A convolutional single-parity-check concatenated coding scheme forhigh-data-rate applications

The coding scheme uses a set of n convolutional codes multiplexed into an inner code and a (n,n-1) single-parity-check code serving as the outer code. Each of the inner convolutional codes is decoded independently, with maximum-likelihood decoding being achieved using n parallel implementations of the Viterbi algorithm. The Viterbi decoding is followed by additional outer soft-decision single-parity-check decoding. Considering n=12 and the set of short constraint length K=3, rate 1/2 convolutional codes, it is shown that the performance of the concatenated scheme is comparable to the performance of the constraint length K=7, rate 1/2 convolutional code with standard soft-decision Viterbi decoding. Simulation results are presented for the K=3, rate 1/2 as well as for the punctured K=3, rate 2/3 and rate 3/4 inner convolutional codes. The performance of the proposed concatenated scheme using a set of K=7, rate 1/2 inner convolutional codes is given     

Multistage decoding of multilevel block M-PSK modulationcodes and its performance analysis

Multistage decoding of multilevel block multilevel phase-shift keying (M-PSK) modulation codes for the additive white Gaussian noise (AWGN) channel is investigated. Several types of multistage decoding, including a suboptimum soft-decision decoding scheme, are devised and analyzed. Upper bounds on the probability of an incorrect decoding of a code are derived for the proposed multistage decoding schemes. Error probabilities of some specific multilevel block 8-PSK modulation codes are evaluated and simulated. The computation and simulation results for these codes show that with multistage decoding, significant coding gains can be achieved with large reduction in decoding complexity. In one example, it is shown that the difference in performance between the proposed suboptimum multistage soft-decision decoding and the single-stage optimum decoding is small, only a fraction of a dB loss in SNR at the block error probability of 10^-6     

BER analysis of convolution coded QDPSK, in digital mobile radio

The bit error rate (BER) performance of convolutional coded quaternary differential phase-shift keying (QDPSK) with Viterbi decoding is theoretically investigated in Rayleigh fading environments. The probability density functions of the path and branch metric values of Viterbi decoding are derived. The BERs after decoding due to additive white Gaussian noise and cochannel interference are theoretically analyzed. Rate 1/2 codes and their symbol punctured high-rate codes are considered, and the symbol positions for deletion to minimize the BER after decoding are presented for the codes with a constraint length K=3-7. It is shown that Viterbi decoding considerably reduces the desired signal-to-interference power ratio as well as the signal energy per information bit-to-noise power spectrum density ratio necessary to achieve a certain BER. The spectrum efficiency of the cellular mobile radio system, achievable by the use of the symbol punctured codes, is also evaluated     

Decoding performance of linear block codes using a trellis indigital mobile radio

Trellis decoding of linear block codes in a Rayleigh fading channel is discussed. Two methods for calculating metric values for each bit in a received block are considered: the values are calculated from the received signal envelope sample and from the demodulator output. Bit error rate (BER) performances of hard decision and trellis decoding are compared using Hamming (7, 4) and Golay (24, 12) codes in computer simulations and laboratory experiments. A simplified trellis decoding algorithm, in which the hard decision output of a bit with an envelope sample greater than the threshold value is accepted as correct, is presented. Laboratory experimental results for trellis decoding in combination with Gaussian minimum-shift-keying (GMSK) modulation and frequency detection are shown. The effect of n-bit A/D-conversion in signal envelope sampling is investigated experimentally. The results show that the trellis decoding algorithm improves BER performance     

Constructions of rate (n-1)/n puncturedconvolutional codes with minimum required SNR criterion

Rate (n-1)/n punctured convolutional codes (n up to 10 and memory length up to 8) are constructed which minimize the required signal-to-noise ratio (SNR) for a bit-error rate (BER) of 10^-9 with and without the restriction of using only four different code generators. Many of these codes improve the free distance and reduce the required SNR more than half a decibel over previously reported codes with the same parameters. This is equivalent to reducing the decoder complexity by one half for the same performance. These codes have many potential applications in systems that require performance improvements with little room for coding overhead     

Multilevel trellis MPSK modulation codes for the Rayleigh fadingchannel

The multilevel coding technique is used for constructing multilevel trellis M-ary phase-shift-keying (MPSK) modulation codes for the Rayleigh fading channel. In the construction of a code, all the factors which affect the code performance and its decoding complexity are considered. The error performance of some of these codes based on both one-stage optimum decoding and multistage suboptimum decoding has been simulated. The simulation results show that these codes achieve good error performance with small decoding complexity     

High-rate punctured convolutional codes for Viterbi and sequentialdecoding

An investigation is conducted of the high-rate punctured convolutional codes suitable for Viterbi and sequential decoding. Results on known short-memory codes M⩽8 discovered by others are extended. Weight spectra and upper bounds on the bit error probability of the best known punctured codes having memory 2⩽M ⩽8, and coding rates 2/3⩽R⩽7/8 are provided. Newly discovered rate-2/3 and -3/4 long-memory punctured convolutional codes with 9⩽M⩽23 are provided together with the leading terms of their weight spectra and their bit error performance bounds. Some results of simulation with sequential decoding are given     

Performance of RS coded M-ary modulation with and without symbol overlapping

In this paper, we present analytical bit error probability results for M-ary modulation concatenated with Reed Solomon (RS) codes. The analysis of bit error probability is nontrivial as the number of bits per symbol for the RS codes may not be an integer multiple of the number of bits per symbol for a modulation symbol. We propose a Markov chain technique which allows analytical evaluation of the bit error probability for such cases. The performance of RS coding with coherent biorthogonal, coherent/non-coherent orthogonal modulation over an additive white Gaussian noise (AWGN) channel is evaluated. Simulation of the bit error probability of RS code concatenated with a Nordstrom Robinson (NR) code as an inner code is performed and compared with the case of biorthogonal modulation. From the results, we notice that a stronger inner code gives better bit error probability. In addition, the throughput of the coded system with biorthogonal modulation over an AWGN channel is discussed. For a Rayleigh flat fading and block fading channel, we analyze the bit error probability of RS codes concatenated with biorthogonal modulation. From the result, we notice that a stronger outer code gives a better bit error probability for the case of Rayleigh flat fading channel.     

Performance analysis of punctured convolutional codes andturbo-codes

Digital radio transmission techniques offer the prospect of improved reception compared with analogue signals and are being introduced for radio broadcasting in the short-wave bands. The coding scheme adopted plays an important part in achieving a high quality in the presence of noise and fading, which can be particularly severe for Digital Radio Mondiale (DRM) receivers. This paper compares the performance of turbo-codes and punctured convolutional codes over the radio broadcast transmission channels proposed in ITU-R Circular Letter 10/LCCE/39. The results show that the bit error ratio for binary communication over both additive white Gaussian noise (AWGN) and Rayleigh fading channels is low for turbo-codes in comparison with that for punctured convolutional codes having the same code rate. This result holds over a wide range of bit energy to noise power ratios. The results, which are evaluated in terms of the efficiency for Gaussian and Rayleigh fading channels, show that system efficiency increases with decreasing code rate     

Performance comparison of different decoding strategies for abandwidth-efficient block-coded scheme on mobile radio channels

The performance of bandwidth-efficient Reed-Solomon (RS)-coded MPSK schemes is evaluated on a shadowed Rician fading channel using different decoding strategies, namely, errors-only, errors-and-erasures, and soft-decision decoding. The lower bounds of the bit error probability are found for errors-only and for errors-and-erasures decoding. For the soft-decision decoding the upper bound of the bit error rate is derived. The error bounds are calculated and examined by simulation for some RS-coded MPSK schemes on a shadowed Rician channel. It is shown that their performance is significantly improved compared to uncoded QPSK. The amount of improvement depends on the signal-to-noise ratio (SNR), the decoding strategy, and the degree of shadowing. A comparison between different decoding techniques, for one of the RS-coded schemes, for different degrees of shadowing shows that the use of channel measurement information in the decoding process is more effective for heavy shadowed channels     

Bandwidth efficient coding for fading channels: code constructionand performance analysis

The authors apply a general method of bounding the event error probability of TCM (trellis-coded modulation) schemes to fading channels and use the effective length and the minimum-squared-product distance to replace the minimum-free-squared-Euclidean distance as code design parameters for Rayleigh and Rician fading channels with a substantial multipath component. They present 8-PSK (phase-shift-keying) trellis codes specifically constructed for fading channels that outperform equivalent codes designed for the AWGN (additive white Gaussian noise) channel when v⩾5. For quasiregular trellis codes there exists an efficient algorithm for evaluating event error probability, and numerical results which demonstrate the importance of the effective length as a code design parameter for fading channels with or without side information have been obtained. This is consistent with the case for binary signaling, where the Hamming distance remains the best code design parameter for fading channels. The authors show that the use of Reed-Solomon block codes with expanded signal sets becomes interesting only for large value of E_s/N₀, where they begin to outperform trellis codes     

On multilevel block modulation codes

The multilevel technique for combining block coding and modulation is investigated. A general formulation is presented for multilevel modulation codes in terms of component codes with appropriate distance measures. A specific method for constructing multilevel block modulation codes with interdependency among component codes is proposed. Given a multilevel block modulation code C with no interdependency among the binary component codes, the proposed method gives a multilevel block modulation code C' that has the same rate as C, a minimum squared Euclidean distance not less than that of C, a trellis diagram with the same number of states as that of C, and a smaller number of nearest neighbor codewords than that of C . Finally, a technique is presented for analyzing the error performance of block modulation codes for an additive white Gaussian noise (AWGN) channel based on soft-decision maximum likelihood decoding. Error probabilities of some specific codes are evaluated by simulation and upper bounds based on their Euclidean weight distributions     

An Adaptive Hybrid ARQ Scheme

A hybrid ARQ in which the transmitter adaptively selects an FEC code according to the channel condition is presented and analyzed. The code is selected according to the past transmissions and acknowledgements by an algorithm which is a generalization of that in [1]. The throughput is obtained as a function of the frame error rate for a general system employing the adaptive hybrid ARQ with acknowledgements that arrive instantly on an error-free return channel. The throughput is obtained as a function of the signal-to-noise ratio for an example quad rate system employing convolutional codes with non-coherent frequency shift keying over the uncorrelated Rayleigh fading channel. This allows the best choice for the parameters of the algorithm to be made. In the case that the channel bit errors are independent, the generalization offers performance improvement of less than 10% over that in [1]. But when the channel errors are bursty, as in the case of Rayleigh fading with finite bit interleaving, the generalization offers throughput improvement as high as 24%. We go on to consider incorporating code combining with the adaptive scheme to form an adaptive memory hybrid ARQ. Simulation of a system using complementary punctured convolutional codes with 4 code rates shows that 2-level code combining can extend the adaptive scheme's useful throughput into the low SNR region by approximately 4 dB.     

Performance of convolutional codes in a direct-detection opticalPPM channel

Several strategies for combining convolutional codes (CCs) and pulse position modulation (PPM) in a direct-detection optical channel are considered, including binary and 2ⁿ-ary PPM with rate-1/ n CCs, 2^m-ary PPM with a dual-m CC, and interleaved 2^L-ary PPM with a rate-1/n CC. In the latter case, J.L. Massey's (1981) concept of coding of L separate component channels constituting a 2^L-ary PPM erasure channel is carried out for both the ideal photon counting and the avalanche photodetection (APD) cases, providing a comparative evaluation of the various strategies in terms of bit error probability and achievable coding gain for different code structures and decoding schemes     

Burst error statistics of simulated Viterbi decoded BPSK on fadingand scintillating channels

The author presents and analyzes burst error statistics of a soft-decision Viterbi decoder when the transmitted signal is encoded with the 313 (3, 1/2) or 31123 (5, 1/2) convolutional codes, modulated via coherent binary phase-shift keying (BPSK) for the additive white Gaussian noise (AWGN) channel, and subjected to slow and nonselective scintillation/fading modeled by the Nakagami-m distribution. These statistics were generated by Monte-Carlo simulations, and presented in terms of burst error length average and quantile (90 and 99%) statistics versus SNR (E_b/N₀) parameterized by the fading intensity parameter m. The results indicate how Viterbi decoder burst error statistics vary with the fading/scintillation intensity m for Nakagami-m channels, and, consequently, provide information important to the design of interleaved or noninterleaved concatenated coding schemes for such channel environments     

Joint Source and Channel Coding using Punctured Ring Convolutional Coded CPM

In this paper, a novel trellis source encoding scheme based on punctured ring convolutional codes is presented. Joint source and channel coding (JSCC) using trellis coded continuous phase modulation (CPM) with punctured convolutional codes over rings is investigated. The channels considered are the additive white gaussian noise (AWGN) channel and the Rayleigh fading channel. Optimal soft decoding for the proposed JSCC scheme is studied. The soft decoder is based on the a posteriori probability (APP) algorithm for trellis coded CPM with punctured ring convolutional codes. It is shown that these systems with soft decoding outperform the same systems with hard decoding especially when the systems operate at low to medium signal-to-noise ratio (SNR). Furthermore, adaptive JSCC approaches based on the proposed source coding scheme are investigated. Compared with JSCC schemes with fixed source coding rates, the proposed adaptive approaches can achieve much better performance in the high SNR region. The novelties of this work are the development of a trellis source encoding method based on punctured ring convolutional codes, the use of a soft decoder, the APP algorithm for the combined systems and the adaptive approaches to the JSCC problem.     

A new approach to the construction of high-rate convolutional codes

This letter presents a new technique to construct high-rate convolutional codes using a structure formed by a high-rate block code and a simpler convolutional code. The goal is to obtain good convolutional codes in terms of free distance and number of nearest neighbors, with better performance than punctured codes. The obtained codes improve over the best known high-rate punctured codes with the same rate and memory in terms of both bit error probability and computational decoding complexity     

Performance of FH/BFSK with generalized fading in worst casepartial-band Gaussian interference

For frequency-hopped (noncoherent) binary frequency shift keying (FH/BFSK) on a worst-case partial-band Gaussian interference channel, the bit error probability results are well known for the extreme cases where the signal is either nonfading or Rayleigh fading. In this work, the region between these extremes is filled in by considering the general Nakagami-m fading model. The worst-case partial-band Gaussian interference results are given by a one-parameter family which for m→∞ gives the Viterbi-Jacobs nonfading result, and for m=1 gives the Rayleigh fading result. In the latter case, a broadband interference strategy is optimal. Thus, the Nakagami- m results provide a smooth one-parameter bridge between the Viterbi-Jacobs channel and the Rayleigh fading channel. The results show that the worst-case interference fraction ρ increases as the fading variance increases, up to Rayleigh fading. Any fading less severe than Rayleigh, however slight the departure from Rayleigh, requires a partial-band strategy for sufficiently large E_b/N_I     

Errors and erasures decoding of BCH and Reed-Solomon codes for reduced M-ary orthogonal signaling

The paper presents a comparison of communication systems using different signal constellation sizes and Reed-Solomon or Bose-Chaudhuri-Hocquengem codes with different rates so that the overall required bandwidth is the same for each system. In these comparisons, the channel symbol size is smaller than the code symbol size, so that a code symbol contains parts of multiple channel symbols. Thus, the normal assumption of independent code symbols does not apply. Instead, consideration must be taken to obtain the best arrangement of channel symbols in each code symbol. Analytical expressions are developed to compare the bit error probability performance of comparable systems, based on individual codewords using errors-only decoding and errors and erasures decoding with transmission over a Rayleigh fading channel.     

Error-correcting codes for list decoding

In the list-of-L decoding of a block code the receiver of a noisy sequence lists L possible transmitted messages, and is in error only if the correct message is not on the list. Consideration is given to (n,e,L) codes, which correct all sets of e or fewer errors in a block of n bits under list-of-L decoding. New geometric relations between the number of errors corrected under list-of-1 decoding and the (larger) number corrected under list-of-L decoding of the same code lead to new lower bounds on the maximum rate of (n,e,L) codes. They show that a jammer who can change a fixed fraction p<1/2 of the bits in an n-bit linear block code cannot prevent reliable communication at a positive rate using list-of- L decoding for sufficiently large n and an L⩽n. The new bounds are stronger for small n , but weaker for fixed e/n in the limit of large n and L than known random coding bounds