Error probability of coded STBC systems in block fading environments

In this letter, a union bound on the error probability of coded multi-antenna systems over block fading channels is derived. The bound is based on uniform interleaving of the coded sequence prior to transmission over the channel. Using this argument the distribution of error bits over the fading blocks is computed and the corresponding pair wise error probability (PEP) is derived. We consider coded systems that concatenate a binary code with a space-time block code (STBC). Coherent detection is assumed with perfect and imperfect channel state information (CSI) at the receiver, where imperfect CSI is obtained using pilot-aided estimation. Under channel estimation environments, the tradeoff between channel diversity and channel estimation is investigated and the optimal channel memory is approximated analytically. Results show that the performance degradation due to channel memory decreases as the number of transmit antennas is increased. Moreover, the optimal channel memory increases with increasing the number of transmit antennas.     

Performance of antenna diversity multiuser receivers in CDMA channels with imperfect fading estimation

The performance of antenna diversity coherent and differentially coherent linear multiuser receivers is analyzed in frequency-nonselective Rayleigh fading CDMA channels with memory. The estimates of the complex fading processes are utilized for maximal-ratio combining and carrier recovery of the coherent multiuser receiver. To analyze the impact of channel estimation errors on the receiver performance, error probability is assessed directly in terms of the fading rate and the number of active users, showing the penalty imposed by imperfect channel estimation as well as the fading-induced error probability floor. The impact of fading dynamics on the differentially coherent decorrelating receiver with equal-gain combining is quantified. While performance of multiuser receivers at lower SNR is determined by both the fading dynamics and the number of active CDMA users, performance at higher SNR is given by an error probability floor which is due to fading only and has the same value as in a single-user case. The comparison of the two receiver structures indicates that the coherent decorrelating receiver with diversity reception may be preferable to the differentially coherent one in nonselective fading CDMA channels with memory.     

Capacity regions and optimal power allocation for groupwise multiuser detection

In this letter, optimal power allocation and capacity regions are derived for groupwise successive interference cancellation (GSIC) systems operating in multipath fading channels, under imperfect channel estimation conditions. It is shown that the impact of channel estimation errors on the system capacity is two-fold: It affects the receiver performance within a group of users, as well as the cancellation performance (through cancellation errors). An iterative power allocation algorithm is derived, based on which it can be shown that that the total required received power is minimized when the groups are ordered according to their cancellation errors, and the first detected group has the smallest cancellation error. Performance/complexity tradeoff issues are also discussed by directly comparing the system capacity for different implementations: GSIC with linear minimum-mean-square error (LMMSE) receivers within the detection groups; GSIC with matched filter (MF) receivers; multicode LMMSE systems; and simple all MF receivers systems.     

The effects of channel estimation errors on a nonlinear precoder for multiple antenna downlink channels

The effects of channel estimation errors on the performance of a system employing Tomlinson-Harashima precoding and the QR decomposition, operating over multiple antenna frequency-flat fading channels with decentralized receivers, is evaluated. The QR decomposition of the channel matrix is employed to arrive at an equivalent channel where successive interference cancellation at the transmitter can be used to remove the effect of the multiuser interference. However, in the case of imperfect channel estimation, it is not possible to remove the effect of the multiuser interference due to a mismatch between the precoder and the channel. Consequently, it is necessary to include the estimation error in deriving the probability of symbol error. We also provide simulations to corroborate the analytical results.     

Multisampling decision-feedback linear prediction receivers for differential space-time modulation over Rayleigh fast-fading channels

Novel decision-feedback (DF) linear prediction (LP) receivers, which process multiple samples per symbol interval in conjunction with optimal sample combining, are proposed for differential space-time modulation (DSTM) over Rayleigh fast-fading channels. Performance analysis demonstrates that multisampling DF-LP receivers outperform their symbol-rate sampling counterpart in fast fading substantially. In addition, an asymptotically tight upper bound on the pairwise error probability is derived. In view of this bound, the design criterion of DSTM for fast fading is the same as that for block-wise static fading. To avoid the estimation of the second-order statistics of the channel, a polynomial-model-based DF-LP receiver is proposed. It can approach the performance of the optimum DF-LP receiver at high signal-to noise ratios, provided fading is moderate.     

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.     

Performance Analysis of Maximum-Likelihood Multiuser Detection in Space–Time-Coded CDMA With Imperfect Channel Estimation

In this paper, the performance of maximum-likelihood multiuser detection in space-time-coded code-division multiple-access (CDMA) systems with imperfect channel estimation is analyzed. A K-user synchronous CDMA system that employs orthogonal space-time block codes with M transmit antennas and N receive antennas is considered. A least-squares estimate of the channel matrix is obtained by sending a sequence of pilot bits from each user. The channel matrix is perturbed by an error matrix that depends on the thermal noise and the correlation between the signature waveforms of different users. Because of the linearity of the channel estimation technique, the characteristic function of the decision variable is used to obtain an exact expression for the pairwise error probability, and by using it, an upper bound on the bit error rate (BER) is obtained. The analytical BER bounds are compared with the BER obtained through simulations. The BER bounds are shown to be increasingly tight for large SNR values. It is shown that the degradation in BER performance due to imperfect channel estimation can be compensated by using a larger number of transmit/receive antennas     

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     

Turbo码在最大比合并分集接收系统中的性能分析

该文提出了turbo码在最大比合并分集接收无线通信系统中纠错性能的联合上界。基于误差积分函数的简化计算式,推导了分集接收的多路信道分别具有不同衰落特性时,成对差错概率的精确表达式。以瑞利衰落信道为例,对采用turbo码的分集接收无线系统的性能进行了数值计算和仿真。结果表明:turbo码纠错性能联合上界数值计算式与仿真结果一致,可普遍用于信道衰落统计特性各不相同的分集接收系统。     

Error performance of mismatched receivers for linear-codedmodulation

Optimum demodulation of intersymbol interference may be performed using maximum-likelihood sequence estimation receivers. Standard performance analysis of these receivers assume that the receiver has perfect knowledge of the channel. We assume instead a mismatched receiver that uses an imperfect estimate of the channel to decode the information sequence. A free-distance expression for a mismatched ML receiver is derived and applied to the case of linear-coded modulation. A truncated union bound approximation accurately predicts the error probability     

Lower bound on training-based channel estimation error for frequency-selective block-fading Rayleigh MIMO channels

A lower bound on the error correlation matrix of training-based channel estimators is derived for multiple-input multiple-output (MIMO) systems over block-fading frequency-selective channels with symbol-spaced receivers. The bound is obtained in a constructive way by evaluating the asymptotic performance of an estimator that fully exploits the algebraic structure of the multipath channel. In particular, the estimator is assumed to be able to estimate the long-term features of the channel consistently (e.g., second order statistics of fading, delays, angles) while tracking the fast-varying fading fluctuations by Wiener filtering. Known estimators that are able to attain the bound under simplified settings are referred to, and general guidelines for designing novel estimators are discussed. Based on the simple analytical expression of the bound, the impact of channel estimation error on the link capacity is investigated for different system parameters and channel characteristics (e.g., Doppler shift, spatial correlation of fading). Numerical results are provided to corroborate the analysis.     

Error performance of selection combining and switched combining systems in Rayleigh fading channels with imperfect channel estimation

In this paper, we present a general analysis of the performance of selection combining (SC), switch-and-stay combining (SSC), and switch-and-examine combining (SEC) systems in Rayleigh fading channels with imperfect channel estimation (ICE). The complex channel estimate and the actual fading are modeled as jointly Gaussian random variables. For SC systems with channel estimation error, closed-form expressions are obtained for the error rates of M/sub s/-ary pulse amplitude modulation (PAM) and rectangular-quadrature amplitude modulation (QAM), and simple single integral formulas with finite integration limits are derived for the symbol error probability of arbitrary two-dimensional (2-D) modulation formats. These error probability expressions are then applied to three types of channel estimation errors potentially encountered in practical systems to study their impact on the performance of selection diversity. Moreover, single integral formulas with finite integration limits are derived for the performance of SSC and SEC systems with minimum mean square error (MMSE) channel estimation. Optimum switching thresholds for 2-D modulation formats with MMSE based switched combining are acquired through numerical computation.     

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.     

Performance Analysis of Rate-Adaptive Modulation with Imperfect Estimation in Space?CTime Coded MIMO System

Based on the imperfect estimation information, the performance analysis of multi-input multi-out (MIMO) systems with rate-adaptive modulation and space?Ctime coding over flat Rayleigh fading channels is presented in this paper. The fading gain value is partitioned into a number of regions by which the modulation is adapted according to the region the fading gain falls in. Under a target bit error rate (BER) and constant power constraint, the fading gain region boundaries are given. By utilizing the minimum mean squared error estimation, the correlation between the channel gains and their estimates, which contributes to imperfect channel information, is evaluated. With this correlation, the probability density function of fading gain is obtained. Based on these results, the closed-form expressions for the SE and average BER are derived in detail, respectively. The theoretical expressions will be more accurate than the existing schemes, and they include perfect estimation as a special case. Numerical results show that the SE and BER of the system with imperfect estimation are worse than those with perfect estimation due to the estimation error. Moreover, the simulation results for SE and BER are in good agreement with the theoretical analysis.     

Performance analysis of an improved MMSE multiuser receiver formismatched delay channels

Motivated by the fact that time delays in a practical direct-sequence code-division multiple-access (DS-CDMA) system can never be perfectly estimated, an improved minimum-mean squared-error (MMSE)-based receiver is proposed and analyzed. Via the simple assumption of a probability distribution for the delay estimation errors, the proposed receiver can achieve a performance superior to that of the conventional MMSE (CMMSE) receiver. The performances of this improved receiver and the CMMSE receiver are compared in terms of the mean squared error (MSE), probability of error, and asymptotic multiuser efficiency (AME). As the original definition of AME does not consider mismatched channels, the behavior of three single-user receivers bearing imperfect delay estimation is also investigated. These single-user receivers are employed to define a more appropriate AME. Finally, an efficient update mechanism to accommodate dynamic channel statistics, and thus practical implementation, is proposed     

Performance analysis of LMMSE receivers for M-ary QAM in Rayleigh faded CDMA channels

We develop approximations for the symbol error rate in a wireless code-division multiple-access channel. We assume that each user employs spectrally efficient M-ary quadrature amplitude modulation and undergoes independent Rayleigh fading. We study the performance of linear minimum mean-squared error receivers in situations where: i) the channels of all users are known perfectly; ii) the receiver knows only the average powers of the interferers but the channel of the user of interest is still assumed to be perfectly known; iii) the channels of the interferers are unknown and there is an error in the channel estimation of the user of interest. In the last of these cases, the symbol error rate is a function of the variance of the channel estimation error. We also determine an expression for this error variance when the channel estimate is obtained from optimal linear smoothing of a sequence of pilot symbols. Recent results on the performance of linear receivers in large systems with random spreading play an important role in our developments.     

Feedback gain in multiple antenna systems

Multiple antenna transmission and reception have been shown to significantly increase the achievable data rates of wireless systems. However, most of the existing analysis assumes perfect or no channel information at the receiver and transmitter. The performance gap between these extreme channel assumptions is large and most practical systems lie in between. Therefore, it is important to analyze multiple antenna systems in the presence of partial channel information. We upper bound the outage probability performance of multiple antenna systems with preamble-based channel estimation and quantized feedback. We design causal feedback and power control schemes to minimize this upper bound on outage probability. We consider the following practical issues in our analysis and design: (1) the channel information is imperfect both at the receiver and at the transmitter and (2) part of the total available resources for the system need to be used for estimation and feedback. Our results demonstrate that for block fading channels, sending a periodic preamble and causally receiving channel state information via a feedback channel can lead to substantial gains in the outage performance over any nonfeedback scheme. Most of the gains achieved by perfect feedback can be achieved by very few bits of feedback. Furthermore, it is demonstrated that these outage probability gains can be translated into improvements in frame error rate performance of systems using space-time codes. Thus, implementing a power control, even at the cost of reduced spectral resources for the forward channel is beneficial for block fading channels     

The effects of phase estimation errors on RAKE receiver performance

As larger diversity and coding gains are introduced into spread-spectrum systems to mitigate the effects of multipath fading and thermal noise, system performance is increasingly being limited by channel estimation error. To properly design these systems, performance measures incorporating the effects of estimation errors are needed. A lower bound on the probability of bit error for a RAKE receiver operating with imperfect phase estimates is presented along with a method to obtain a closed-form expression for its evaluation. Further, this lower bound is compared to an upper bound so that an accurate assessment of system performance can be obtained     

Analysis and optimization of pilot symbol-assisted Rake receivers for DS-CDMA systems

The effect of imperfect channel estimation (CE) on the performance of pilot-symbol-assisted modulation (PSAM) and MRC Rake reception over time- or frequency-selective fading channels with either a uniform power delay profile (UPDP) or a nonuniform power delay profile (NPDP) is investigated. For time-selective channels, a Wiener filter or linear minimum mean square error (LMMSE) filter for CE is considered, and a closed-form asymptotic expression for the mean square error (MSE) when the number of pilots used for CE approaches infinity is derived. In high signal-to-noise ratio (SNR), the MSE becomes independent of the channel Doppler spectrum. A characteristic function method is used to derive new closed-form expressions for the bit error rate (BER) of Rake receivers in UPDP and NPDP channels. The results are extended to two-dimensional (2-D) Rake receivers. The pilot-symbol spacing and pilot-to-data power ratio are optimized by minimizing the BER. For UPDP channels, elegant results are obtained in the asymptotic case. Furthermore, robust spacing design criteria are derived for the maximum Doppler frequency.     

Performance Evaluation of Generalized Selection Combiners Over Slow Fading with Estimation Errors

In this paper we derive closed-form expressions for the single-user adaptive capacity of generalized selection combining (GSC) system, taking into account the effect of imperfect channel estimation at the receiver. The channel considered is a slowly varying spatially independent flat Rayleigh fading channel. The complex channel estimate and the actual channel are modelled as jointly Gaussian random variables with a correlation that depends on the estimation quality. Three adaptive transmission schemes are analyzed: (1) optimal power and rate adaptation; and (2) constant power with optimal rate adaptation, and (3) channel inversion with fixed rate. In addition to deriving an exact expression for the capacity of the aforementioned adaptive schemes, we analyze the impact of channel estimation error on the capacity statistics and the symbol error rate for GSC systems. The capacity statistics derived in this paper are the moment generating function, complementary cumulative distribution function and probability density function for arbitrary number of receive antennas. Moreover, exact closed-form expressions for M-PAM/PSK/QAM employing GSC are derived. As expected, the channel estimation error has a significant impact on the system performance.