改进交织的单层极化码高阶编码调制系统

极化码作为一种新型编码方式,被采纳为5G通信中的短码方案。本文将极化码应用到比特交织编码调制（Bit-Interleaved Coded Modulation,BICM）系统,优化交织器的设计,提出了一种新型交织算法。相比于现有的交织算法,新型交织算法的提出是基于比特信道可靠性衡量参数,将高可靠性的比特信道与低可靠性的比特信道交错设计,按照高可靠性信道对低可靠性信道辅助译码的方式,提高极化码的纠错性能。由于新型交织算法只存在于比特信道可靠度参数的简单排序,在复杂度上没有明显增加。仿真结果表明:新型交织算法具有优异的性能,当误码率为10-5,码长为256时,采用新型交织算法的极化码BICM系统与LDPC码的BICM系统相比大约有1.51 dB的增益。      

Adaptive interleaver based on rate-compatible punctured convolutional codes

This letter focuses on the design of an adaptive Bit- Interleaved Coded Modulation (BICM) scheme for frequency selective slow fading channels where the transmitter has certain knowledge of the channel response. In particular, we consider the design of the interleaver stage for a specific convolutional code operating with OFDM modulation. The adaptive interleaver uses the puncturing tables of the Rate-Compatible Punctured Convolutional Codes (RCPC codes) to rearrange the bits as a function of the fading values and the specific constellation. The performance of different interleavers are compared, revealing that the adaptive RCPC-based interleaver produces larger Euclidean distances between the received codewords and reduces the packet error rate (PER), specially when the number of deep-faded subcarriers increases. Numerical results also evidence the importance of the interleaver choice when comparing the performance of different power allocation strategies.     

Performance Bounds for Unequal Error Protecting Turbo Codes

In many communications systems, data can be divided into different importance levels. For these systems, unequal error protection (UEP) techniques are used to guarantee lower BER for the more important classes. In particular, if the precise characteristics of the channel are not known, UEP can be used to recover the more important classes even in poor receiving conditions. In this paper, we derive bounds on the performance of unequal error protecting turbo codes. These bounds serve as an important tool in predicting the performance of these codes. In order to derive the bounds, we introduce the notion of UEPuniform interleaver which is a random interleaver that does not change the order of classes in the turbo code frame. We also present a method to derive the weight enumerating function for UEP turbo codes.     

Design and performance analysis of a new class of rate compatible serially concatenated convolutional codes

In this paper, a novel class of serially concatenated convolutional codes (SCCCs) is addressed. In contrast to standard SCCCs, where high rates are obtained by puncturing the outer code, the heavy puncturing is moved to the inner code, which can be punctured beyond the unitary rate. We derive analytical upper bounds on the error probability of this code structure by considering an equivalent code construction consisting of the parallel concatenation of two codes, and address suitable design guidelines for code optimization. It is shown that the optimal puncturing of the inner code depends on the outer code, i.e., it is interleaver dependent. This dependence cannot be tracked by the analysis for standard SCCCs, which fails in predicting code performance. Based on the considerations arising from the bounds analysis, we construct a family of rate-compatible SCCCs with a high level of flexibility and a good performance over a wide range of code rates, using simple constituent codes. The error rate performance of the proposed codes is found to be better than that of standard SCCCs, especially for high rates, and comparable to the performance of more complex turbo codes.     

A logarithmic upper bound on the minimum distance of turbo codes

We derive new upper bounds on the minimum distance, which turbo codes can maximally attain with the optimum interleaver of a given length. The new bounds grow approximately logarithmically with the interleaver length, and they are tighter than all previously derived bounds for medium-length and long interleavers. An extensive discussion highlights the impacts of the new bounds in the context of interleaver design and provides some new design guidelines.     

Serial concatenation of interleaved codes: performance analysis,design, and iterative decoding

A serially concatenated code with interleaver consists of the cascade of an outer encoder, an interleaver permuting the outer codewords bits, and an inner encoder whose input words are the permuted outer codewords. The construction can be generalized to h cascaded encoders separated by h-1 interleavers. We obtain upper bounds to the average maximum-likelihood bit error probability of serially concatenated block and convolutional coding schemes. Then, we derive design guidelines for the outer and inner encoders that maximize the interleaver gain and the asymptotic slope of the error probability curves. Finally, we propose a new, low-complexity iterative decoding algorithm. Throughout the paper, extensive comparisons with parallel concatenated convolutional codes known as “turbo codes” are performed, showing that the new scheme can offer superior performance     

Analysis, design, and iterative decoding of double seriallyconcatenated codes with interleavers

A double serially concatenated code with two interleavers consists of the cascade of an outer encoder, an interleaver permuting the outer codeword bits, a middle encoder, another interleaver permuting the middle codeword bits, and an inner encoder whose input words are the permuted middle codewords. The construction can be generalized to h cascaded encoders separated by h-1 interleavers, where h>3. We obtain upper bounds to the average maximum likelihood bit-error probability of double serially concatenated block and convolutional coding schemes. Then, we derive design guidelines for the outer, middle, and inner codes that maximize the interleaver gain and the asymptotic slope of the error probability curves. Finally, we propose a low-complexity iterative decoding algorithm. Comparisons with parallel concatenated convolutional codes, known as “turbo codes”, and with the proposed serially concatenated convolutional codes are also presented, showing that in some cases, the new schemes offer better performance     

Combined turbo codes and interleaver design

The impact of the distance spectrum and interleaver structure on the bit error probability of turbo codes is considered. A new turbo code design method for Gaussian channels is presented. The proposed method combines a search for good component codes with interleaver design. The optimal distance spectrum is used as the design criterion to construct good turbo component codes at low signal-to-noise ratios (SNRs). In addition, an interleaver design method is proposed. This design improves the code performance at high SNR. Search for good component codes at low SNR is combined with a code matched interleaver design. This results in new turbo codes with a superior error performance relative to the best known codes at both low and high SNR. The performance is verified by both analysis and simulation     

A code-matched interleaver design for turbo codes

A code-matched interleaver design for turbo codes in which a particular interleaver is constructed to match the code weight distribution is proposed. The design method is based on the code distance spectrum. The low weight paths in the code trellis which give large contributions to the error probability in the signal-to-noise ratio region of interest for practical communication systems are eliminated so that they do not appear in the overall code trellis after interleaving. The proposed interleaver improves the code error performance at moderate to high signal-to-noise ratio and considerably increases the asymptotic slope of the error probability curves     

Combined MMSE interference suppression and turbo coding for acoherent DS-CDMA system

The performance of a turbo-coded code division multiaccess system with a minimum mean-square error (MMSE) receiver for interference suppression is analyzed on a Rayleigh fading channel. In order to accurately estimate the performance of the turbo coding, two improvements are proposed on the conventional union bounds: the information of the minimum distance of a particular turbo interleaver is used to modify the average weight spectra, and the tangential bound is extended to the Rayleigh fading channel. Theoretical results are derived based on the optimum tap weights of the MMSE receiver and maximum-likelihood decoding. Simulation results incorporating iterative decoding, RLS adaptation, and the effects of finite interleaving are also presented. The results show that in the majority of the scenarios that we are concerned with, the MMSE receiver with a rate-1/2 turbo code will outperform a rate-1/4 turbo code. They also show that, for a bit error rate lower than 10^-3, the capacity of the system is increased by using turbo codes over convolutional codes, even with small block sizes     

Adaptive Transmission with Variable-Rate Turbo Bit-Interleaved Coded Modulation

We study an adaptive transmission scheme based on variable-rate turbo bit-interleaved coded modulation (VR- Turbo-BICM). The proposed coding scheme employs punctured turbo codes. A continuously varying transmission rate can be obtained by changing the code rate through both puncturing of the coded bits and adapting of the modulation constellation size. The main results are elaborated in two parts. First, we derive a closed-form expression for a set of achievable rate bounds (called rate thresholds) for VR-Turbo-BICM by employing recent results on the parallel channel performance of turbo code ensembles and the BICM parallel channel analysis model. The derived rate threshold is expressed as a fraction of the capacity of BICM with Gray mapping, where this fraction is a turbo code weight spectrum parameter. Simulation results illustrate that introduced rate thresholds predict well the rate versus SNR performance of VR-Turbo-BICM for a wide range of codeword error probabilities and codeword lengths. Next, based on a simplified rate threshold, we derive a power, puncturing rate, and modulation constellation size assignment policy for a slow fading channel.     

Reliability-based coded modulation with low-density parity-check codes

In this letter, we consider the interleaver design in bit-interleaved coded modulation (BICM) with low-density parity-check (LDPC) codes. The design paradigm is to provide more coding protection through iterative decoding to bits that are less protected by modulation (and are thus less reliable at the output of the demodulator). The design is carried out by an ad hoc search algorithm over the column permutations of the parity-check matrix. Our simulations show that the proposed reliability-based coded modulation scheme can improve the error-rate performance of conventional BICM schemes based on regular LDPC codes by a few tenths of a decibel, with no added complexity.     

Benchmarking Iterative Receivers for BICM Multi-Carrier Systems Against Capacity Bounds

In this paper, we analyze iterative receivers for bit-interleaved coded modulation (BICM) multi-carrier systems and compare them against theoretical capacity bounds for the channel, coded modulation, and BICM. We map the theoretical capacity bounds into bit-error rate (BER) versus average signal-to-noise ratio per bit plots to simplify the comparison between the theoretical capacity bounds and simulated BER curves. As BER simulations show, iterative receivers with code doping or spreading reach the turbo-cliff within 1 or 0.3dB of the independent Rayleigh fading channel capacity. While the iterative receiver with spreading is closer to the channel capacity than the one with code doping, the later one can eliminate the residual bit-errors after the turbo-cliff. We further present a combinatorial analysis of the distribution of the spread symbol constellation for Walsh-Hadamard spreading codes used in a BICM multi-carrier system to explain the above results.     

Zigzag codes and concatenated zigzag codes

This paper introduces a family of error-correcting codes called zigzag codes. A zigzag code is described by a highly structured zigzag graph. Due to the structural properties of the graph, very low-complexity soft-in/soft-out decoding rules can be implemented. We present a decoding rule, based on the Max-Log-APP (MLA) formulation, which requires a total of only 20 addition-equivalent operations per information bit, per iteration. Simulation of a rate-1/2 concatenated zigzag code with four constituent encoders with interleaver length 65 536, yields a bit error rate (BER) of 10^-5 at 0.9 dB and 1.3 dB away from the Shannon limit by optimal (APP) and low-cost suboptimal (MLA) decoders, respectively. A union bound analysis of the bit error probability of the zigzag code is presented. It is shown that the union bounds for these codes can be generated very efficiently. It is also illustrated that, for a fixed interleaver size, the concatenated code has increased code potential as the number of constituent encoders increases. Finally, the analysis shows that zigzag codes with four or more constituent encoders have lower error floors than comparable turbo codes with two constituent encoders     

Interleaver design for serially concatenated convolutional codes: theory and application

This paper addresses the problem of interleaver design for serially concatenated convolutional codes (SCCCs) tailored to the constituent codes of the SCCC configuration. We present a theoretical framework for interleaver optimization based on a cost function closely tied to the asymptotic bit-error rate (BER) of the block code C/sub s/ resulting from proper termination of the constituent codes in the SCCC code. We define a canonical form of the interleaving engine denoted as the finite state permuter (FSP) and using its structural property, develop a systematic iterative technique for construction of interleavers. The core theoretical results focus on the asymptotic behavior of a class of cost functions and their martingale property, which is then used to develop an order recursive interleaver optimization algorithm. We address the issue of the complexity of the interleaver growth algorithm presented in the paper and demonstrate that it has polynomial complexity. Subsequently, we provide details about the application of the proposed technique and present a modification of the algorithm that employs error pattern feedback for improved performance at a reduced complexity. Sample experimental results are provided for an SCCC code of rate 1/3 and information block length 320 that achieves a minimum distance of d/sub min/=44.     

Product accumulate codes: a class of codes with near-capacity performance and low decoding complexity

We propose a novel class of provably good codes which are a serial concatenation of a single-parity-check (SPC)-based product code, an interleaver, and a rate-1 recursive convolutional code. The proposed codes, termed product accumulate (PA) codes, are linear time encodable and linear time decodable. We show that the product code by itself does not have a positive threshold, but a PA code can provide arbitrarily low bit-error rate (BER) under both maximum-likelihood (ML) decoding and iterative decoding. Two message-passing decoding algorithms are proposed and it is shown that a particular update schedule for these message-passing algorithms is equivalent to conventional turbo decoding of the serial concatenated code, but with significantly lower complexity. Tight upper bounds on the ML performance using Divsalar's (1999) simple bound and thresholds under density evolution (DE) show that these codes are capable of performance within a few tenths of a decibel away from the Shannon limit. Simulation results confirm these claims and show that these codes provide performance similar to turbo codes but with significantly less decoding complexity and with a lower error floor. Hence, we propose PA codes as a class of prospective codes with good performance, low decoding complexity, regular structure, and flexible rate adaptivity for all rates above 1/2.     

On error bounds and turbo-codes

Turbo-codes have been hailed as the ultimate step toward achieving the capacity limit Shannon established some 50 years ago. We look at the performance of turbo-codes with respect to various information theoretic error bounds. This comparison suggests that, if (block, or) frame error rates are considered, careful interleaver design is necessary to ensure an error performance within a fraction of a decibel of the theoretical limit for large block sizes, while random interleavers perform well for block sizes smaller than about 2 K. If the bit error performance is considered, interleaver design seems to have only a minor effect, and the codes perform close to the limit for all block sizes considered     

Source-channel optimized trellis codes for bitonal imagetransmission over AWGN channels

We consider the design of trellis codes for transmission of binary images over additive white Gaussian noise (AWGN) channels. We first model the image as a binary asymmetric Markov source (BAMS) and then design source-channel optimized (SCO) trellis codes for the BAMS and AWGN channel. The SCO codes are shown to be superior to Ungerboeck's codes by approximately 1.1 dB (64-state code, 10^-5 bit error probability), We also show that a simple “mapping conversion” method can be used to improve the performance of Ungerboeck's codes by approximately 0.4 dB (also 64-state code and 10 ^-5 bit error probability). We compare the proposed SCO system with a traditional tandem system consisting of a Huffman code, a convolutional code, an interleaver, and an Ungerboeck trellis code. The SCO system significantly outperforms the tandem system. Finally, using a facsimile image, we compare the image quality of an SCO code, an Ungerboeck code, and the tandem code, The SCO code yields the best reconstructed image quality at 4-5 dB channel SNR     

On the theory and performance of trellis termination methods forturbo codes

The performance of a turbo code can be severely degraded if no trellis termination is employed. This paper investigates the implications of the choice of trellis termination method for turbo codes, and explains the origin of the performance degradation often experienced without trellis termination. An efficient method to derive the distance spectrum of turbo codes for different trellis termination methods is presented. Further, we present interleaver design rules that are tailored to each termination method. Using interleavers designed with these restrictions, we demonstrate that the performance difference between various termination methods is very small, including no trellis termination at all. For example, we demonstrate a turbo code with a 500-bit interleaver that exhibits no sign of an error floor for frame error rates as low as 10^-8, even though no trellis termination is employed     

Multilevel turbo coding with short interleavers

The impact of the interleaver, embedded in the encoder for a parallel concatenated code, called the turbo code, is studied. The known turbo codes consist of long random interleavers, whose purpose is to reduce the value of the error coefficients. It is shown that an increased minimum Hamming distance can be obtained by using a structured interleaver. For low bit-error rates (BERs), we show that the performance of turbo codes with a structured interleaver is better than that obtained with a random interleaver. Another important advantage of the structured interleaver is the short length required, which yields a short decoding delay and reduced decoding complexity (in terms of memory). We also consider the use of turbo codes as component codes in multilevel codes. Powerful coding structures that consist of two component codes are suggested. Computer simulations are performed in order to evaluate the reduction in coding gain due to suboptimal iterative decoding. From the results of these simulations we deduce that the degradation in the performance (due to suboptimal decoding) is very small