Limited feedback unitary precoding for spatial multiplexing systems

Multiple-input multiple-output (MIMO) wireless systems use antenna arrays at both the transmitter and receiver to provide communication links with substantial diversity and capacity. Spatial multiplexing is a common space-time modulation technique for MIMO communication systems where independent information streams are sent over different transmit antennas. Unfortunately, spatial multiplexing is sensitive to ill-conditioning of the channel matrix. Precoding can improve the resilience of spatial multiplexing at the expense of full channel knowledge at the transmitter-which is often not realistic. This correspondence proposes a quantized precoding system where the optimal precoder is chosen from a finite codebook known to both receiver and transmitter. The index of the optimal precoder is conveyed from the receiver to the transmitter over a low-delay feedback link. Criteria are presented for selecting the optimal precoding matrix based on the error rate and mutual information for different receiver designs. Codebook design criteria are proposed for each selection criterion by minimizing a bound on the average distortion assuming a Rayleigh-fading matrix channel. The design criteria are shown to be equivalent to packing subspaces in the Grassmann manifold using the projection two-norm and Fubini-Study distances. Simulation results show that the proposed system outperforms antenna subset selection and performs close to optimal unitary precoding with a minimal amount of feedback.     

Quantized Principal Component Selection Precoding for Spatial Multiplexing with Limited Feedback

In this paper, for spatial multiplexing with limited feedback, a quantized principal component selection (QPCS) precoding scheme is proposed that achieves comparable capacity to the closed-loop multiple-input multiple-output (MIMO) and furthermore adapts to various fading channel conditions without any additional feedback bits and transmit channel state information (CSI). We propose a systematic design method for a codebook consisting of a finite number of unitary matrices based on a maximizing minimum distance criterion in the one- dimensional angular domain and show that the method outperforms the Grassmannian subspace packing method in various fading channel conditions. The proposed QPCS precoding scheme allows for adjustment of the precoding matrix based on limited feedback information on the principal vectors approximating a MIMO channel in the angular domain according to various channel conditions. Furthermore, for practical implementation of the QPCS precoding scheme, we propose a structured precoder optimization procedure and show that the proposed procedure induces a negligible capacity loss compared with the exhaustive precoder optimization, even with considerably reduced complexity.     

基于遗传搜索和有限反馈的酉阵设计

在MIMO系统中,酉阵主要应用于差分空时调制（DUSTM）及有限反馈预编码两个方面。DUSTM主要应用于无法获得信道状态参数（CSI）情况下,而有限反馈预编码则应用于完全或部分可获得CSI情况下。文中提出了一种基于实型＂频率＂系数的遗传搜索方法,即使在系统维数很大的情况下,运用该方法获得酉阵设计的码字性能优于由整型＂频率＂系数穷举搜索而获得的码字。     

Limited feedback unitary precoding for orthogonal space-time block codes

Orthogonal space-time block codes (OSTBCs) are a class of easily decoded space-time codes that achieve full diversity order in Rayleigh fading channels. OSTBCs exist only for certain numbers of transmit antennas and do not provide array gain like diversity techniques that exploit transmit channel information. When channel state information is available at the transmitter, though, precoding the space-time codeword can be used to support different numbers of transmit antennas and to improve array gain. Unfortunately, transmitters in many wireless systems have no knowledge about current channel conditions. This motivates limited feedback precoding methods such as channel quantization or antenna subset selection. This paper investigates a limited feedback approach that uses a codebook of precoding matrices known a priori to both the transmitter and receiver. The receiver chooses a matrix from the codebook based on current channel conditions and conveys the optimal codebook matrix to the transmitter over an error-free, zero-delay feedback channel. A criterion for choosing the optimal precoding matrix in the codebook is proposed that relates directly to minimizing the probability of symbol error of the precoded system. Low average distortion codebooks are derived based on the optimal codeword selection criterion. The resulting design is found to relate to the famous applied mathematics problem of subspace packing in the Grassmann manifold. Codebooks designed by this method are proven to provide full diversity order in Rayleigh fading channels. Monte Carlo simulations show that limited feedback precoding performs better than antenna subset selection.     

What is the value of limited feedback for MIMO channels?

Feedback in a communications system can enable the transmitter to exploit channel conditions and avoid interference. In the case of a multiple-input multiple-output channel, feedback can be used to specify a precoding matrix at the transmitter, which activates the strongest channel modes. In situations where the feedback is severely limited, important issues are how to quantize the information needed at the transmitter and how much improvement in associated performance can be obtained as a function of the amount of feedback available. We give an overview of some recent work in this area. Methods are presented for constructing a set of possible precoding matrices, from which a particular choice can be relayed to the transmitter. Performance results show that even a few bits of feedback can provide performance close to that with full channel knowledge at the transmitter.     

一种OFDM中继系统的有限反馈预编码方法

基于正交频分复用的放大转发协作中继系统,为了减少系统误比特率、增加信道容量和增强实用性,该文提出一种实用的源节点预编码和中继节点预编码的联合优化有限比特反馈的预编码方法。利用3节点构成的两个下行链路,实现两跳协作通信方式;采用SVD分解和QR(orthogonal triangular)分解相结合设计预编码,其中源节点预编码每帧只要一个;优选量化码本,把预编码矩阵量化后反馈到发射端。仿真结果表明,该方案能提高平均和速率、降低误码率和改善中断概率特性,且反馈比特数较少具有更好的实用价值。     

Multimode basis selection for spatial multiplexing with limited feedback

Proposed is multimode basis selection in which the active bases are selected at the receiver from the common unitary basis set known at both receiving and transmitting ends, conveyed to the transmitter using limited feedback, and assembled into a precoding matrix at the transmitter. It is shown that the proposed multimode basis selection scheme outperforms both multimode antenna selection and Grassmannian multimode precoding in capacity in correlated fading channels without additional complexity and feedback.     

Channel prediction for precoded spatial multiplexing multiple-input multiple-output systems in time-varying fading channels

Conventional precoded spatial multiplexing multiple-input multiple-output (MIMO) systems using limited feedback are mainly based on the notion of time invariant channels throughout transmission. Consequently, the precoding matrix can be found during the training symbols and used over the subsequent data symbols. In this study, the authors consider a more practical system where the channel varies from one block of symbols to another. In such a scenario, the precoding matrix designed at the receiver based on the previous training symbols becomes outdated, which results in significant system performance degradation. In order to avoid this problem and reduce performance degradation, the authors propose the use of a Kalman filter linear predictor at the receiver to provide the transmitter with the precoding matrix for the next block of symbols. The performance of this method is assessed using computer simulation, and the obtained results for the proposed channel prediction demonstrate improved bit error rate performance for time-varying Rayleigh fading channels.     

Tomlinson-Harashima precoding minimizing total MSE with transmitter spatial correlation

This paper proposes a Tomlinson-Harashima precoding(THP) transceiver for multiple-input multiple-output(MIMO) system,where the spatial correlation information at the transmitter is included in the channel state information(CSI) model.It derives the total mean square error(MSE) and its lower bound as a function of precoding matrix.Then,a precoding matrix and the closed-form expression of minimum MSE lower bound are obtained by use of optimization and matrix theory.By right-multiplying a proper unitary matrix to the above precoding matrix,the paper develops the optimal precoding matrix,thus the optimal transceiver matrices are achieved.Simulation results show that the total MSE performance of the proposed method outperforms the existing linear method and the naive THP method.     

Precoder Partitioning in Closed-loop MIMO Systems

We study unitary precoding for multistream MIMO systems with partial channel state information at the transmitter. We introduce a quantization scheme in which the full space of non-equivalent precoding matrices is partitioned into Grassmannian and orthogonalization parts. The Grassmannian part is used for maximizing the power after precoding and the orthogonalization part is used for removing cross talk between the data streams. We show that orthogonalization improves the attainable capacity when the receiver is linear. We give a parametrization for the non-equivalent orhogonalization matrices and a metric which measures the orthogonality of the transmission. Optimal orthogonalization codebooks for two-stream transmission are presented. When feedback is limited, the optimal partitioning of feedback bits between Grassmannian and orthogonalization parts becomes an issue. In correlated scenarios, the number of feedback bits may be significantly reduced by investing bits into the orthogonalization part.     

Interpolation based transmit beamforming for MIMO-OFDM with limited feedback

Transmit beamforming and receive combining are simple methods for exploiting spatial diversity in multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) system. Optimal beamforming requires channel state information in the form of the beamforming vectors for each OFDM subcarrier. This paper proposes a limited feedback architecture that combines beamforming vector quantization and smart vector interpolation. In the proposed system, the receiver sends a fraction of information about the optimal beamforming vectors to the transmitter and the transmitter computes the beamforming vectors for all subcarriers through interpolation. A new spherical interpolator is developed that exploits parameters for phase rotation to satisfy the phase invariance and unit norm properties of the transmitted beamforming vectors. The beamforming vectors and phase parameters are quantized at the receiver and the quantized information is provided to the transmitter. The proposed quantization system provides only a moderate increase in complexity versus over comparable approaches. Numerical simulations show that the proposed scheme performs better than existing diversity techniques with the same feedback data rate.     

Precoding for Orthogonal Space–Time Block-Coded OFDM Downlink: Mean or Covariance Feedback?

This paper presents linear and nonlinear precoding design for error-rate improvement in orthogonal space–time block-coded (OSTBC) multiple-input–multiple-output (MIMO) orthogonal frequency-division-multiplexed (OFDM) downlink, where both the conditional mean of the channel gain matrix and the channel gain covariance matrix may be available at the transmitter. The conditional means of the channel matrix are derived for a general transmit-antenna-correlated frequency-selective fading MIMO channel with estimation errors and feedback delay. Mean-feedback linear precoding and nonlinear Tomlinson–Harashima precoding (THP) are developed to maximize the signal-to-noise power ratio (SNR). The intuition that when the mean feedback becomes accurate the mean-feedback precoding outperforms covariance precoding is confirmed. Dual-mode precoding is also proposed, in which the novel mean-feedback precoding or covariance precoding is adaptively chosen at the receiver. The precoding-mode switching metric is the maximized SNR, which is an indicator of the error rate. The receiver calculates its metric, selects the mode that achieves a higher SNR, and decides whether mean feedback is necessary. Our proposed precoders (both mean feedback and adaptive) significantly reduce the system error rate. Nonlinear precoding is shown to outperform linear precoding. Adaptive precoding outperforms both mean-feedback precoding and covariance precoding if individually applied in OSTBC OFDM.      

Approximate Minimum BER Power Allocation for MIMO Spatial Multiplexing Systems

Precoding for multiple-input multiple-output (MIMO) spatial multiplexing generally requires high feedback overhead and/or high-complexity processing. Simultaneous reduction in transmitter complexity and feedback overhead is proposed by imposing a diagonal structural constraint to precoding, i.e., power allocation. Minimum bit-error rate (MBER) is employed as the optimization criterion, and an approximate MBER (AMBER) power-allocation algorithm is proposed for a variety of receivers, including zero-forcing (ZF), successive interference cancellation (SIC), and ordered SIC (OSIC). While previously proposed precoding schemes either require ZF equalization for MBER, or use a minimum mean-squared error (MMSE) criterion, we provide a unified MBER solution to power allocation for ZF, SIC, and OSIC receiver structures. Improved error-rate performance is shown both analytically and by simulation. Simulation results also indicate that SIC and OSIC with AMBER power allocation offer superior performance over previously proposed MBER precoding with ZF equalization, as well as over MMSE precoding/decoding. Performance under noisy channels and power feedback is analyzed. A modified AMBER algorithm that mitigates error propagation in interference cancellation is developed. Compared with existing precoding methods, the proposed schemes significantly reduce both transmit processing complexity and feedback overhead, and improve error-rate performance     

Adaptive precoding and power allocation in distributed antenna systems with limited feedback

This paper introduces the limited feedback precoding into the distributed antenna system and proposes to adapt the predetermined orthogonal space time block codes to the available channel state information at the transmitter. The optimal representation of precoding information, namely the precoder, with least bits therefore becomes the key problem. Inspired by the characteristics of the distributed antenna system, we focus our work on the precoder construction, adaptable in response to the large and small scale fading, such that the symbol error probability is significantly reduced over that of a fixed, non‐adaptive, independent and identically distributed precoder codebook design. Furthermore, a suboptimal power‐loading strategy is presented by minimizing the derived tight upper bound on the average pairwise error probability of the precoded orthogonal space time block codes, which approaches the optimal performance asymptotically without additional channel knowledge other than the available feedback information. We prove that the proposed precoded orthogonal space time transmission scheme can achieve full diversity order. In particular, the robustness of our proposed transmission scheme to channel estimation error and feedback delay is respectively investigated in some detail, and numerical results show that it obviously improves the link reliability and obtains substantial gains even with few bits of feedback in comparison with conventional antenna selection scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

Non-unitary precoder design for reducing feedback overhead in MIMO-OFDM systems

The precoder feedback overhead in MIMO-OFDM systems is heavy as the number of subcarriers is usually large. Most of the current limited feedback works focus on unitary precoding schemes. Based on the fact that the channel matrices of neighboring subcarriers in MIMO-OFDM systems are correlated, an alternative transceiver is proposed here. In this paper, non-unitary precoders shared by several adjacent subcarriers, and the corresponding individual equalizers are jointly designed under partial CSI. As the number of precoding matrices is smaller, the feedback overhead can be reduced. The transceivers are designed to minimize the detection mean-squared error (MSE) under the total transmit power constraint. A convergent iterative algorithm based on the Lagrange multipliers method is proposed. The necessary conditions for an optimal transceiver, Karush–Kuhn–Tucker (KKT) conditions, are satisfied in each iteration. Numerical results under the frequency-selective MIMO-OFDM channel, which is generated by the classical Zheng and Xiao's model, prove that the proposed transceiver can significantly reduce the feedback overhead without severe performance degradation.     

基于码本的有限反馈非酉矩阵预编码多用户MIMO系统

该文提出了一种基于码本的有限反馈非酉矩阵预编码下行多用户MIMO系统。该方案根据用户反馈的信道信息SINR在发送端进行调度和预编码来提高系统容量。预编码的码本设计依据格拉斯曼空间装箱原理,并将码本中的向量按其相关性构成非酉矩阵来提高预编码增益和抑制多用户共道干扰。新方案反馈量少、复杂度低,在相同情况下比传统的单用户MIMO系统和基于码本的酉矩阵预编码多用户MIMO系统都具有更好的性能。     

多用户MIMO系统下行链路的随机多波束复用技术

该文提出了一种随机多波束多用户复用技术,充分利用多用户分集以及基站多天线的空间自由度来提高系统吞吐量。不同于传统的随机波束形成技术,该技术首先在给定预编码码本内随机选取一个码字,然后调度多个空分复用用户以及其余预编码矩阵。该文采用了一种逐次调度的方式,第一次训练调度一个主发送用户并确定一个次发送预编码矩阵,通过第二次训练选择次发送用户,这种方式能以很小的反馈开销有效控制复用用户之间的相互干扰。同时,该文所提技术能进一步推广到用户具有不同天线配置的异构情形。仿真结果表明,该文技术在具有不同相关性的信道环境下都能获得较优的系统吞吐量。     

Improved Multiuser MIMO Unitary Precoding Using Partial Channel State Information and Insights from the Riemannian Manifold

Multiple-input multiple-output (MIMO) systems can be leveraged to increase capacity in fading channels. Especially in multiuser downlink communication systems, it has been shown that knowledge of channel state information at the transmitter (CSIT) is critical to leverage the capacity gain available from multiple antennas. When duplexing is performed using time division, CSIT can often be successfully obtained when channel reciprocity is available. CSIT acquisition, however, is much more difficult in frequency division duplexing. Sending feedback on the uplink has been shown to be a powerful technique to improve downlink performance in single user MIMO systems. The basic idea is to restrict the CSIT to a B bit codebook so that the mobiles can easily transmit these bits on the uplink. In this paper, we consider the multiuser downlink model with unitary precoding when there is a codebook consisting of 2^B unitary matrices that the precoder is restricted to lie in. This codebook is designed offline and known to both the basestation and all users. Each user sends back signal-to-interference plus noise ratio (SINR) information along with binary feedback about the unitary precoder. Based on the CSIT received on the uplink, the basestation selects one of the unitary matrices in the codebook to maximize the sum-rate. For this set-up, we first analyze the sum-rate performance of the unitary precoding scheme. We then show that the codebook of unitary precoders represents a collection of points in a special kind of manifold and show how the achievable sum-rate performance relates to the minimum distance of the codebook points in this space. Finally, we present a framework for constructing the codebook to maximize this minimum distance. Monte Carlo simulation results are presented to show the sum-rate performance of the proposed codebook design.     

Channel Feedback Quantization for High Data Rate MIMO Systems

In this work, we study a multiple-input multiple-output (MIMO) wireless system where the channel state information is partially available at the transmitter through a feedback link. Based on singular value decomposition, the MIMO channel is split into independent sub-channels. Effective feedback of the required spatial channel information entails efficient quantization/encoding of a unitary matrix. We propose two schemes for quantizing unitary matrices via Givens rotations and examine the performance for a scenario where the rates allocated to the sub-channels are selected according to their corresponding gains. Numerical results show that the proposed schemes offer a significant performance improvement as compared to that of MIMO systems without feedback, with a negligible increase in the complexity     

Robust Tomlinson-Harashima Precoding for Two-Way Relaying

Most of the non-linear transceivers, which are based on Tomlinson-Harashima (TH) precoding and have been proposed in the literature for two-way relay systems, assume perfect channel state information (CSI). In this paper, a novel and robust TH precoding scheme has been investigated for two-way relay systems, with multiple antennas at the transceiver and relay nodes. We assume imperfect CSI and the channel uncertainty is bounded by a spherical region. The minimum mean square error at the destinations is used as the design criterion and we develop an iterative method to solve this nonconvex problem to obtain TH precoding matrices at the transmitter, linear precoding matrix at the relay and linear equalizer at the destination nodes, where each subproblem is convex. Simulations are provided to evaluate the performance and to validate the efficiency of the proposed scheme.