Spatial precoder design for space–time coded MIMO systems: based on fixed parameters of MIMO channels

In realistic channel environments the performance of space–time coded multiple-input multiple output (MIMO) systems is significantly reduced due to non-ideal antenna placement and non-isotropic scattering. In this paper, by exploiting the spatial dimension of a MIMO channel we introduce the novel idea of linear spatial precoding (or power-loading) based on fixed and known parameters of MIMO channels to ameliorate the effects of non-ideal antenna placement on the performance of coherent (channel is known at the receiver) and non-coherent (channel is un-known at the receiver) space–time codes. Antenna spacing and antenna placement (geometry) are considered as fixed parameters of MIMO channels, which are readily known at the transmitter. With this design, the precoder is fixed for fixed antenna placement and the transmitter does not require any feedback of channel state information (partial or full) from the receiver. We also derive precoding schemes to exploit non-isotropic scattering distribution parameters of the scattering channel to improve the performance of space–time codes applied on MIMO systems. However, these schemes require the receiver to estimate the non-isotropic parameters and feed them back to the transmitter. Closed form solutions for precoding schemes are presented for systems with up to three receive antennas. A generalized method is proposed for more than three receive antennas.     

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.     

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.     

Optimal minimum distance-based precoder for MIMO spatial multiplexing systems

We describe a new precoder based on optimization of the minimum Euclidean distance d/sub min/ between signal points at the receiver side and for use in multiple-input multiple-output (MIMO) spatial multiplexing systems. Assuming that channel state information (CSI) can be made available at the transmitter, the three steps ( noise whitening, channel diagonalization and dimension reduction), which are currently used in investigations on MIMO systems, are performed. Thanks to this representation, an optimal d/sub min/ precoder is derived in the case of two different transmitted data streams. For quadrature phase-shift keying (QPSK) modulation, a numerical approach shows that the precoder design depends on the channel characteristics. Comparisons with maximum signal-to-noise ratio (SNR) strategy and other precoders based on criteria, such as water-filling (WF), minimum mean square error (MMSE), and maximization of the minimum singular value of the global channel matrix, are performed to illustrate the significant bit-error-rate (BER) improvement of the proposed precoder.     

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.     

Interpolation Based Unitary Precoding for Spatial Multiplexing MIMO-OFDM With Limited Feedback

Spatial multiplexing with linear precoding is a simple technique for achieving high spectral efficiency in multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems. Linear precoding requires channel state information for each OFDM subcarrier, which can be achieved using feedback. To reduce the amount of feedback, this paper proposes a limited feedback architecture that combines precoder quantization with a special matrix interpolator. In the proposed system, the receiver sends information about a fraction of the precoding matrices to the transmitter and the transmitter reconstructs the precoding matrices for all the subcarriers. A new interpolator is proposed inspired by spherical interpolation that respects the orthogonal columns of the precoding matrices and the performance invariance to right multiplication by a unitary matrix. The interpolator is parameterized by a set of unitary matrices; a construction of a suitable set is briefly described. Simulations illustrate the performance of limited feedback precoding with coding, estimation or prediction error, and time variation for bit error rate (BER), mutual information, and mean squared error (MSE)     

Capacity of a Multiple-Antenna Fading Channel With a Quantized Precoding Matrix

Given a multiple-input multiple-output (MIMO) channel, feedback from the receiver can be used to specify a transmit precoding matrix, which selectively activates the strongest channel modes. Here we analyze the performance of random vector quantization (RVQ), in which the precoding matrix is selected from a random codebook containing independent, isotropically distributed entries. We assume that channel elements are independent and identically distributed (i.i.d.) and known to the receiver, which relays the optimal (rate-maximizing) precoder codebook index to the transmitter using $B$ bits. We first derive the large system capacity of beamforming (rank-one precoding matrix) as a function of $B$, where large system refers to the limit as $B$ and the number of transmit and receive antennas all go to infinity with fixed ratios. RVQ for beamforming is asymptotically optimal, i.e., no other quantization scheme can achieve a larger asymptotic rate. We subsequently consider a precoding matrix with arbitrary rank, and approximate the asymptotic RVQ performance with optimal and linear receivers (matched filter and minimum mean squared error (MMSE)). Numerical examples show that these approximations accurately predict the performance of finite-size systems of interest. Given a target spectral efficiency, numerical examples show that the amount of feedback required by the linear MMSE receiver is only slightly more than that required by the optimal receiver, whereas the matched filter can require significantly more feedback.      

Joint linear precoder and power allocation design for uplink MIMO systems with limited feedback

In this article, the authors consider joint design of a linear precoder and power allocation for uplink multiuser multiple input multiple output (MIMO) communication systems with limited feedback to improve the bit error rate (BER) performance for all users. Precoder selection from the codebook set is directly based on the exact BER performance, instead of other suboptimal criteria, to achieve the optimal precoder matrix, but closed-form expressions may not exist in the view of power allocation based directly on the BER criterion. From this perspective, the authors propose the joint transmitter optimization algorithm for the consideration ofprecoder design, with total power constraint for asymptotic MBER (AMBER) criterion. In this AMBER criterion, a closed-form solution has been derived for power allocation with an optimal precoder. The simulation results show that the proposed joint design algorithm can achieve a much better performance than precoding with uniform power allocation and only consideration of power allocation.     

Design of reduced complexity feedback precoding for MIMO‐OFDM

Beamforming technique is applied to significantly increase the performance of a MIMO system, if the channel information (CI) of the communication system is available at the transmitter. For the transmitter to obtain the entire CI, however, a considerable reverse‐link bandwidth is required. To save the bandwidth, a limited‐rate closed‐loop system, therefore, uses a predetermined codebook which is derived from the CI. The codebook consists of a finite number of precoders out of which the index of the best one is transmitted from the receiver to the transmitter using only a few bits saving substantial bandwidth. However, the amount of bits that need to be fed back can still be significantly large for MIMO‐OFDM systems when the precoding matrix index (PMI) for each subcarrier should be transmitted. Such per‐subcarrier precoding scheme has high feedback overhead and also incurs huge computational cost to determine the best PMI for each subcarrier. We, therefore, propose a per‐band precoding scheme that precodes a band (group) of subcarriers by only one precoder. More importantly, we develop, for the proposed per‐band scheme, reduced‐complexity precoding selection methods that lead to the design of efficient receivers. The effectiveness of the proposed scheme is investigated through computer simulations. Copyright © 2014 John Wiley & Sons, Ltd.     

A design framework for limited feedback MIMO systems with zero-forcing DFE

We consider the design of multiple-input multiple-output communication systems with a linear precoder at the transmitter, zero-forcing decision feedback equalization (ZFDFE) at the receiver, and a low-rate feedback channel that enables communication from the receiver to the transmitter. The channel state information (CSI) available at the receiver is assumed to be perfect, and based on this information the receiver selects a suitable precoder from a codebook and feeds back the index of this precoder to the transmitter. Our approach to the design of the components of this limited feedback scheme is based on the development, herein, of a unified framework for the joint design of the precoder and the ZF-DFE under the assumption that perfect CSI is available at both the transmitter and the receiver. The framework is general and embraces a wide range of design criteria. This framework enables us to characterize the statistical distribution of the optimal precoder in a standard Rayleigh fading environment. Using this distribution, we show that codebooks constructed from Grassmann packings minimize an upper bound on an average distortion measure, and hence are natural candidates for the codebook in limited feedback systems. Our simulation studies show that the proposed limited feedback scheme can provide significantly better performance at a lower feedback rate than existing schemes in which the detection order is fed back to the transmitter.     

Exploiting Quantized Channel Norm Feedback Through Conditional Statistics in Arbitrarily Correlated MIMO Systems

In the design of narrowband multi-antenna systems, a limiting factor is the amount of channel state information (CSI) available at the transmitter. This is especially evident in multi-user systems, where the spatial user separability determines the multiplexing gain, but it is also important for transmission-rate adaptation in single-user systems. To limit the feedback load, the unknown and multi-dimensional channel needs to be represented by a limited number of bits. When combined with long-term channel statistics, the norm of the channel matrix has been shown to provide substantial CSI that permits efficient user selection, linear precoder design, and rate adaptation. Herein, we consider quantized feedback of the squared Frobenius norm in a Rayleigh fading environment with arbitrary spatial correlation. The conditional channel statistics are characterized and their moments are derived for both identical, distinct, and sets of repeated eigenvalues. These results are applied for minimum mean square error (MMSE) estimation of signal and interference powers in single- and multi-user systems, for the purpose of reliable rate adaptation and resource allocation. The problem of efficient feedback quantization is discussed and an entropy-maximizing framework is developed where the post-user-selection distribution can be taken into account in the design of the quantization levels. The analytic results of this paper are directly applicable in many widely used communication techniques, such as space-time block codes, linear precoding, space division multiple access (SDMA), and scheduling.     

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。     

有限反馈预编码优化中基于性能准则的码字选择

多输入多输出(MIMO)空分复用系统中,采用有限比特反馈方式设计最优预编码矩阵,使用较少的反馈开销,可以有效地提高系统性能。对于有限反馈预编码设计方案,提出了一种基于Qos准则的码字选择方法,以最优化系统MSE性能和BER性能为目标选择最佳码字。仿真结果表明,与已有的码字选择方式相比,本文提出的方法更加直接,对于各种码本具有通用性,而且进一步改善了有限比特反馈预编码的性能,系统BER值降低了1.5 dB。     

多小区MIMO系统中基于泄漏的多用户下行链路预编码新算法

针对多小区MIMO系统的多用户下行链路预编码设计是未来移动通信系统研究的重要内容之一。本文在蜂窝多小区MIMO通信环境下,基于信漏噪声比(Signal-to-Leakage-Noise Ratio,SLNR)最大准则,提出了改进的多用户下行链路线性预编码算法。通过在优化目标中考虑接收端白化滤波器的影响,该预编码算法实现了对用户间干扰和小区间干扰(OCI)的同步抑制消除;在求解预编码矩阵优化问题时,本文基于QR分解提出了一种新的低复杂度求解算法,该算法能以较低的计算复杂度实现系统性能的有效提升。仿真实验结果表明,本文的改进SLNR线性预编码算法能有效抑制小区间干扰(OCI)的影响,使多小区MIMO系统获得更高的和速率(Sum Rate)容量。      

Progressive linear precoder optimization for MIMO packet retransmissions

This paper investigates the optimal linear precoder design for packet retransmissions in multi-input-multi-output (MIMO) systems. To fully utilize the time diversity provided by automatic repeat request (ARQ), we derive a sequence of successive optimal linear ARQ precoders for flat fading MIMO channels, which minimize the mean-square error between the transmitted data and the joint receiver output. The optimization is subject to an overall transmit power constraint. This progressive linear ARQ precoder combines the appropriate power loading and the optimal pairing of channel matrix singular values in the current retransmission with previous transmissions. This optimal pairing is a special feature unique to our sequential ARQ precoding approach. Simulation results demonstrate the effectiveness of this optimized ARQ precoding in reducing symbol MSE and detection bit-error rate.     

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.     

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.      

Optimal precoding for orthogonalized spatial multiplexing in closed-loop MIMO systems

In this paper, we propose a new precoding algorithm for orthogonalized spatial multiplexing (OSM) systems over flat-fading multiple-input multiple-output (MIMO) channels. The OSM scheme was recently introduced for closed-loop MIMO systems which allows single symbol decodable maximum likelihood detection. To further improve the performance of the OSM system, we propose a new precoding method by maximizing the minimum Euclidean distance between constellation points in the effective channel. In order to efficiently identify the parameters of a precoder which maximizes the minimum distance, we introduce a partitioning approach. Through analysis, it is shown that one real value parameter and two bits are required for feedback information for precoding in 16-QAM systems. Simulation results demonstrate that our algorithm provides 9 dB and 7.5 dB gains at a bit error rate (BER) of 10^-4 over the conventional OSM systems for 4-QAM and 16-QAM, respectively. We also confirm that the performance of the proposed scheme is the same as that of the optimum closed-loop MIMO systems in terms of the minimum distance. Consequently, our precoding algorithm significantly improves the system performance with a small increase of feedback amount.     

Linear block precoding for OFDM systems based on maximization of mean cutoff rate

A new linear block precoding technique is proposed to improve the performance of orthogonal frequency division multiplexing (OFDM) communication systems. The design of our precoder is based on the maximization of the mean cutoff rate and requires only the knowledge of the average relative channel multipath powers and delays at the transmitter. Simulation results show an improved performance of the proposed precoder relative to other known linear block precoding techniques.     

Antenna Selection for MIMO–OFDM WLAN Systems

This paper discusses the packet error rate (PER) performance of multiple-input multiple-output (MIMO) wireless systems. We focus our discussion on communication systems based on the IEEE 802.11a/g standard. In particular, we study the performance of spatial multiplexing systems with joint encoding at the transmitter and linear detection at the receiver. We show that spatial multiplexing systems based on minimum mean square error (MMSE) or zero forcing (ZF) demultiplexing benefit greatly from antenna subset selection. These results agree with recent analytical results showing the equivalence in diversity order between a full system (all receive antennas) and a system with antenna selection.