On the capacity-achieving distribution of the discrete-time noncoherent and partially coherent AWGN channels

We investigate the communication limits over rapid phase-varying channels and consider the capacity of a discrete- time noncoherent additive white Gaussian noise (NCAWGN) channel under the average power constraint. We obtain necessary and sufficient conditions for the capacity-achieving input distribution and show that this distribution is discrete and possesses an infinite number of mass points. Using this characterization of the capacity-achieving distribution we compute a tight lower bound on the capacity of the channel based on examining suboptimal input distributions. In addition, we provide some easily computable lower and upper bounds on the channel capacity. Finally, we extend some of these results to the partially coherent channel, where it is assumed that a phase-locked loop (PLL) is used to track the carrier phase at the receiver, and that an ideal interleaver and de-interleaver are employed-rendering the Tikhonov distributed residual phase errors statistically independent from one symbol interval to another.     

脉冲超宽带(UWB)跳时系统的频域信道估计

提出一种基于序列和的最大似然(ML)频域信道估计算法,该算法可适用于对跳时UWB通信系统进行基于训练序列的信道估计和盲信道估计,大大简化了运算的复杂性。在仿真中比较了基于理想信道估计、基于一般的频域信道估计以及基于序列和的ML频域信道估计的性能。结果显示,在频域的估计过程中,对发射信号的频谱设定适当的可信门限后,可以获得与直接进行频域信道估计相似的性能。     

Large system performance of linear multiuser receivers in multipathfading channels

A linear multiuser receiver for a particular user in a code-division multiple-access (CDMA) network gains potential benefits from knowledge of the channels of all users in the system. In fast multipath fading environments we cannot assume that the channel estimates are perfect and the inevitable channel estimation errors will limit this potential gain. We study the impact of channel estimation errors on the performance of linear multiuser receivers, as well as the channel estimation problem itself. Of particular interest are the scalability properties of the channel and data estimation algorithms: what happens to the performance as the system bandwidth and the number of users (and hence channels to estimate) grows? Our main results involve asymptotic expressions for the signal-to-interference ratio of linear multiuser receivers in the limit of large processing gain, with the number of users divided by the processing gain held constant. We employ a random model for the spreading sequences and the limiting signal-to-interference ratio expressions are independent of the actual signature sequences, depending only on the system loading and the channel statistics: background noise power, energy profile of resolvable multipaths, and channel coherence time. The effect of channel uncertainty on the performance of multiuser receivers is succinctly captured by the notion of effective interference     

Channel estimation for ultra-wideband communications

This paper deals with channel estimation in ultra-wideband communications operating in a multipath environment and in the presence of multiaccess interference. The channel parameters are the attenuations and delays incurred by the signal echoes along the propagation paths. Time-hopping modulation with binary symbols is assumed. The estimation method is based on the maximum-likelihood criterion and is applied to two different scenarios: either with known symbols (DA estimation) or with unknown symbols (NDA estimation). The effects of the estimation errors on the performance of a RAKE receiver are assessed by simulation by comparing the receiver bit-error rate with either perfect channel estimates or imperfect estimates as obtained from the proposed algorithms. The results show that the degradations are tolerable as long as the number of users is limited. They also show that the DA method has an edge over the NDA in that it can handle a larger number of users for a fixed degradation. The number of users that can be accommodated in practice is found for some values of the system parameters.     

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.     

Performance of diversity schemes for OFDM systems with frequency offset, phase noise, and channel estimation errors

We provide expressions for the bit error rate of various transmit and receive diversity schemes for orthogonal frequency division multiplexing (OFDM) systems in the presence of frequency offset, phase noise, and channel estimation errors. The derivations are also applicable for a general multiplicative distortion of the received signal. Our results show that with perfect channel estimates, practical values of the phase noise do not significantly degrade the performance of the various diversity methods for binary phase-shift keying modulation. In contrast, the transmit diversity schemes for OFDM are much more sensitive to channel estimation errors than maximal ratio combining receive diversity.     

Space-time trellis codes over rapid rayleigh fading channels with channel estimation?part ii: performance analysis and code design for non-identical channels

We consider the maximum likelihood (ML) receiver design, performance analysis and code design for space-time trellis codes (STTC) over non-identical, rapid fading channels with imperfect channel state information (CSI). The exact pairwise error probability (PEP) and PEP bounds for the ML receiver are obtained. A new code design criterion exploiting the statistical information of the channel estimates is proposed, which can minimize the performance loss caused by channel estimation error. New codes are obtained via an iterative search algorithm with reduced complexity. Under actual channel estimation conditions, our codes perform better than the existing codes in the literature which are designed on the assumption of identical channels, and perfect CSI at the receiver. More performance gain can be achieved by our codes when the degree of imbalance among the links is higher.     

Iterative maximum-likelihood sequence estimation for space-timecoded systems

In previous work on decoding space-time codes, it is either assumed that perfect channel state information (CSI) is present, or a channel estimate is obtained using pilot symbols and then used as if it were perfect to extract symbol estimates. In the latter case, a loss in performance is incurred, since the resulting overall receiver is not optimal. We look at maximum-likelihood (ML) sequence estimation for space-time coded systems without assuming CSI. The log-likelihood function is presented for both-quasi-static and nonstatic fading channels, and an expectation-maximization (EM)-based algorithm is introduced for producing ML data estimates, whose complexity is much smaller than a direct evaluation of the log-likelihood function. Simulation results indicate the EM-based algorithm achieves a performance close to that of a receiver which knows the channel perfectly     

Multiple Antenna Enhancements for a High Rate CDMA Packet Data System

A High Data Rate (HDR) system has been proposed for providing downlink wireless packet service by using a channel-aware scheduling algorithm to transmit to users in a time-division multiplexed manner. In this paper, we propose using multiple antennas at the transmitter and/or at the receiver to improve performance of an HDR system. We consider the design tradeoffs between scheduling and multi-antenna transmission/detection strategies and investigate the average Shannon capacity throughput as a function of the number of antennas assuming ideal channel estimates and rate feedback. The highest capacities are achieved using multiple antennas at both the transmitter and receiver. For such systems, the best performance is achieved using a multi-input multi-output capacity-achieving transmission scheme such as BLAST (Bell Labs Layered Space-Time) in which the transmitted signal is coded in space and time, and the receive antennas are used to resolve the spatial interference. In the second part of the paper, we discuss practical transmitter and receiver architectures using BLAST for approaching the theoretical gains promised by the capacity analysis. Because the terminal receivers will be portable devices with limited computational and battery power, we perform a computational complexity analysis of the receiver and make high-level assessments on its feasibility. We conclude that the overall computational requirements are within the reach of current hardware technology.     

Multiple-Antenna Signaling Over Fading Channels With Estimated Channel State Information: Capacity Analysis

Multiple-antenna concepts for wireless communication systems promise high spectral efficiencies and improved error rate performance by proper exploitation of the randomness in multipath propagation. In this paper, we investigate the impact of channel uncertainty caused by channel estimation errors on the capacity of Rayleigh and Ricean block-fading channels. We consider a training-based multiple-antenna system that reserves a portion of time to sound the channel. The training symbols are used to estimate the channel state information (CSI) at the receiver by means of an arbitrary linear estimation filter. No CSI is assumed at the transmitter. Our analysis is based on an equivalent system model for training-based multiple-antenna systems which specifies the channel by the estimated (and hence, known) channel coefficients and an uncorrelated, data-dependent, multiplicative noise. This model includes the special cases of perfect CSI and no CSI. We present new upper and lower bounds on the maximum instantaneous mutual information to compute ergodic and outage capacities, and extend previous results to arbitrary (and possibly mismatched) linear channel estimators and to correlated Ricean fading. Several numerical results for single- and multiple-antenna systems with estimated CSI are included as illustration.     

Capacity Analysis for Continuous Alphabet Channels With Side Information, Part II: MIMO Channels

In this second part of our two-part paper, we consider the capacity analysis for wireless mobile systems with multiple-antenna architectures. We apply the results of the first part to a commonly known baseband, discrete-time multiple-antenna system where both the transmitter and receiver know the channel's statistical law. We analyze the capacity for additive white Gaussian noise (AWGN) channels, fading channels with full channel state information (CSI) at the receiver, fading channels with no CSI, and fading channels with partial CSI at the receiver. For each type of channels, we study the capacity value as well as issues such as the existence, uniqueness, and characterization of the capacity-achieving measures for different types of moment constraints. The results are applicable to both Rayleigh and Rician fading channels in the presence of arbitrary line-of-sight and correlation profiles.     

BLIND CHANNEL ESTIMATION IN DELAY DIVERSITY FOR FREQUENCY SELECTIVE CHANNELS1

Delay diversity is an effective transmit diversity technique to combat adverse effects of fading. Thus far, previous work in delay diversity assumed that perfect estimates of current channel fading conditions are available at the receiver and training symbols are required to estimate the channel from the transmitter to the receiver. However, increasing the number of the antennas increases the required training interval and reduces the available time with in whichdata may be transmitted. Learning the channel coefficients becomes increasingly difficult for the frequency selective channels. In this paper, with the subspace method and the delay character of delay diversity, a channel estimation method is proposed, which does not use training symbols. It addresses the transmit diversity for a frequency selective channel from a single carrier perspective in the form of a simple equivalent flat fading model. Monte Carlo simulations give the performance of channel estimation and the performance comparison of our channel-estimation-based detector with decision feedback equalization, which uses the perfect channel information.     

Effects of imperfect channel sensing and estimation on the performance of cognitive radio systems

In realistic scenarios of cognitive radio (CR) systems, imperfect channel sensing may occur due to false alarms and miss detections. Channel estimation between the secondary user transmitter and another secondary user receiver is another challenge in CR systems, especially for frequency‐selective fading channels. In this context, this paper presents a study of the effects of imperfect channel sensing and channel estimation on the performance of CR systems. In particular, different methods of channel estimation are analyzed under channel sensing imperfections. Initially, a CR system model with channel sensing errors is described. Then, the expectation maximization (EM) algorithm is implemented in order to learn the channel fading coefficients. By exploiting the pilot symbols and the detected symbols at the secondary user receiver, we can estimate the channel coefficients. We further compare the proposed EM estimation algorithm with different estimation algorithms such as the least squares (LS) and linear minimum mean square error (LMMSE). The expressions of channel estimates and mean squared errors (MSE) are determined, and their dependencies on channel sensing uncertainty are investigated. Finally, to reduce the complexity of EM algorithm, a sub‐optimal algorithm is also proposed. The obtained results show that the proposed sub‐optimal algorithm provides a comparable bit error rate (BER) performance with that of the optimal one yet with less computational complexity.     

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.     

Characterization and computation of optimal distributions for channel coding

This paper concerns the structure of capacity-achieving input distributions for stochastic channel models, and a renewed look at their computational aspects. The following conclusions are obtained under general assumptions on the channel statistics. i) The capacity-achieving input distribution is binary for low signal-to-noise ratio (SNR). The proof is obtained on comparing the optimization equations that determine channel capacity with a linear program over the space of probability measures. ii) Simple discrete approximations can nearly reach capacity even in cases where the optimal distribution is known to be absolutely continuous with respect to Lebesgue measure. iii) A new class of algorithms is introduced based on the cutting-plane method to iteratively construct discrete distributions, along with upper and lower bounds on channel capacity. It is shown that the bounds converge to the true channel capacity, and that the distributions converge weakly to a capacity-achieving distribution.     

Decision-directed estimation of MIMO time-varying Rayleigh fading channels

This paper presents a decision-directed (DD) maximum a posteriori probability (MAP) channel-estimation scheme for multiple-input multiple-output (MIMO) time-varying fading channels. With the estimate of the channel matrix for the current symbol interval, a zero-forcing (ZF) receiver is applied to detect the spatially multiplexed data on a symbol-by-symbol basis. Symbol decisions are then fed to the channel predictor for estimation of channel coefficients in future symbol intervals. Simulated error performance of a ZF receiver with the DD MAP and perfect channel estimates is provided and compared.     

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     

Capacity-achieving input covariance for single-user multi-antenna channels

We characterize the capacity-achieving input covariance for multi-antenna channels known instantaneously at the receiver and in distribution at the transmitter. Our characterization, valid for arbitrary numbers of antennas, encompasses both the eigenvectors and the eigenvalues. The eigenvectors are found for zero-mean channels with arbitrary fading profiles and a wide range of correlation and keyhole structures. For the eigenvalues, in turn, we present necessary and sufficient conditions as well as an iterative algorithm that exhibits remarkable properties: universal applicability, robustness and rapid convergence. In addition, we identify channel structures for which an isotropic input achieves capacity.     

Capacity-achieving probability measure for conditionally Gaussian channels with bounded inputs

A conditionally Gaussian channel is a vector channel in which the channel output, given the channel input, has a Gaussian distribution with (well-behaved) input-dependent mean and covariance. We study the capacity-achieving probability measure for conditionally Gaussian channels subject to bounded-input constraints and average cost constraints. Many practical communication systems, including additive Gaussian noise channels, certain optical channels, fading channels, and interference channels fall within this framework. Subject to bounded-input constraint (and average cost constraints), we show that the channel capacity is achievable and we derive a necessary and sufficient condition for a probability measure to be capacity achieving. Under certain conditions, the capacity-achieving measure is proved to be discrete.     

Performance Analysis of Optimum and Suboptimum Selection Diversity Schemes on Rayleigh Fading Channels With Imperfect Channel Estimates

In this paper, we present a comparative analysis on the effects of channel estimation errors on the performance of optimum and suboptimum selection diversity (SD) receivers on Rayleigh-fading channels. By modeling the estimation errors as independent complex Gaussian random variables, we derive simple closed-form expressions for the average probability of error for both optimum and suboptimum SD schemes with noisy channel estimates. With dual diversity and imperfect estimates, we establish a connection between optimum SD and maximal-ratio combining (MRC), and between suboptimum SD and equal-gain combining diversity schemes. Interestingly, we show that the optimum SD receiver structure and the resulting performance for differential binary coherent phase-shift keying (DBPSK) signaling can be obtained, in a straightforward way, as a special case of the performance of the optimum SD scheme with binary PSK signaling and channel estimation errors. For a fixed average power and bit duration, in conjunction with pilot-assisted minimum mean-square error channel estimation, we show that the optimum coherent SD scheme coincides with that of the optimum noncoherent SD scheme with binary frequency-shift keying (BFSK) signaling, whereas the coherent MRC scheme coincides with the optimum noncoherent receiver (i.e., the square-law combiner) for BFSK. The optimum number of diversity channels, under an energy-sharing mode of operation, is also studied. Finally, we formulate the problem of optimal pilot placement, consider channel estimation with a practical pilot-symbol-assisted modulation technique, and present some numerical results illustrating the comparative performances of various SD receivers