The capacity of discrete-time memoryless Rayleigh-fading channels

We consider transmission over a discrete-time Rayleigh fading channel, in which successive symbols face independent fading, and where neither the transmitter nor the receiver has channel state information. Subject to an average power constraint, we study the capacity-achieving distribution of this channel and prove it to be discrete with a finite number of mass points, one of them located at the origin. We numerically compute the capacity and the corresponding optimal distribution as a function of the signal-to-noise ratio (SNR). The behavior of the channel at low SNR is studied and finally a comparison is drawn with the ideal additive white Gaussian noise channel     

On the asymptotic capacity of stationary Gaussian fading channels

We consider a peak-power-limited single-antenna flat complex-Gaussian fading channel where the receiver and transmitter, while fully cognizant of the distribution of the fading process, have no knowledge of its realization. Upper and lower bounds on channel capacity are derived, with special emphasis on tightness in the high signal-to-noise ratio (SNR) regime. Necessary and sufficient conditions (in terms of the autocorrelation of the fading process) are derived for capacity to grow double-logarithmically in the SNR. For cases in which capacity increases logarithmically in the SNR, we provide an expression for the "pre-log", i.e., for the asymptotic ratio between channel capacity and the logarithm of the SNR. This ratio is given by the Lebesgue measure of the set of harmonics where the spectral density of the fading process is zero. We finally demonstrate that the asymptotic dependence of channel capacity on the SNR need not be limited to logarithmic or double-logarithmic behaviors. We exhibit power spectra for which capacity grows as a fractional power of the logarithm of the SNR     

Optimal Throughput-Diversity-Delay Tradeoff in MIMO ARQ Block-Fading Channels

In this paper, we consider an automatic-repeat-request (ARQ) retransmission protocol signaling over a block-fading multiple-input–multiple-output (MIMO) channel. Unlike previous work, we allow for multiple fading blocks within each transmission (ARQ round), and we constrain the transmitter to fixed rate codes constructed over complex signal constellations. In particular, we examine the general case of average input-power-constrained constellations with a fixed signaling alphabet of finite cardinality. This scenario is a suitable model for practical wireless communications systems employing orthogonal frequency division multiplexing (OFDM) techniques over a MIMO ARQ channel. Two cases of fading dynamics are considered, namely, short-term static fading where channel fading gains change randomly for each ARQ round, and long-term static fading where channel fading gains remain constant over all ARQ rounds pertaining to a given message. As our main result, we prove that for the block-fading MIMO ARQ channel with a fixed signaling alphabet satisfying a short-term power constraint, the optimal signal-to-noise ratio (SNR) exponent is given by a modified Singleton bound, relating all the system parameters. To demonstrate the practical significance of the theoretical analysis, we present numerical results showing that practical Singleton-bound-achieving maximum distance separable codes achieve the optimal SNR exponent.      

Model-based pilot and data power adaptation in psam with periodic delayed feedback

We consider the optimum design of pilot-symbolassisted modulation (PSAM) schemes with feedback. The received signal is periodically fed back to the transmitter through a noiseless delayed link and the time-varying channel is modeled as a Gauss-Markov process. We optimize a lower bound on the channel capacity which incorporates the PSAM parameters and Kalman-based channel estimation and prediction. The parameters available for the capacity optimization are the data power adaptation strategy, pilot spacing and pilot power ratio, subject to an average power constraint. Compared to the optimized open-loop PSAM (i.e., the case where no feedback is provided from the receiver), our results show that even in the presence of feedback delay, the optimized power adaptation provides higher information rates at low signal-to-noise ratios (SNR) in mediumrate fading channels. However, in fast fading channels, even the presence of modest feedback delay dissipates the advantages of power adaptation.     

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.     

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.      

一种平坦衰落信道下的盲信噪比估计算法

针对基于子空间分解信噪比估计算法中信号子空间维数估计复杂度高、小样本条件下估计偏差大的问题,提出了一种改进的盲信噪比估计算法.该算法首先构造接收信号的自相关矩阵,然后从矩阵奇异值序列的尾部开始,间隔两项依次进行差分得到梯度序列,再以梯度序列相邻两项均值大干特定阈值为条件确定信号子空间的维数,最后求得信噪比.仿真结果表明:信噪比范围为-5～+15 dB时,平坦衰落信道下常用调制信号的信噪比估计标准差小于0.1 dB,与MDL,AIC方法相比,该算法计算量小,且能适应更低的信噪比和更短的数据长度.     

On Achievable Data Rates and Optimal Power Allocation in Fading Channels with Imperfect Channel State Information

Achievable rates of wireless communication systems with pilot-based channel estimation are investigated for the case of time-selective fading. Novel analytical expressions for the maximum achievable rates of such systems are derived in terms of the system signal-to-noise ratio (SNR), fading rate and estimation scheme deployed. The frame size is optimized jointly based on the SNR and the fading rate. The maximum rate achieving coding scheme is suggested and shown to be a modified version of the classical water-filling algorithm that accounts for imperfect channel state information (CSI) at the transmitter. The impact of the estimation scheme and the angular spread of the received signal on the quality of estimation and achievable rates is evaluated. A number of numerical simulations are provided to illustrate the dependence of the optimal block length and achievable rates on SNR, fading rate, estimation scheme and angular spread of the channel.

Non Data Aided SNR Estimation for OFDM Signals in Frequency Selective Fading Channels

This paper deals with the problem of non data aided (NDA) signal to noise ratio (SNR) estimation of OFDM signals transmitted through unknown multipath fading channel. Most of present day’s SNR estimators are based on the knowledge of pilot sequences which is not applicable in some contexts such as cognitive radio for example. Moreover in Multipath fading channels SNR also depends on frequency offset which is caused by mismatch between the oscillator in the transmitter and that in the receiver. Previous NDA SNR estimation schemes assumed a perfect synchronization at reception (i.e. τ = 0 and ${\varepsilon = 0}$ ) which results estimation of SNR with less accuracy. The frequency offset attenuates the desired signal and causes intercarrier interference, thus reducing the SNR. In this paper we propose a new NDA SNR estimator which uses periodic redundancy induced by the cyclic prefix, considering SNR degradation due to frequency offset ( ${\varepsilon}$ ).     

Communication on the Grassmann manifold: a geometric approach tothe noncoherent multiple-antenna channel

We study the capacity of multiple-antenna fading channels. We focus on the scenario where the fading coefficients vary quickly; thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver. We use a noncoherent block fading model proposed by Marzetta and Hochwald (see ibid. vol.45, p.139-57, 1999). The model does not assume any channel side information at the receiver or at the transmitter, but assumes that the coefficients remain constant for a coherence interval of length T symbol periods. We compute the asymptotic capacity of this channel at high signal-to-noise ratio (SNR) in terms of the coherence time T, the number of transmit antennas M, and the number of receive antennas N. While the capacity gain of the coherent multiple antenna channel is min{M, N} bits per second per Hertz for every 3-dB increase in SNR, the corresponding gain for the noncoherent channel turns out to be M* (1 - M*/T) bits per second per Hertz, where M*=min{M, N, [T/2]}. The capacity expression has a geometric interpretation as sphere packing in the Grassmann manifold     

Optimizing the transmit power for slow fading channels

We consider the design of power-adaptive systems for minimizing the average bit-error rate over flat fading channels. Channel state information, obtained through estimation at the receiver, is sent to the transmitter over a feedback channel, where it is used to optimally adapt the transmit power. We consider finite-state optimal policies to reflect the limitations of the feedback channel. We develop an iterative algorithm that determines the optimal finite-state power control policy given the probability density function (PDF) of the fading. Next, we present a discretized formulation of the problem and obtain a suboptimal solution via standard dynamic programming techniques. The discretization of the problem enables us to obtain a suboptimal policy for arbitrary fading channels for which the analytic expression of the fading probability density function is not available. Simulation results are used to draw conclusions regarding the effects of limited feedback channel capacity, delay and number of states on the bit-error rate performance of the proposed policies under slow and moderate fading conditions     

Transmit signal design for optimal estimation of correlated MIMO channels

We address optimal estimation of correlated multiple-input multiple-output (MIMO) channels using pilot signals, assuming knowledge of the second-order channel statistics at the transmitter. Assuming a block fading channel model and minimum mean square error (MMSE) estimation at the receiver, we design the transmitted signal to optimize two criteria: MMSE and the conditional mutual information between the MIMO channel and the received signal. Our analysis is based on the recently proposed virtual channel representation, which corresponds to beamforming in fixed virtual directions and exposes the structure and the true degrees of freedom in the correlated channel. However, our design framework is applicable to more general channel models, which include known channel models, such as the transmit and receive correlated model, as special cases. We show that optimal signaling is in a block form, where the block length depends on the signal-to-noise ratio (SNR) as well as the channel correlation matrix. The block signal corresponds to transmitting beams in successive symbol intervals along fixed virtual transmit angles, whose powers are determined by (nonidentical) water filling solutions based on the optimization criteria. Our analysis shows that these water filling solutions identify exactly which virtual transmit angles are important for channel estimation. In particular, at low SNR, the block length reduces to one, and all the power is transmitted on the beam corresponding to the strongest transmit angle, whereas at high SNR, the block length has a maximum length equal to the number of active virtual transmit angles, and the power is assigned equally to all active transmit angles. Consequently, from a channel estimation viewpoint, a faster fading rate can be tolerated at low SNRs relative to higher SNRs.     

多普勒域上稀疏的双向中继信道估计

信道估计技术作为获得信道衰落信息的方法,是提高无线信道传输接收性能的关键技术。本文针对放大转发双向中继系统的时间选择性平坦衰落信道,利用信道在多普勒域的稀疏性进行压缩信道估计。相比于传统的线性估计方法,压缩信道估计考虑了信道的固有稀疏性,降低了导频的开销,改善了信道估计性能,提高了频谱利用率及系统吞吐量。文中通过对双向中继信道进行多普勒域的稀疏建模,仿真分析了信道估计性能随着导频数量增加、信噪比增加,得到不断改善;而不同的导频分布将影响观测矩阵的相关度,从而对信道估计产生影响。仿真表明,当导频随机分布时,信道估计效果最佳。同时,文中还仿真分析了最大多普勒频移对信道估计性能的影响。      

Outage analysis of the hybrid free-space optical and radio-frequency channel

We study the hybrid free-space optical (FSO) and radio-frequency (RF) channel from an information theoretic perspective. Since both links operate at vastly different carrier frequencies, we model the hybrid channel as a pair of parallel channels. Moreover, since the FSO channel signals at a higher rate than the RF channel, we incorporate this key feature in the parallel channel model. Both channels experience fading due to scintillation, which is slow compared to typical signalling rates. Under this framework, we study the fundamental limits of the hybrid channel. In particular, we analyse the outage probability in the large signal-to-noise ratio (SNR) regime, and obtain the outage diversity or SNR exponent of the hybrid system. First we consider the case when only the receiver has perfect channel state information (CSIR case), and obtain the exponents for general scintillation distributions. These exponents relate key system design parameters to the asymptotic outage performance and illustrate the benefits of using hybrid systems with respect to independent FSO or RF links. We next consider the case when perfect CSI is known at both the receiver and transmitter, and derive the optimal power allocation strategy that minimises the outage probability subject to peak and average power constraints. The optimal solution involves non-convex optimisation, which is intractable in practical systems. We therefore propose a suboptimal algorithm that achieves significant power savings (on the order of tens of dBs) over uniform power allocation. We show that the suboptimal algorithm has the same diversity as the optimal power allocation strategy.     

Tomlinson-Harashima precoding with partial channel knowledge

We consider minimum mean-square error Tomlinson-Harashima (MMSE-TH) precoding for time-varying frequency-selective channels. We assume that the receiver estimates the channel and sends the channel state information (CSI) estimate to the transmitter through a lossless feedback channel that introduces a certain delay. Thus, the CSI mismatch at the receiver is due to estimation errors, while the CSI mismatch at the transmitter is due to both estimation errors and channel time variations. We exploit a priori statistical channel knowledge, and we derive an optimal TH precoder, adopting a Bayesian approach. We use simulations to compare the performance of the so-derived TH precoder with that of the same-complexity MMSE decision-feedback equalizer (DFE). We observe that for low signal-to-noise ratios (SNRs) and sufficiently slow channel time variations, the optimal TH precoder outperforms the DFE, while at high SNR, the opposite happens.     

How much training is needed in multiple-antenna wireless links?

Multiple-antenna wireless communication links promise very high data rates with low error probabilities, especially when the wireless channel response is known at the receiver. In practice, knowledge of the channel is often obtained by sending known training symbols to the receiver. We show how training affects the capacity of a fading channel-too little training and the channel is improperly learned, too much training and there is no time left for data transmission before the channel changes. We compute a lower bound on the capacity of a channel that is learned by training, and maximize the bound as a function of the received signal-to-noise ratio (SNR), fading coherence time, and number of transmitter antennas. When the training and data powers are allowed to vary, we show that the optimal number of training symbols is equal to the number of transmit antennas-this number is also the smallest training interval length that guarantees meaningful estimates of the channel matrix. When the training and data powers are instead required to be equal, the optimal number of symbols may be larger than the number of antennas. We show that training-based schemes can be optimal at high SNR, but suboptimal at low SNR.     

Joint channel estimation and resource allocation for MIMO systems-part I: single-user analysis

Multiple antenna systems are known to provide very large data rates, when the perfect channel state information (CSI) is available at the receiver. However, this requires the receiver to perform a noise-free, multi-dimensional channel estimation, without using communication resources. In practice, any channel estimation is noisy and uses system resources. We shall examine the trade-off between improving channel estimation and increasing the achievable data rate. We consider transmitside correlated multi-input multi-output (MIMO) channels with block fading, where each block is divided into training and data transmission phases. The receiver has a noisy CSI that it obtains through a channel estimation process, while the transmitter has partial CSI in the form of covariance feedback. In Part I of this two-part paper, we consider the single-user case, and optimize the achievable rate jointly over parameters associated with the training phase and data transmission phase. In particular, we first choose the training signal to minimize the channel estimation error, and then, develop an iterative algorithm to solve for the optimum system resources such as time, power and spatial dimensions. Specifically, the algorithm finds the optimum training duration, the optimum allocation of power between training and data transmission phases, the optimum allocation of power over the antennas during the data transmission phase.     

Turbo space-time processing to improve wireless channel capacity

By deriving a generalized Shannon capacity formula for multiple-input, multiple-output Rayleigh fading channels, and by suggesting a layered space-time architecture concept that attains a tight lower bound on the capacity achievable. Foschini (see Wireless Pers. Commun., vol.6, no.3, p.311-35, 1998) has shown a potential enormous increase in the information capacity of a wireless system employing multiple-element antenna arrays at both the transmitter and receiver. The layered space-time architecture allows signal processing complexity to grow linearly, rather than exponentially, with the promised capacity increase. This paper includes two important contributions. First, we show that Foschini's lower bound is, in fact, the Shannon bound when the output signal-to-noise ratio (SNR) of the space-time processing in each layer is represented by the corresponding "matched filter" bound. This proves the optimality of the layered space-time concept. Second, we present an embodiment of this concept for a coded system operating at a low average SNR and in the presence of possible intersymbol interference. This embodiment utilizes the already advanced space-time filtering, coding and turbo processing techniques to provide yet a practical solution to the processing needed. Performance results are provided for quasi-static Rayleigh fading channels with no channel estimation errors. We see for the first time that the Shannon capacity for wireless communications can be both increased by N times (where N is the number of the antenna elements at the transmitter and receiver) and achieved within about 3 dB in average SNR about 2 dB of which is a loss due to the practical coding scheme we assume-the layered space-time processing itself is nearly information-lossless.     

Characterizing the statistical properties of mutual information in MIMO channels

We consider Gaussian multiple-input multiple-output (MIMO) frequency-selective spatially correlated fading channels, assuming that the channel is unknown at the transmitter and perfectly known at the receiver. For Gaussian codebooks, using results from multivariate statistics, we derive an analytical expression for a tight lower bound on the ergodic capacity of such channels at any signal-to-noise ratio (SNR). We show that our bound is tighter than previously reported analytical lower bounds, and we proceed to analytically quantify the impact of spatial fading correlation on ergodic capacity. Based on a closed-form approximation of the variance of mutual information in correlated flat-fading MIMO channels, we provide insights into the multiplexing-diversity tradeoff for Gaussian code books. Furthermore, for a given total number of antennas, we consider the problem of finding the optimal (ergodic capacity maximizing) number of transmit and receive antennas, and we reveal the SNR-dependent nature of the maximization strategy. Finally, we present numerical results and comparisons between our capacity bounds and previously reported bounds.     

Transmit Diversity Schemes for WCDMA

Previous work on transmitter antenna diversity has shown that the use of multiple transmitter antennas at the base station results in improved performance due to increase of diversity (path diversity). This happens with no bandwidth cost, even when the signal quality along several paths is poor, but the receiver estimates the channel with accuracy. This paper evaluates the effects of channel estimation errors in the performance of the schemes designated as Space-Time Transmitter Diversity (STTD) and Selective Transmitter Diversity (STD). We consider low to medium signal to noise ratios characteristic of Wideband Code Division Multiple Access (WCDMA) networks. Different fading channel models are considered for comparison between the two schemes. The improvement due to convolutional coding associated with these techniques is also evaluated.