Quantization Methods for Equal Gain Transmission With Finite Rate Feedback

We consider the design and analysis of quantizers for equal gain transmission (EGT) systems with finite rate feedback-based communication in flat-fading multiple input single output (MISO) systems. EGT is a beamforming technique that maximizes the MISO channel capacity when there is an equal power-per-antenna constraint at the transmitter, and requires the feedback of t-1 phase angles, when there are t antennas at the transmitter. In this paper, we contrast two popular approaches for quantizing the phase angles: vector quantization (VQ) and scalar quantization (SQ). On the VQ side, using the capacity loss with respect to EGT with perfect channel information at transmitter as performance metric, we develop a criterion for designing the beamforming codebook for quantized EGT (Q-EGT). We also propose an iterative algorithm based on the well-known generalized Lloyd algorithm, for computing the beamforming vector codebook. On the analytical side, we study the performance of Q-EGT and derive closed-form expressions for the performance in terms of capacity loss and outage probability in the case of i.i.d. Rayleigh flat-fading channels. On the SQ side, assuming uniform scalar quantization and i.i.d. Rayleigh flat-fading channels, we derive the high-resolution performance of quantized EGT and contrast the performance with that of VQ. We find that although both VQ and SQ achieve the same rate of convergence (to the capacity with perfect feedback) as the number of feedback bits B increases, there exists a fixed gap between the two     

Performance analysis of MIMO beamforming with imperfect feedback

The quality of channel state information at the transmitter (CSIT) is critical to MIMO beamforming systems. However, in practical wireless systems, CSIT suffers from imperfections originating from quantization effects, feedback error and feedback delay. In this paper, we study the impact of feedback error and delay on the symbol error rate of MIMO beamforming systems with finite rate feedback. The feedback channel is modeled as a uniform symmetric channel. We derive an symbol error rate upper bound that is tight for a good beamformer. We also quantify the diversity gain and array gain loss due to the feedback error and delay. The codebook design method that is applicable to the beamforming systems with error or delay feedback is discussed. Both analytical and simulation results show that feedback error and delay will make the system behave badly at high signal‐to‐noise ratios. Copyright © 2012 John Wiley & Sons, Ltd.     

Outage probability of multiple-input single-output (MISO) systems with delayed feedback

We investigate the effect of feedback delay on the outage probability of multiple-input single-output (MISO) fading channels. Channel state information at the transmitter (CSIT) is a delayed version of the channel state information available at the receiver (CSIR). We consider two cases of CSIR: (a) perfect CSIR and (b) CSI estimated at the receiver using training symbols. With perfect CSIR, under a short-term power constraint, we determine: (a) the outage probability for beamforming with imperfect CSIT (BF-IC) analytically, and (b) the optimal spatial power allocation (OSPA) scheme that minimizes outage numerically. Results show that, for delayed CSIT, BF-IC is close to optimal for low SNR and uniform spatial power allocation (USPA) is close to optimal at high SNR. Similarly, under a longterm power constraint, we show that BF-IC is better for low SNR and USPA is better at high SNR. With imperfect CSIR, we obtain an upper bound on the outage probability with USPA and BF-IC. Results show that the loss in performance due to imperfection in CSIR is not significant, if the training power is chosen appropriately.     

Throughput-Optimal Precoding and Rate Allocation for MISO Systems With Noisy Feedback Channels

A throughput metric is considered for a multiple-input single-output (MISO) system with noisy feedback of channel state information (CSI). The goal is to optimize a precoding matrix with a medium-access control layer metric. The problem is a nonlinear multidimensional optimization. Results show that the optimal precoding turns into beamforming when the signal-to-noise ratio (SNR) of CSI feedback is sufficiently large. A necessary condition for the optimality of beamforming under the throughput metric is determined, and the necessary and sufficient condition is numerically found based on the Gauss-Chebyshev Quadrature method. Next, the rate allocation for beamforming and spatial diversity is analyzed. Then, a two-mode transmission scheme is proposed such that the transmitter is engaged in either the beamforming mode or the spatial diversity mode depending on the SNR of the CSI feedback. It is shown that at a fairly high SNR of CSI feedback, the rate allocation needs to be reduced, while at a low SNR of CSI feedback, the allocated rate should be increased. It is shown that when the SNR of CSI feedback is lower than a threshold, there always exists an SNR of the transmitted signal such that the CSI feedback can be viewed as the real CSI solely for the purpose of rate allocation. The result also shows that the throughput of two-mode transmission is almost the same as the throughput of the optimal precoding scheme, even with a low SNR and large feedback delay.     

Exponential Diversity Achieving Spatio–Temporal Power Allocation Scheme for Fading Channels

In this paper, we analyze optimal (in space and time) adaptive power transmission policies for fading channels when the channel-state information (CSI) at the transmitter (CSIT) and the receiver (CSIR) is available. The transmitter has a long-term (time) average power constraint. There can be multiple antennas at the transmitter and at the receiver. The channel experiences Rayleigh fading. We consider beamforming and space-time coded systems with perfect/imperfect CSIT and CSIR. The performance measure is the bit error rate (BER). We show that in both coded and uncoded systems, our power allocation policy provides exponential diversity order if perfect CSIT is available. We also show that, if the quality of CSIT degrades then the exponential diversity is retained in the low SNR region but we get only polynomial diversity in the high SNR region. Another interesting conclusion is that in case of imperfect CSIT and CSIR, knowledge of CSIT at the receiver is very important. Finally, for the optimal power control policy of the uncoded system we find the error-exponents which provide the rate versus diversity-order tradeoff for this policy. This tradeoff is of an entirely different nature than the well-known Zheng-Tse tradeoff.     

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.     

On transceiver design and channel quantization for downlink multiuser MIMO systems with limited feedback

We consider a MIMO broadcast channel where both the transmitter and receivers are equipped with multiple antennas. Channel state information at the transmitter (CSIT) is obtained through limited (i.e., finite-bandwidth) feedback from the receivers that index a set of precoding vectors contained in a predefined codebook. We propose a novel transceiver architecture based on zero-forcing beamforming and linear receiver combining. The receiver combining and quantization for CSIT feedback are jointly designed in order to maximize the expected SINR for each user. We provide an analytic characterization of the achievable throughput in the case of many users and show how additional receive antennas or higher multiuser diversity can reduce the required feedback rate to achieve a target throughput.We also propose a design methodology for generating codebooks tailored for arbitrary spatial correlation statistics. The resulting codebooks have a tree structure that can be utilized in time-correlated MIMO channels to significantly reduce feedback overhead. Simulation results show the effectiveness of the overall transceiver design strategy and codebook design methodology compared to prior techniques in a variety of correlation environments.     

部分信道状态信息下多天线系统的最优发送

在实际的单用户多天线平坦衰落通信系统中,接收端往往具有理想的信道状态信息,而发送端只有来自接收端的部分信道状态信息反馈,因此在发送端信道模型假设为复高斯随机矩阵.在发射端具有信道协反差反馈或者均值反馈的情形下,对达到最大的信道容量即信息论角度的最优化问题进行了理论分析,研究了系统的最优发送方案.对目前的关于单方向发射的最优条件进行扩展,进一步推导了沿任意多个方向发送达到信道容量的条件.数值结果验证了分析结论.     

On the capacity of some channels with channel state information

We study the capacity of some channels whose conditional output probability distribution depends on a state process independent of the channel input and where channel state information (CSI) signals are available both at the transmitter (CSIT) and at the receiver (CSIR). When the channel state and the CSI signals are jointly independent and identically distributed (i.i.d.), the channel reduces to a case studied by Shannon (1958). In this case, we show that when the CSIT is a deterministic function of the CSIR, optimal coding is particularly simple. When the state process has memory, we provide a general capacity formula and we give some more restrictive conditions under which the capacity has still a simple single-letter characterization, allowing simple optimal coding. Finally, we turn to the additive white Gaussian noise (AWGN) channel with fading and we provide a generalization of some results about capacity with CSI for this channel. In particular, we show that variable-rate coding (or multiplexing of several codebooks) is not needed to achieve capacity and, even when the CSIT is not perfect, the capacity achieving power allocation is of the waferfilling type     

CSIT获得中带有错误呈现的鲁棒THP设计

In this paper, a robust THP is proposed for broadcast channel in multi-user multi-input multi-output (MU-MIMO) system, in which the channel uncertainty caused by feedback delay and mobile speed is taken into account. In order to acquire the imperfect channel state information at transmitter (CSIT), a dynamic CSIT model is introduced where the channel temporal and spatial correlations are considered. By utilizing the unchanged channel statistics and accurate outdated feedback channel information jointly, the...     

Limited Feedback Beamforming Over Temporally-Correlated Channels

Feedback of quantized channel state information (CSI), called limited feedback, enables transmit beamforming in multiple-input-multiple-output (MIMO) wireless systems with a small amount of overhead. Due to its efficiency, beamforming with limited feedback has been adopted in several wireless communication standards. Prior work on limited feedback commonly adopts the block fading channel model where temporal correlation in wireless channels is neglected. In this paper, we consider temporally correlated channels and design single-user transmit beamforming with limited feedback. Analytical results concerning CSI feedback are derived by modeling quantized CSI as a first-order finite-state Markov chain. These results include the information rate of the CSI quantizer output, the bit rate a CSI feedback channel is required to support, and the effect of feedback delay on throughput. In particular, based on the theory of Markov chain convergence rate, feedback delay is proved to reduce the throughput gain due to CSI feedback at least exponentially. Furthermore, an algorithm is proposed for CSI feedback compression in time. Combining the results in this work leads to a new method for designing limited feedback beamforming as demonstrated by a design example.     

Practical robust uniform channel decomposition scheme for CSI mismatch in limited feedback MIMO systems

Uniform channel decomposition (UCD) has been proven to be optimal in bit error rate (BER) performance and strictly capacity lossless when perfect channel state information (CSI) is assumed to be available at both the transmitter and receiver side. In practice, CSI can be obtained by channel estimation at receiver and conveyed to transmitter via a limited-rate feedback channel. In such case, the implementation of traditional UCD by treating the imperfect CSI as perfect CSI cause significant performance degradation due to inevitable channel estimation error and vector quantization error. To overcome this problem, a practical robust UCD scheme was proposed in this paper, which includes two steps, firstly, a matching architecture was proposed to eliminate the mismatch between CSI at receiver (CSIR) and CSI at transmitter (CSIT), secondly, an MMSE based robust UCD scheme considering channel estimation error and vector quantization error as an integral part of the design was derived. Simulation results show that the proposed practical robust UCD scheme is capable of improving the BER performance greatly in the context of channel estimation error and vector quantization error compared with the traditional UCD scheme.     

Systematic Codebook Designs for Quantized Beamforming in Correlated MIMO Channels

The full diversity gain provided by a multi-antenna channel can be achieved by transmit beamforming and receive combining. This requires the knowledge of channel state information (CSI) at the transmitter which is difficult to obtain in practice. Quantized beamforming where fixed codebooks known at both the transmitter and the receiver are used to quantize the CSI has been proposed to solve this problem. Most recent works focus attention on limited feedback codebook design for the uncorrelated Rayleigh fading channel. Such designs are sub-optimal when used in correlated channels. In this paper, we propose systematic codebook design for correlated channels when channel statistical information is known at the transmitter. This design is motivated by studying the performance of pure statistical beamforming in correlated channels and is implemented by maps that can rotate and scale spherical caps on the Grassmannian manifold. Based on this study, we show that even statistical beamforming is near-optimal if the transmitter covariance matrix is ill-conditioned and receiver covariance matrix is well-conditioned. This leads to a partitioning of the transmit and receive covariance spaces based on their conditioning with variable feedback requirements to achieve an operational performance level in the different partitions. When channel statistics are difficult to obtain at the transmitter, we propose a universal codebook design (also implemented by the rotation-scaling maps) that is robust to channel statistics. Numerical studies show that even few bits of feedback, when applied with our designs, lead to near perfect CSI performance in a variety of correlated channel conditions.     

多用户MISO干扰信道中的安全协同波束成形

针对多用户多输入单输出(MISO, multiple-input single-output)干扰信道中保密信息泄露问题,提出了理想信道状态信息(CSI, channel state information)下的安全协同波束成形(SCB, secure coordinated beamforming)方案和非理想CSI下的顽健安全协同波束成形(RSCB, robust secure coordinated beamforming)方案。对于理想CSI情况,联合设计最优的协同波束成形向量,最大化最小安全速率。采用半定松弛(SDR, semidefinite relaxation)技术和连续的凸估计(SCA, successive convex approximation)算法得到原始非凸问题的局部最优解。进一步,将该框架扩展到信道向量和信道协方差矩阵存在确定误差的情况,提出的RSCB方案能够最大化最差情况的安全速率。仿真结果验证了所提方案的有效性和顽健性。     

Robust beamforming in the MISO downlink with quadratic channel estimation and optimal training

Estimation of the channel state information (CSI) in quadratic form (i.e., quadratic channel estimation) in the downlink can be performed at the base station by using the relayed signals from the mobile users, which facilitates optimization with transmitter CSI. In this letter, the condition for the optimal training sequence for quadratic channel estimation in a multiuser multiple-input single-output (MISO) antenna system in the downlink is first obtained. The mean-square-error (MSE) in the CSI estimate is then analyzed. Based on the quadratic CSI estimates, a robust beamforming optimization algorithm to minimize the base station power while achieving individual users' quality-of-service (QoS) constraints, measured by the MSE in data reception, is proposed.     

Degrees of freedom of MIMO interference channel with distributed space-time interference alignment

In the context of K-user MIMO interference channel (IC),achievable degrees of freedom (DoF) were investigated with distributed space-time interference alignment (DSTIA).By precoding with distributed current and outdated channel state information at the transmitters (CSIT),new tradeoff regions between achievable DoF and CSI feedback delay/frequency were achieved for MISO system.The impact of the number of transmit antennas on achievable DoF in the MISO system was analyzed,revealing that DoF results approach to the outer bound as the number of transmit antennas increases.Further,the impact of the number of receive antennas on achievable DoF was characterized,deriving the range of CSI feedback delay that preserves achievable DoF in the MIMO system.Theoretical and numerical analyses show that,the proposed DSTIA scheme can achieve better sum-DoFs by eliminating inter-user interference perfectly,tighten the gap between achievable DoF and outer bound,as well as improve the achievable rate of the system.     

On the Information Rate of MIMO Systems With Finite Rate Channel State Feedback Using Beamforming and Power On/Off Strategy

It is well known that multiple-input multiple-output (MIMO) systems have high spectral efficiency, especially when channel state information at the transmitter (CSIT) is available. In many practical systems, it is reasonable to assume that the CSIT is obtained by a limited (i.e., finite rate) feedback and is therefore imperfect. We consider the design problem of how to use the limited feedback resource to maximize the achievable information rate. In particular, we develop a low complexity power on/off strategy with beamforming (or Grassmann precoding), and analytically characterize its performance. Given the eigenvalue decomposition of the covariance matrix of the transmitted signal, refer to the eigenvectors as beams, and to the corresponding eigenvalues as the beam's power. A power on/off strategy means that a beam is either turned on with a constant power, or turned off. We will first assume that the beams match the channel perfectly and show that the ratio between the optimal number of beams turned on and the number of antennas converges to a constant when the numbers of transmit and receive antennas approach infinity proportionally. This motivates our power on/off strategy where the number of beams turned on is independent of channel realizations but is a function of the signal-to-noise ratio (SNR). When the feedback rate is finite, beamforming cannot be perfect, and we characterize the effect of imperfect beamforming by quantization bounds on the Grassmann manifold. By combining the results for power on/off and beamforming, a good approximation to the achievable information rate is derived. Simulations show that the proposed strategy is near optimal and the performance approximation is accurate for all experimented SNRs.     

非理想信道状态信息下RIS辅助的安全通信

利用可重构智能表面(Reconfigurable Intelligent Surface, RIS)辅助无线发射机,可提高多用户无线网络的安全传输能力。在非理想信道状态信息(Channel State Information, CSI)下提出了鲁棒波束形成优化方法来提高系统对抗干扰和窃听攻击的能力。具体地,使用RIS辅助发射机,对RIS的相位波束形成和基站的传输功率进行联合优化,在分别满足有界CSI的最坏情况速率约束和统计CSI的速率中断概率约束来最小化系统的总传输功率。由于存在CSI误差,针对有界CSI和统计CSI误差约束,分别利用S-procedure来松弛保密速率约束和大偏差不等式(Large Deviation Inequality, LDI)来松弛保密速率中断概率约束。仿真结果表明,相比于无源反射法和传统波束形成方案,该方法可分别降低约88%和93%总传输功率,同时提高约15 dBm和12 dBm的干扰容限。     

Achievable Rates with Imperfect Transmitter Side Information Using a Broadcast Transmission Strategy

rdquoWe investigate the performance of the broadcast approach for various fading distributions, which correspond to different models of partial transmit channel state information (CSI). The first model considered is the quantized limited feedback. In this model, the receiver can send as feedback only a finite number of bits describing the fading gain. We derive the optimal power allocation for the broadcast approach for the quantized feedback model. For a Rayleigh fading channel, numerical results here show that if the feedback word can be longer than one bit, the broadcasting gain becomes negligible, due to diminished channel uncertainty. The second partial transmit CSI model is a stochastic Gaussian model with mean and variance information, which is commonly used for modeling the channel estimation error. In a single-input single-output (SISO) channel, this model also corresponds to the Ricean fading distribution, for which we derive maximal achievable broadcasting rates. We further consider a multiple-input single-output (MISO) channel, and derive the optimal power allocation strategy in a broadcast approach. Numerical results here show that uniform power allocation is preferable over beamforming power allocation in the region where broadcasting gain over single level coding is non-negligible.     

Opportunistic Beamforming over Rayleigh Channels with Partial Side Information

In recent years, diversity techniques have evolved into highly attractive technology for wireless communications in different forms. For instance, the channel fluctuations of the users in a network are exploited as multiuser diversity by scheduling the user with the best signal-to-noise ratio (SNR). When fading is slow, beamforming at a multiple antenna transmitter is used to induce artificial channel fluctuations to ensure multiuser diversity in the network. Such a beamforming scheme is called opportunistic beamforming since the transmitter uses random beamforming to artificially induce opportunism in the network [1]. Opportunism requires a large number of users in the system in order to reach the performance of the true beamforming that uses perfect channel state information (CSI). In this paper we investigate the benefit of having partial CSI at an opportunistic transmitter. In the investigation, we focus on the maximum normalized SNR scheduling where user?s feedback consists of SNR relative to its channel gain. We show that opportunism can be beneficially used to increase the average throughput of the system. Simulations support the analytical average throughput results obtained as the amount of CSI and the number of users vary.