On the ergodic capacity as a function of the correlation properties in systems with multiple transmit antennas without CSI at the transmitter

In this letter, the impact of correlation of the transmit antennas of a multiple-input single-output (MISO) system, with no channel state information (CSI) at the transmitter and perfect CSI at the receiver is analyzed. We show that the ergodic capacity for the single-user MISO system is Schur-concave with respect to the vector with eigenvalues of the channel covariance matrix, i.e., the more correlation that exists between the transmit antennas, the less is the achievable capacity. Furthermore, the capacity loss for fully correlated transmit antennas in comparison with the uncorrelated case is derived. The results for the ergodic capacity are compared with the impact of correlation on the outage probability. The relationship between correlation properties and outage probability is more complicated than the relationship between the correlation properties and the ergodic capacity. It is shown that the outage probability is Schur-convex in the high signal-to-noise ratio (SNR) regime, and Schur-concave in the low SNR regime.     

Ergodic capacity and outage probability optimization for secondary user in cognitive radio networks under interference outage constraint

This paper considers a cognitive radio network where a secondary user (SU) coexists with a primary user (PU). The interference outage constraint is applied to protect the primary transmission. The power allocation problem to jointly maximize the ergodic capacity and minimize the outage probability of the SU, subject to the average transmit power constraint and the interference outage constraint, is studied. Suppose that the perfect knowledge of the instantaneous channel state information (CSI) of the interference link between the SU transmitter and the PU receiver is available at the SU, the optimal power allocation strategy is then proposed. Additionally, to manage more practical situations, we further assume only the interference link channel distribution is known and derive the corresponding optimal power allocation strategy. Extensive simulation results are given to verify the effectiveness of the proposed strategies. It is shown that the proposed strategies achieve high ergodic capacity and low outage probability simultaneously, whereas optimizing the ergodic capacity (or outage probability) only leads to much higher outage probability (or lower ergodic capacity). It is also shown that the SU performance is not degraded due to partial knowledge of the interference link CSI if tight transmit power constraint is applied.     

Multiple-antenna capacity in correlated Rayleigh fading with channel covariance information

We analyze a mobile multiple input multiple output wireless link with M transmit and N receive antennas operating in a spatially correlated Rayleigh flat fading environment. Only the correlations between the channel coefficients are assumed to be known at the transmitter and the receiver. The channel coefficients are correlated in space and uncorrelated in time from one coherence interval to another. These coefficients remain constant for a coherence interval of T symbol periods after which they change to another independent realization according to the spatial correlation model. For this system we characterize the structure of the input signal that achieves capacity. The capacity achieving transmit signal is expressed as the product of an isotropically distributed unitary matrix, an independent nonnegative diagonal matrix and a unitary matrix whose columns are the eigenvectors of the transmit fade covariance matrix. For the case where the number of transmit antennas M is larger than the channel coherence interval T, we show that the channel capacity is independent of the smallest M-T eigenvalues of the transmit fade covariance matrix. In contrast to the previously reported results for the spatially white fading model where adding more transmit antennas beyond the coherence interval length (M>T) does not increase capacity, we find that additional transmit antennas always increase capacity as long as their channel fading coefficients are spatially correlated with the other antennas. We show that for fast hopping or fast fading systems (T=1) with only channel covariance information available to the transmitter and receiver, transmit fade correlations are beneficial. Mathematically, we prove this by showing that capacity is a Schur-convex function of the vector of eigenvalues of the transmit fade correlation matrix. We also show that the maximum possible capacity gain due to transmitter fade correlations is 10logM dB.     

相关信道下MIMO信道容量优化算法的研究

该文研究MIMO系统收发端天线采用均匀线阵且放置空间有限,存在相关衰落时信道容量的优化方法。采用规范化信道模型,分析了信道相关性对平均信道容量和最优信号协方差矩阵的影响,推导了最优协方差矩阵的一阶条件;利用Jensen's不等式确定了信道容量的上界,给出了闭式解,并对相关信道下信号的传输模式进行了讨论。仿真结果表明,采用该优化方法,在各种SNR下,其平均容量接近Jensen's上界;得出信道相关程度对信道平均容量的影响依赖于信噪比的结论。     

Capacity of correlated MIMO Rayleigh channels

Abstract-This paper presents some exact results on the capacity of multiple-input-multiple-output (MIMO) channels subject to correlated Rayleigh fading when perfect channel state information (CSI) is known at the receiver. The authors focus on the semicorrelated scenario in which correlation exists either at the transmitter or at the receiver., They consider two cases: 1) the transmitter does not have any CSI and as such allocates power equally among transmitter antennas and 2) the transmitter only knows the statistical distribution of the channel. The first case derives the moment generating function (MGF) of the mutual information (MI) and then deduces from this MGF the mean MI. The authors also study the cumulative distribution function (CDF) of the MI, which can serve as an upper bound to the outage probability under the capacity versus outage formulation when the channel is nonergodic. The second case studies the capacity achieved by optimum power-loading and beamforming schemes based on covariance feedback. Numerical results illustrate that the full capacity of MIMO systems can be preserved even for relatively high values of correlation coefficients.     

On the Ergodic Capacity and Energy Efficiency for Fading MIMO NOMA Systems

Non-orthogonal multiple access (NOMA) is expected to be a promising multiple access techniques for 5G networks due to its superior spectral efficiency (SE). Previous research mainly focus on the design to improve the SE performance with instantaneous channel state information (CSI). In this paper, we consider the fading MIMO channels with only statistical CSI at the transmitter, and explore the potential gains of MIMO NOMA scheme in terms of both ergodic capacity and energy efficiency (EE). The ergodic capacity maximization problem is first studied for the fading multiple-input multiple-output (MIMO) NOMA systems. We derive the optimal input covariance structure and propose both optimal and low complexity suboptimal power allocation schemes to maximize the ergodic capacity of MIMO NOMA system. For the EE maximization, the optimization problem is formulated to maximize the system EE (defined by ergodic capacity under unit power consumption) under the total transmit power constraint and the minimum rate constraint of the weak user. By transforming the EE maximization problem into an equivalent one-dimensional optimization problem, the optimal power allocation for EE design is proposed. To further reduce the computation complexity, a near-optimal solution based on golden section search and suboptimal closed form solution are proposed as well. Numerical results show that the proposed NOMA schemes significantly outperform the traditional orthogonal multiple access scheme with traditional orthogonal multiple access transmission in terms of both SE and EE.     

Capacity of multiple-antenna systems with both receiver and transmitter channel state information

The capacity of multiple-antenna systems operating in Rayleigh flat fading is considered under the assumptions that channel state information (CSI) is available at both transmitter and receiver, and that the transmitter is subjected to an average power constraint. First, the capacity of such systems is derived for the special case of multiple transmit antennas and a single receive antenna. The optimal power-allocation scheme for such a system is shown to be a water-filling algorithm, and the corresponding capacity is seen to be the same as that of a system having multiple receive antennas (with a single transmitter antenna) whose outputs are combined via maximal ratio combining. A suboptimal adaptive transmission technique that transmits only over the antenna having the best channel is also proposed for this special case. It is shown that the capacity of such a system under the proposed suboptimal adaptive transmission scheme is the same as the capacity of a system having multiple receiver antennas (with a single transmitter antenna) combined via selection combining. Next, the capacity of a general system of multiple transmitter and receiver antennas is derived together with an equation that determines the cutoff value for such a system. The optimal power allocation scheme for such a multiple-antenna system is given by a matrix water-filling algorithm. In order to eliminate the need for cumbersome numerical techniques in solving the cutoff equation, approximate expressions for the cutoff transmission value are also provided. It is shown that, compared to the case in which there is only receiver CSI, large capacity gains are available with optimal power and rate adaptation schemes. The increased capacity is shown to come at the price of channel outage, and bounds are derived for this outage probability.     

Power allocation and code construction for space-time turbo trellis codes with partial CSI feedback

Conventional space-time turbo trellis coding (STTuTC) schemes allocate transmit power equally to the available antennas. This is not optimal if channel state information (CSI) is available at the transmitter. A STTuTC scheme is considered with partial CSI knowledge that optimally allocates power to the transmit antennas. This scheme is referred to as STTuTC with dynamic transmit power allocation (STTuTC/DTPA). The optimum four-state constituent code and power allocation are presented. Simulation results show that the STTuTC/DTPA scheme with the new code outperforms conventional STTuTC schemes.     

Optimizing antenna configuration for MIMO systems with imperfect channel estimation

We study the optimal antenna configuration (i.e. number of transmit and receive antennas) for multiple-input multiple-output systems in pilot-symbol-assisted modulation schemes with imperfect channel estimation. We assume block flat-fading channels and focus on a practical range of high signal-to-noise ratio. An ergodic capacity lower bound is used as the objective function to be maximized. We analytically study the capacity gain from adding extra antennas to the transmitter or to the receiver in two different scenarios. Our numerical results show that the optimal antenna configuration under imperfect channel estimation can be significantly different from that under perfect channel estimation assumption. In addition, we investigate the capacity gain from optimizing antenna configuration and find that the gain can be larger than that achieved by optimizing transmit power over pilot and data symbols, particularly for large block lengths.     

Cooperative power allocation with partial channel information in multi‐cell multi‐user dual‐hop MIMO relay systems

This paper considers cooperative power allocation with the use of partial channel state information (CSI) in a multi‐user dual‐hop relay system with multiple antennas. The end‐to‐end capacity can be improved by dynamically allocating the transmit power of the base station and relay according to co‐channel interference caused by the adjacent relays. The proposed scheme allocates the transmit power in association with the eigenvalues and angle difference between the eigenvectors of transmit correlation matrices of the desired and interference channel. It is shown by means of upper‐bound analysis that the end‐to‐end capacity of the proposed scheme can be maximized in highly correlated channel environments when the principal eigenvectors of transmit correlation matrices of the desired and interference channel are orthogonal to each other. It is also shown that the proposed scheme is robust to the channel estimation error. Finally, the performance of the proposed scheme is verified by the computer simulation. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimum Power Allocation for Single-User MIMO and Multi-User MIMO-MAC with Partial CSI

We consider both the single-user and the multi-user power allocation problems in MIMO systems, where the receiver side has the perfect channel state information (CSI), and the transmitter side has partial CSI, which is in the form of covariance feedback. In a single-user MIMO system, we consider an iterative algorithm that solves for the eigenvalues of the optimum transmit covariance matrix that maximizes the rate. The algorithm is based on enforcing the Karush-Kuhn-Tucker (KKT) optimality conditions of the optimization problem at each iteration. We prove that this algorithm converges to the unique global optimum power allocation when initiated at an arbitrary point. We, then, consider the multi-user generalization of the problem, which is to find the eigenvalues of the optimum transmit covariance matrices of all users that maximize the sum rate of the MIMO multiple access channel (MIMO-MAC). For this problem, we propose an algorithm that finds the unique optimum power allocation policies of all users. At a given iteration, the multi-user algorithm updates the power allocation of one user, given the power allocations of the rest of the users, and iterates over all users in a round-robin fashion. Finally, we make several suggestions that significantly improve the convergence rate of the proposed algorithms.     

Stochastic power allocation using causal channel information for delay-limited communications

Without delay constraints, ergodic capacity can be achieved through channel coding over time frames spanning the whole fading process of the channel without the need of channel state information (CSI) at the transmitter. If the channel capacity is delay-constrained, then causal CSI at the transmitter becomes important. In this letter, our aim is to resolve the optimal power-and-rate allocation in maximizing the expected capacity subject to total power constraint, given the causal CSI at the transmitter. The proposed solution is adaptive to the current and past CSI, and also the statistics of the future channels. Asymptotically in both the high and low signal-to-noise ratio (SNR) regimes, we show that at a particular constant in time, the optimal power policy has an interpretation of water-filling among the current channel and some estimates of the future channels     

Gallager?s Exponent for MIMO Channels: A Reliability-Rate Tradeoff

In this paper, we derive Gallager's random coding error exponent for multiple-input multiple-output (MIMO) Rayleigh block-fading channels, assuming no channel-state information (CSI) at the transmitter and perfect CSI at the receiver. This measure gives insight into a fundamental tradeoff between the communication reliability and information rate of MIMO channels, enabling to determine the required codeword length to achieve a prescribed error probability at a given rate below the channel capacity. We quantify the effects of the number of antennas, channel coherence time, and spatial fading correlation on the MIMO exponent. In addition, the general formulae for the ergodic capacity and the cutoff rate in the presence of spatial correlation are deduced from the exponent expressions. These formulae are applicable to arbitrary structures of transmit and receive correlation, encompassing all the previously known results as special cases of our expressions.     

Transmit antenna selection based strategies in MISO communication systems with low-rate channel state feedback

The performance of multiple-antenna communication systems is known to critically depend on the amount of channel state information (CSI) available at the transmitter. In the low-rate CSI feedback case, an important problem is what kind of information should be submitted to the transmitter in each feedback cycle and what is the optimal transmission strategy in this case. In this paper, we address this problem in the multiple-input single-output (MISO) case by analytically comparing the bit error rate (BER) performance of different low-rate feedback based transmitter strategies involving various combinations of transmit antenna selection, Alamouti's spacetime coding, and adaptive power allocation.     

基于Schmidt正交化的多天线选择复用

该文从提高信道传输有效性的角度,提出了一种新的多天线选择发送策略--贪婪搜索(GS)法,使用这种方法挑选出使容量最大化的发送天线组合作为传输信号的天线。此法在结合注水(waterfilling)法进行天线功率分配的情况下,与理想的全天线注水法相比,可以减小发射机的复杂度,在独立信道下传输容量略有损失,在相关信道下可以提高传输容量,并且所付出的代价是仅需要对信道矩阵进行Schmidt正交化变换。这种方法一般用于信道矩阵列不满秩,即发送端天线数大于接收端天线数,发送端已知信道矩阵的情况(与注水法结合)或者接收端已知信道矩阵的情况(等功率分配)。     

Robust transmit eigen beamforming based on imperfect channel state information

Transmit beamforming is a powerful technique for enhancing the performance and increasing the throughput of wireless communication systems that employ multiple antennas at the transmitter. A major drawback of most existing transmit beamforming techniques is that they require nearly perfect knowledge of the channel at the transmitter, which is typically not available in practice. Transmitter designs that address the imperfect channel state information (CSI) problem commonly use statistical models for the channel and/or mismatch between the presumed and actual transmitter CSI. Since these approaches are model based, they can suffer from mismodeling. In this paper, a more robust framework is proposed in which no statistical assumptions are made about the CSI mismatch or the channel. The goal is to design a transmitter that has the best performance under the worst-case CSI mismatch. The transmitter designed herein achieves this goal for all CSI mismatches below a certain threshold level. The proposed design combines beamforming along the eigenvectors of the (deterministic) autocorrelation of the channel matrix perceived by the transmitter and power loading across those beams. While the power-loading algorithm resembles conventional water-filling to some degree, it explicitly incorporates robustness to the CSI mismatch, and the water level can be determined in a simple systematic way.     

On the Secrecy Capacity of Fading Channels

We consider the secure transmission of information over an ergodic fading channel in the presence of an eavesdropper. Our eavesdropper can be viewed as the wireless counterpart of Wyner's wiretapper. The secrecy capacity of such a system is characterized under the assumption of asymptotically long coherence intervals. We first consider the full channel state information (CSI) case, where the transmitter has access to the channel gains of the legitimate receiver and the eavesdropper. The secrecy capacity under this full CSI assumption serves as an upper bound for the secrecy capacity when only the CSI of the legitimate receiver is known at the transmitter, which is characterized next. In each scenario, the perfect secrecy capacity is obtained along with the optimal power and rate allocation strategies. We then propose a low-complexity on/off power allocation strategy that achieves near-optimal performance with only the main channel CSI. More specifically, this scheme is shown to be asymptotically optimal as the average signal-to-noise ratio (SNR) goes to infinity, and interestingly, is shown to attain the secrecy capacity under the full CSI assumption. Overall, channel fading has a positive impact on the secrecy capacity and rate adaptation, based on the main channel CSI, is critical in facilitating secure communications over slow fading channels.      

协方差反馈时空时编码系统的线性预编码设计

本文讨论了在空间相关的多输入-多输出(MIMO)系统中利用发射端已知的信道协方差矩阵最优设计线性预编码矩阵的问题。推导出最优的预编码矩阵的方向与发射相关矩阵和空时编码的结构相关,而最优的功率分配方案与发射相关矩阵、接收相关矩阵和空时编码都相关。仿真结果表明本文推导出的最优功率分配方案的性能优于平均功率分配和单方向波束形成方案。     

MIMO Precoder Designs for Frequency-Selective Fading Channels Using Spatial and Path Correlation

Multiple-input–multiple-output (MIMO) precoder design for frequency-selective fading channels using partial channel information based on the spatial and path correlation matrices is presented. By representing a frequency-selective fading channel as a multipath model with $L$ effective paths, a general precoding structure is proposed and used to derive optimum precoding designs that maximize Jensen's upper bound on the channel ergodic capacity under the transmitted power constraint for two cases, i.e., uncorrelated and correlated channel paths. Analytical results show that, in the uncorrelated case, the precoder structure consists of a number of parallel precoders for frequency-flat fading channels. The power assignment to each precoder and the power allocation over the eigenmodes of each precoder are calculated based on the power of channel paths and the eigenvalues of the transmit correlation matrix. In the correlated case, the precoder structure is an eigenbeamformer with the beams referred to a function of eigenvectors of the Kronecker product of path and transmit correlation matrices. Furthermore, the power allocated to each eigenmode can be obtained from a statistical water-pouring policy that is specified by the product of eigenvalues of the transmit antenna and path correlation matrices. Simulation results for different scenarios indicate that the proposed precoder can increase the ergodic capacity of MIMO systems in a frequency-selective fading environment with spatial and path correlations, and its offered capacity gain is increased with the level of correlation and numbers of antennas and channel paths.      

Some results on the sum-rate capacity of MIMO fading broadcast channels

We study the ergodic sum-rate capacity of the fading MIMO broadcast channel which is used to model the downlink of a cellular system with N/sub t/ transmit antennas at the,base and K mobile users each having N/sub r/ receive antennas. Assuming perfect channel state information (CSI) for all users is available at the transmitter and the receivers, we evaluate the sum-rate capacity numerically using the duality between uplink and downlink. Assuming Nt K, we also derive both upper and lower bounds on the sum-rate capacity to study its increase rate due to multi-user diversity. Finally, we compare three transmission schemes which use the single-user-MIMO scheme (SU-MIMO), ranked known interference (RKI) and zero-forcing beamforming (ZFB), respectively, to transmit to a selected set of users in order to approach the sum-rate capacity. We show that both ZFB and RKI outperform SU-MIMO in a cellular downlink scenario. when many mobile users are present.