Efficient expression and bound for pairwise error probability in Rayleigh fading channels, with application to union bounds for turbo codes

This work provides a new exact expression for the pairwise error probability under perfect channel state information. Using this expression, the average union upper bounds for (1,7/5,7/5) and (1,5/7,5/7) turbo coding schemes are presented based on transfer function approach assuming fully interleaved Rayleigh fading channels. Derivation for the pairwise error probability leads to a compact approximation which overlaps with the exact expression. This new approximation is computationally more efficient than the existing ones, and leads to a much quicker general ion of union bounds for performance analysis.     

Turbo 码在无信道状态信息的Nakagami 衰落信道中的性能分析

讨论了Turbo码在无法获取信道状态信息的移动通信系统中的纠错性能联合上界，提出无信道状态信息时，Nakagami慢衰落信道中成对差错概率的2个精确计算式和2个近似计算式。数值分析结果表明，成对差错概率的2个精确计算式的数值积分曲线完全吻合；封闭形式近似计算式的计算结果与精确值相近。仿真结果说明，理论联合上界在高信噪比处逼近仿真性能。所提出的性能分析式可快速有效地评估无信道状态信息的移动通信系统中Turbo码的性能。     

Dual MRC diversity reception of TCM-MPSK signals over Nakagamifading channels

Space diversity reception and forward error correction coding are powerful techniques for combating the multipath fading encountered in mobile radio communications. In this Letter, the authors analyse the performance of a dual maximal ratio combining (MRC) diversity system using trellis coded modulation-multiple phase shift keying (TCM-MPSK) on slow, nonselective correlated Nakagami fading channels. An alternative exact derivation is introduced for the pairwise error probability, used in calculating average bit error rate analytical upper bounds     

Dual predetection SC diversity reception of TCM-MPSK signals overNakagami fading channels

Space diversity reception and forward error correction coding are powerful techniques for combatting multipath fading encountered in mobile radio communications. The authors analyse the performance of a dual predetection selection combining (SC) diversity system using trellis coded modulation-multiple phase shift keying (TCM-MPSK) on slow, nonselective correlated Nakagami fading channels. An alternative exact derivation for the pairwise error probability, used in calculating average bit error rate analytical upper bounds, is introduced     

Closed form expression and improved bound on pairwise error probability for performance analysis of turbo codes over Rician fading channels

This letter provides derivations for an exact expression and a bound on pair wise error probability over fully interleaved Rician fading channels under the assumption of ideal channel state information. The derivation which is based on the probability distribution of the sum of squared Rician random variables leads to an improved upper bound in comparison with the only known bound in literature. Pairwise error probability plots together with average union upper bounds for turbo codes having (1,7/5,7/5) and (1,5/7,5/7) generator polynomials are presented to demonstrate the effectiveness of the new results.     

Analysis of the error performance of trellis-coded modulations inRayleigh-fading channels

This work presents an exact expression for the pairwise error event probability of trellis-coded modulation (TCM) transmitted over Rayleigh-fading channels. It includes phase shift keying (PSK) and multilevel quadrature amplitude modulation (QAM) codes, as well as coherent and partially coherent (e.g. differential, pilot tone, etc.) detection. Due to the form of the exact pairwise error event probabilities, this calculation technique cannot be used with the transfer function technique to obtain an upper (union) bound on the overall bit error probability. For this reason, the authors estimate the bit error probability by considering only a small number of short error events. Through simulations, they found that the estimation is usually very accurate at high signal-to-noise ratios but not as accurate at lower signal-to-noise ratios. They study several coded modulation schemes this way. Among the results are the fact that TCM provides significant improvement in the error floor when detected differentially, and an asymmetry in the pairwise error event probability for 16 QAM     

On the error performance analysis of space-time trellis codes

We present analytical performance results for space-time trellis codes over spatially correlated Rayleigh fading channels. Bit-error-probability estimates are obtained based on the derivation of an exact pairwise error probability expression using a residue technique combined with a characteristic function approach. We investigate both quasi-static and interleaved channels and demonstrate how the spatial fading correlation affects the performance of space-time codes over these two different channel models. Simulation results are also included to confirm the accuracy of analytical estimates.     

Tight bounds on the error probability of coded modulation schemesin Rayleigh fading channels

This paper presents a tight upper bound on the bit error performance of coded modulation schemes in Rayleigh fading channels. Upper and lower bounds on the pairwise error probability are first derived. The upper bound is then expressed in a product form to be used with the transfer function bounding technique. This upper bound has the same simplicity as the union-Chernoff bound while providing closer results to the exact expression. Examples for the case of four-state and eight-state TCM 8PSK schemes are also given to illustrate the tightness and the application of this upper bound     

Pairwise error probability of space-time codes in frequency selective Rician channels

In this letter, we investigate the performance of space-time codes in frequency selective correlated Rician channels. An exact expression has been derived for the pairwise error probability (PEP) of space-time trellis codes over frequency selective Rician fading channel, which is in the form of a single finite range integral. We also obtain a closed form expression for the PEP when the signal matrices are drawn from some special design and the performance upper bound.     

Evaluation of the exact union bound for trellis-coded modulationsover fading channels

An analytical technique is presented for computing the exact union bound on the average bit error probability of trellis coded modulation schemes over Rayleigh, Rician, or shadowed Rician-fading channels. To this end, an integral expression is derived for the pairwise error event probability (PEP). Existing bounds can be obtained as special cases of this expression. It turns out that a Gauss-Chebyshev quadrature rule offers excellent accuracy for this integral. By extension, the exact union bound (i.e., the weighted sum of an exact PEPs of a code) can readily be evaluated. This method has the same complexity as the union-Chernoff bound, and a few examples are given to show its application     

Analysis of switched diversity TCM-MPSK systems on Nakagami fadingchannels

Space diversity reception and forward-error correction coding are powerful techniques to combat multipath fading encountered in mobile radio communications. In this paper, we analyze the performance of a discrete-time switched diversity system using trellis-coded modulation multiple phase-shift keying (TCM-MPSK) on slow, nonselective correlated Nakagami (1960) fading channels. Analytical upper bounds using the transfer function bounding technique are obtained and illustrated by several numerical examples. A simple integral expression for calculating the exact pairwise error probability is presented. The use of optimum adaptive and fixed switching thresholds is considered. Monte Carlo simulation results, which are more indicative of the exact system performance, are also given     

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     

Bounds on the pairwise error probability of coded DS/SSMAcommunication systems in Rayleigh fading channels

Arbitrarily tight upper and lower bounds on the pairwise error probability (PEP) of a trellis-coded or convolutional-coded direct-sequence spread-spectrum multiple-access (DS/SSMA) communication system over a Rayleigh fading channel are derived. A new set of probability density functions (PDFs) and cumulative distribution functions (CDFs) of the multiple-access interference (MAI) statistic is derived, and a modified bounding technique is proposed to obtain the bounds. The upper bounds and lower bounds together specify the accuracy of the resulting estimation of the PEP, and give an indication of the system error performance. Several suboptimum decoding schemes are proposed and their performances are compared to that of the optimum decoding scheme by the average pairwise error probability (APEP) values. The approach can be used to accurately study the multiple-access capability of the coded DS/SSMA system without numerical integrations     

On the error probability of quasi‐orthogonal space–time block codes

In this paper, we derive the exact pairwise error probabilities (PEPs) of various quasi‐orthogonal space–time block codes (QO‐STBCs) using the moment generating function. By classifying the exact PEPs of QO‐STBCs into three types, we derive the closed‐form expression for each type of PEP. Based on these closed‐form expressions, we obtain the union bounds on the symbol error probability and bit error probability for QPSK modulation. Through simulation, it is shown that these union bounds are quite tight. Copyright © 2008 John Wiley & Sons, Ltd.     

On the robustness of space-time coding techniques based on a general space-time covariance model

The robustness of space-time coding techniques for wireless channels that exhibit both temporal and spatial correlation is investigated. A general space-time covariance model is developed and employed to evaluate the exact pairwise error probability for space-time block codes. The expressions developed for the pairwise error probability are used in conjunction with the union bound to determine an upper bound for the probability of a block error. The block error probability is evaluated for several space-time codes and for wireless channels that exhibit varying degrees of spatial and temporal correlation. Numerical results are presented for a two-dimensional Gaussian scatterer model which has been shown to be consistent with recent field measurements of wireless channels. The results demonstrate that the best-case wireless channel is uncorrelated in both space and time. Correlation between transmission paths, due to insufficient spacing of the transmit antennas or scatterers located in close proximity to the mobile, can result in a significant performance degradation. The conditions that result in uncorrelated transmission paths are quantified in terms of the effective scattering radius and the spacing of the transmit and receive antennas.     

Theoretical analysis and performance limits of noncoherent sequencedetection of coded PSK

A theoretical performance analysis of noncoherent sequence detection schemes previously proposed by the authors for combined detection and decoding of coded M-ary phase-shift keying (M-PSK) is presented. A method for the numerical evaluation of the pairwise error probability-for which no closed-form expressions exist-is described, the classical union bound is computed, and results are compared with computer simulations. An upper bound on this pairwise error probability is also presented. This upper bound may be effectively used for the definition of an equivalent distance, which may be useful in exhaustive searches for optimal codes. Using this bound, it is proven that, in the general coded case, the considered noncoherent decoding schemes perform as close as desired to an optimal coherent receiver when a phase memory parameter is sufficiently large. In the case of differentially encoded M-PSK, a simple expression of the asymptotic bit-error probability is derived, which is in agreement with simulations for high as well as low signal-to-noise ratio (SNR)     

Performance of high-diversity multidimensional constellations

Following the approach introduced by Cavers and Ho (1992), the performance of component-interleaved multidimensional constellations over the Rayleigh fading channel is evaluated analytically. The error probabilities are approximated by the union bound using an exact expression of the pairwise error probability. Simulation results show that this bound can be used effectively as a design criterion for the selection of high-diversity multidimensional constellation over the Rayleigh fading channel     

Performance of RCPC codes with soft-decision decoding overNakagami-m fading channels

The analytical upper bounds on the pairwise error probability of rate compatible punctured convolutional (RCPC) codes, using coherent BPSK signals over slow frequency nonselective Nakagami-m fading channels with AWGN, are evaluated. With perfect channel state information (CSI) assumption, we use a direct integral with the Nakagami-m probability density function to obtain a closed form upper bound. For the case without CSI, we find an approximated upper bound for the high SNR cases and the approximation can be justified for signal to noise ratio (SNR) E _s/N₀ ≫ 1.5 dB     

Performance bounds for optimum and suboptimum reception under Class-A impulsive noise

The transmission over the memoryless additive white Class-A noise (AWCN) channel is considered. For uncoded transmission, an exact expression for the symbol error rate is derived. For coded transmission, the Chernoff bound on the pairwise error probability is calculated and the performance achieved on the real and the complex AWCN channels is compared. Moreover, a low-complexity, suboptimum decoding metric is derived and analyzed employing the cutoff rate as a performance criterion.     

Performance analysis for coded CDMA systems

This correspondence analyzes the bit-error rate (BER) performance of coded synchronous code-division multiple-access (CDMA) systems assuming perfect channel state information (CSI) and optimal joint multiuser detection/decoding (OJMUDD). Our analysis is conducted in the same framework as that of uncoded systems. First, we derive the precise probability of an error event, then we provide an upper bound on the BER based on the sum of pairwise error probabilities, and, finally, we tighten the upper bound by considering decomposable error events. Many new concepts unique to coded systems are introduced. We propose to use quasi parity checks for identifying permissible error events, introduce the concept of compatible probability of error matrices, extend the list of conditions for identifying decomposable error events, and introduce the concept of conjugate sets to explore the symmetry among indecomposable error events. Simulation results are given along with theoretical predictions.