numerically to be a close approximation, and 2) how to reduce the number of active streams -- crucial to the overall complexity -- while preserving the sum rate to within a constant loss. The latter results in two practical algorithms: a stream elimination algorithm and a stream ordering algorithm. Finally, we investigate the constant-gap optimality of linearly precoded RS with respect to the capacity. Our result reveals that, while the achievable rate of linear precoding alone can be arbitrarily far from the capacity, the introduction of RS can help achieve the capacity region to within a constant gap in the two-user case. Nevertheless, we prove that the RS scheme's constant-gap optimality does not extend to the three-user case. Specifically, we show, through a pathological example, that the gap between the sum rate and the sum capacity can be unbounded.

Consider a Gaussian multiple-input multiple-output (MIMO) multiple-access channel (MAC) with channel matrix $\mathbf{H}$ and a Gaussian MIMO broadcast channel (BC) with channel matrix $\mathbf{H}^{\mathsf{T}}$. For the MIMO MAC, the integer-forcing architecture consists of first decoding integer-linear combinations of the transmitted codewords, which are then solved for the original messages. For the MIMO BC, the integer-forcing architecture consists of pre-inverting the integer-linear combinations at the transmitter so that each receiver can obtain its desired codeword by decoding an integer-linear combination. In both cases, integer-forcing offers higher achievable rates than zero-forcing while maintaining a similar implementation complexity. This paper establishes an uplink-downlink duality relationship for integer-forcing, i.e., any sum rate that is achievable via integer-forcing on the MIMO MAC can be achieved via integer-forcing on the MIMO BC with the same sum power and vice versa. Using this duality relationship, it is shown that integer-forcing can operate within a constant gap of the MIMO BC sum capacity. Finally, the paper proposes a duality-based iterative algorithm for the non-convex problem of selecting optimal beamforming and equalization vectors, and establishes that it converges to a local optimum.

In this letter, the diversity order and the coding gain of Receive Space Modulation (RSM) in the Multiple-Input Multiple-Output (MIMO) broadcast channel are derived by taking into account both small and large scale fading. The considered linear precoding method is Zero Forcing (ZF). Note that the proposed framework is directly applicable to the conventional spatially multiplexed broadcast channel. Based on the derived mathematical framework, a theoretical criterion is provided, which determines the superiority between the deployment of RSM and Spatial MultipleXing (SMX) in the shadowing MIMO broadcast channel. Finally, the provided theoretical results are validated via Monte Carlo simulation results.

Precoding schemes in the framework of vector perturbation (VP) for the multiple-input multiple-output (MIMO) Gaussian broadcast channel (GBC) are investigated. The VP scheme, originally a “one-shot” technique, is generalized to encompass processing over multiple time instances. Using lattice-based extended alphabets (“perturbations”), and considering the infinite time-span extension limit, a lower bound on the achievable sum-rate using the generalized VP scheme is analytically obtained, by which it is shown to achieve the optimum sum-rate in the high signal-to-noise ratio (SNR) regime (both in terms of degrees-of-freedom and power offset). The above lower bound is constructively obtained by means of an efficient practically oriented suboptimum transmit energy minimization algorithm, which has merits of its own, and it demonstrates the significant performance enhancement that can be obtained by preprocessing over multiple symbol instances, potentially eliminating the gap to the ultimate performance at high SNRs. For the 2×2 MIMO GBC, the VP scheme is generalized further, and an inner bound for the entire achievable rate region is obtained, by which an interesting correspondence is identified to the ultimate capacity region, as obtained by “dirty paper” coding (DPC).

We provide an alternative proof for the capacity region of the degraded Gaussian multiple-input multiple-output (MIMO) broadcast channel. Our proof does not use the channel enhancement technique as opposed to the original proof of Weingertan and the alternative proof of Liu et al. Our proof starts with the single-letter description of the capacity region of the degraded broadcast channel, and by using it, obtains a tight (i.e., achievable) outer bound for the capacity region of the degraded Gaussian MIMO broadcast channel, by using two main technical tools. The first one is the generalized de Bruijn identity due to Palomar which provides a connection between the differential entropy and the Fisher information matrix. The second tool we use is an inequality due to Dembo which lower bounds the differential entropy in terms of the Fisher information matrix.

This letter investigates the secure degrees of freedom (SDoF) of reconfigurable intelligent surface (RIS) aided 2-user multiple-input multiple-output (MIMO) broadcast channel with confidential messages. In this model, transmitter sends corresponding secure messages to each receiver and ensure the other receiver cannot decode them. Compared to existing studies on broadcast channel with confidential messages, our model employs an active RIS that reduces the rank of channel matrices, enlarging the nullspace for secure transmission to increase the SDoF. In particular, our model can realize secure communication even when the SDoF of original model is zero. According to the different configurations of antennas, the problem is divided into several regimes. We present outer bound and corresponding proof for each regime by utilizing theoretical methods and RIS beamforming. Then we propose universal achievable schemes for different regimes by using zero-forcing decoding. Finally, we determine the exact SDoF of RIS-assisted MIMO broadcast channel with confidential messages and its SDoF gain compared to original model.

In this paper, we present a game-theoretic approach for the purpose of deriving the problem of joint beamforming and power control in cognitive radio (CR) multiple-input multiple-output (MIMO) broadcast channels (CR MIMO-BCs), where the primary users (PUs) coexist with the secondary users (SUs) and they share the same spectrum. The cognitive base station (CBS), which is equipped with multiple antennas, is capable of transmitting data to the SU’s multiple-antenna receiver by employing the technology of beamforming. The proposed approach is an application of separable games, which are formally stated by the subgames of beamforming and power control. Furthermore, based on the model of noncooperative separate games, separable cost functions for the parameters of beamforming and power control are also proposed, showing that these cost functions are convex. Therefore, the convex theory of a noncooperative game can be employed to investigate the best response strategies as well as existence of Nash equilibrium solutions. Finally, we propose an iterative algorithm to achieve the optimal Nash equilibrium of the proposed joint beamforming subgame and power control subgame. Numerical results verify both the convergence and the tracking properties of the proposed algorithm for variant scenarios.