Gallager Bounds for Noncoherent Decoders in Fading Channels

Recently, Gallager's bounding techniques have been used to derive tight performance bounds for coded systems in fading channels. Most works in this field have thus far dealt with coherent decoding. This paper develops Gallager bounds for noncoherent systems in fading channels. Unlike coherent decoding, the exact error probability of a noncoherent decoder/detector conditioned on the fading coefficients does not admit a closed-form expression. This difficulty is overcome in this paper by employing the Chernoff technique. Although it weakens the bounds to some extent, the Chernoff technique enables the derivations of the limit-before-average (LBA) bound and Gallager bounds in closed form for noncoherent fading channels. Numerical examples show that the proposed bounds are convergent and are tighter than the conventional union bound.      

On improved bounds on the decoding error probability of block codesover interleaved fading channels, with applications to turbo-like codes

We derive here improved upper bounds on the decoding error probability of block codes which are transmitted over fully interleaved Rician fading channels, coherently detected and maximum-likelihood (ML) decoded. We assume that the fading coefficients during each symbol are statistically independent (due to a perfect channel interleaver), and that perfect estimates of these fading coefficients are provided to the receiver. The improved upper bounds on the block and bit error probabilities are derived for fully interleaved fading channels with various orders of space diversity, and are found by generalizing some previously introduced upper bounds for the binary-input additive white Gaussian nose (AWGN) channel. The advantage of these bounds over the ubiquitous union bound is demonstrated for some ensembles of turbo codes and low-density parity-check (LDPC) codes, and it is especially pronounced in a portion of the rate region exceeding the cutoff rate. Our generalization of the Duman and Salehi bound (Duman and Salehi 1998, Duman 1998) which is based on certain variations of Gallager's (1965) bounding technique, is demonstrated to be the tightest reported upper bound. We therefore apply it to calculate numerically upper bounds on the thresholds of some ensembles of turbo-like codes, referring to the optimal ML decoding. For certain ensembles of uniformly interleaved turbo codes, the upper bounds derived here also indicate good match with computer simulation results of efficient iterative decoding algorithms     

Bit error probability of distributed Turbo coded relay systems over quasi-static Rayleigh fading channels

As is known, distributed Turbo coding (DTC) performs close to the theoretic outage probability bound of a relay channel when correct decoding is assumed at the relay. However, decoding error is inevitable in practical fading channels due to the error-prone feature of radio channels, and the decoding error propagation in DTC scheme will severely degrade the error performance of the relay system. As a result, it is necessary to evaluate the error performance of the DTC scheme in multi-hop relaying wireless systems in practical fading channels. Moreover, the theoretical method of analysis provides an effective tool for obtaining the error performance besides lengthy simulations. In this article, the concept of equivalent signal-to-noise ratio (SNR) of the two-hop relay channel and the method of computing equivalent SNR are developed, and then the upper bound on the bit error probability (BEP) of DTC relay systems is analyzed by use of Turbo code's distance spectrum, the concept of uniform interleaver, the limit-before-averaging technique, and the union bound method. Both theoretical analysis and numerical simulation are implemented for relay systems with DTC scheme over quasi-static Rayleigh fading channels. The results show that the upper bound approaches the simulation results in the medium to high SNR region.     

Performance Analysis of LDPC-Coded Space-Time Modulation over MIMO Fading Channels

A closed-form upper bound on the error performance is proposed for LDPC-coded space-time modulation over MIMO block/slow fading channels based on the analysis framework developed for the fast fading case. This follows from the observation that the pairwise error probability (PEP) in all these fading cases is determined by a certain metric of codewords, with respect to which we can enumerate all distinct PEPs and thus concisely formulate the union bound. Simulation results indicate that the bound is useful to benchmark the performance of iterative decoding and detection algorithms     

Properties of optimum binary message-passing decoders

We consider a class of message-passing decoders for low-density parity-check (LDPC) codes whose messages are binary valued. We prove that if the channel is symmetric and all codewords are equally likely to be transmitted, an optimum decoding rule (in the sense of minimizing message error rate) should satisfy certain symmetry and isotropy conditions. Using this result, we prove that Gallager's Algorithm B achieves the optimum decoding threshold among all binary message-passing decoding algorithms for regular codes. For irregular codes, we argue that when the nodes of the message-passing decoder do not exploit knowledge of their decoding neighborhood, optimality of Gallager's Algorithm B is preserved. We also consider the problem of designing irregular LDPC codes and find a bound on the achievable rates with Gallager's Algorithm B. Using this bound, we study the case of low error-rate channels and analytically find good degree distributions for them.     

On the performance of rate 1/2 convolutional codes with QPSK onRician fading channels

Consideration is given to the bit error probability performance of rate 1/2 convolutional codes in conjunction with quaternary phase shift keying (QPSK) modulation and maximum-likelihood Viterbi decoding on fully interleaved Rician fading channels. Applying the generating function union bounding approach, an asymptotically tight analytic upper bound on the bit error probability performance is developed under the assumption of using the Viterbi decoder with perfect fading amplitude measurement. Bit error probability performance of constraint length K=3-7 codes with QPSK is numerically evaluated using the developed bound. Tightness of the bound is examined by means of computer simulation. The influence of perfect amplitude measurement on the performance of the Viterbi decoder is observed. A performance comparison with rate 1/2 codes with binary phase shift keying (BPSK) is provided     

On the converse to the coding theorem for discrete memoryless channels (Corresp.)

A new lower bound on the probability of decoding error for the case of rates above capacity is presented. It forms a natural companion to Gallager's random coding bound for rates below capacity. The strong converse to the coding theorem follows immediately from the proposed lower bound.     

A New Union Bound on the Error Probability of Binary Coded OFDM Systems in Wireless Environments

Orthogonal Frequency Division Multiplexing (OFDM) systems are commonly used to mitigate frequency-selective multipath fading and provide high-speed data transmission. In this paper, we derive new union bounds on the error probability of a coded OFDM system in wireless environments. In particular, we consider convolutionally coded OFDM systems employing single and multiple transmit antennas over correlated block fading (CBF) channels with perfect channel state information (CSI). Results show that the new union bound is tight to simulation results. In addition, the bound accurately captures the effect of the correlation between sub-carriers channels. It is shown that as the channel becomes more frequency-selective, the performance get better due to the increased frequency diversity. Moreover, the bound also captures the effect of multi-antenna as space diversity. The proposed bounds can be applied for coded OFDM systems employing different coding schemes over different channel models.     

Hybrid Hard-Decision Iterative Decoding of Irregular Low-Density Parity-Check Codes

Time-invariant hybrid (Hscr_TI) decoding of irregular low-density parity-check (LDPC) codes is studied. Focusing on Hscr_TI algorithms with majority-based (MB) binary message-passing constituents, we use density evolution (DE) and finite-length simulation to analyze the performance and the convergence properties of these algorithms over (memoryless) binary symmetric channels. To apply DE, we generalize degree distributions to have the irregularity of both the code and the decoding algorithm embedded in them. A tight upper bound on the threshold of MB Hscr_TI algorithms is derived, and it is proven that the asymptotic error probability for these algorithms tends to zero, at least exponentially, with the number of iterations. We devise optimal MB Hscr_TI algorithms for irregular LDPC codes, and show that these algorithms outperform Gallager's algorithm A applied to optimized irregular LDPC codes. We also show that compared to switch-type algorithms, such as Gallager's algorithm B, where a comparable improvement is obtained by switching between different MB algorithms, MB Hscr_TI algorithms are more robust and can better cope with unknown channel conditions, and thus can be practically more attractive     

Performance bounds for space-time trellis codes

We derive the union bound for space-time trellis codes over quasi-static fading channels. We first observe that the standard approach for evaluating the union bound yields very loose, in fact divergent, bounds over the quasi-static fading channel. We then develop a method for obtaining a tight bound on the error probability. We derive the union bound by performing expurgation of the standard union bound. In addition, we limit the conditional union bound before averaging over the fading process. We demonstrate that this approach provides a tight bound on the error probability of space-time codes. The bounds can be used for the case when the fading coefficients among different transmit/receive antenna pairs are correlated as well. We present several examples of the bounds to illustrate their usefulness.     

Error performance of MPSK trellis-coded modulation overnonindependent Rician fading channels

This paper presents new upper bounds on the pairwise error probability (PEP) of trellis-coded modulation (TCM) schemes over nonindependent Rician fading channels. Cases considered are coherent and pilot-tone-aided detection and differential detection of trellis-coded multilevel phase-shift keying (TC-MPSK) systems. The average bit-error probability P_b can be approximated by truncating the union bound. This method does not necessarily lead to an upper bound on P_b, and, hence, the approximation must be used with simulation results. In addition, for Rayleigh fading channels with an exponential autocovariance function, bounds resembling those for memoryless channels have been derived. The bounds are substantially more accurate than Chernoff bounds and hence allow for accurate estimation of system performance when the assumption of ideal interleaving is relaxed     

New Gallager Bounds in Block-Fading Channels

In this paper, we propose a new upper bound on the error performance of binary linear codes over block-fading channels by employing Gallager's first- and second-bounding techniques. As the proposed bound is numerically intensive in its general form, we consider two special cases, namely, the spherical bound and the DS2-exponential bound, which are found to be tight in nonergodic and near-ergodic block-fading channels, respectively. The tightness of the proposed bounds is demonstrated for turbo codes. Many existing bounds for quasistatic or fully interleaved fading channels can be viewed as special cases of the proposed Gallager bound     

On space-time coding for free-space optical systems

Atmospheric turbulence-induced fading is one of the main impairments affecting free-space optics (FSO) communications. In this paper, we consider FSO systems with intensity modulation and direct detection (IM/DD) and derive a closed- form expression for the asymptotic pairwise error probability of general FSO space-time codes (STCs) for two lasers and an arbitrary number of photodetectors for channels suffering from Gamma-Gamma fading. Furthermore, we provide a simple design criterion for FSO STCs which is used to establish the quasi-optimality of previously proposed FSO repetition codes. We also show that STCs optimized for RF systems achieve full diversity in FSO systems but are suboptimal as far as the coding gain is concerned. Simulation results confirm the analytical findings of this paper.     

Performance and design of space-time coding in fading channels

The pairwise-error probability upper bounds of space-time codes (STCs) in independent Rician fading channels are derived. Based on the performance analysis, novel code design criteria for slow and fast Rayleigh fading channels are developed. It is found that, in fading channels, the STC design criteria depend on the value of the possible diversity gain of the system. In slow fading channels, when the diversity gain is smaller than four, the code error performance is dominated by the minimum rank and the minimum determinant of the codeword distance matrix. However, when the diversity gain is larger than, or equal to, four, the performance is dominated by the minimum squared Euclidean distance. Based on the proposed design criteria, new codes are designed and evaluated by simulation.     

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.     

A union bound on the error probability of binary codes over block-fading channels

Block-fading is a popular channel model that approximates the behavior of different wireless communication systems. In this paper, a union bound on the error probability of binary-coded systems over block-fading channels is proposed. The bound is based on uniform interleaving of the coded sequence prior to transmission over the channel. The distribution of error bits over the fading blocks is computed. For a specific distribution pattern, the pairwise error probability is derived. Block-fading channels modeled as Rician and Nakagami distributions are studied. We consider coherent receivers with perfect and imperfect channel side information (SI) as well as noncoherent receivers employing square-law combining. Throughout the paper, imperfect SI is obtained using pilot-aided estimation. A lower bound on the performance of iterative receivers that perform joint decoding and channel estimation is obtained assuming the receiver knows the correct data and uses them as pilots. From this, the tradeoff between channel diversity and channel estimation is investigated and the optimal channel memory is approximated analytically. Furthermore, the optimal energy allocation for pilot signals is found for different channel memory lengths.     

慢变平坦衰落信道下空时码设计准则的改进

对用于慢变平坦瑞利衰落信道的空时码设计准则进行了研究。考察了不同长度的首事件差错概率，并基于这些概率推出误帧率的上界，进而提出一种使误帧率的上界最小化的新准则。另外，证明了空时码的几何一致性并使用该特性大大地减少了空时码搜索的复杂度。依据该准则搜索得到一些新码，并与现有好码进行性能比较。     

Exact expression and tight bound on pairwise error probability for performance analysis of turbo codes over nakagami-m fading channels

This letter presents derivation for an exact and efficient expression on pairwise error probability over fully interleaved Nakagami-m fading channels under ideal channel state information at the decoder. As an outcome, this derivation also leads to a tight upper bound on pairwise error probability which is close to the exact expression. Pairwise error probability plots for different values of Nakagami parameter m along with an already existing numerically computable expression are provided. As an application of pairwise error probability, average union upper bounds for turbo codes having (1,7/5,7/5) and (1,5/7,5/7) generator polynomials employing transfer function approach are presented to illustrate the usefulness of the new efficient results     

A Tight BER Upper Bound for Bit-Interleaved Coded Modulation with Square QAM and Gray Labeling

Bit-interleaved coded modulation (BICM) is a bandwidth-efficient coded system with diversity order higher than that of Ungerboeck's trellis-coded modulation on fading channels. In this paper, we investigate the BER (bit error rate) performance of BICM in the additive white Gaussian and Rayleigh fading channels. A new upper bound is given for the square QAM constellation with gray labeling, which constitutes a large portion of practical applications of BICM systems. The new upper bound is tighter than the well-known BICM union bound proposed in G. Caire et al. (1998)     

BICM-ID with signal space diversity over cascaded rayleigh fading channels [transactions letters]

Bit-interleaved coded modulation with iterative decoding (BICM-ID) using signal space diversity (SSD) is considered for cascaded Rayleigh fading channels. A tight bound on the asymptotic error probability is derived to determine the optimal rotation matrix for SSD design and to identify the key parameters that influence the system performance. It is shown that, for small modulation constellation, a cascaded Rayleigh fading causes a much more severe performance degradation than a conventional Rayleigh fading. However, BICM-ID employing SSD with a sufficiently large constellation can close the performance gap between the conventional and cascaded Rayleigh fading channels, and their performance can closely approach that over an AWGN channel. Illustrative simulation results for various scenarios are in a good agreement with analytical derivations.