In this paper we investigate the decoding of parallel turbo codes over the binary erasure channel suited for upper-layer error correction. The proposed algorithm performs on-the-fly decoding, i.e. it starts decoding as soon as the first symbols are received. This algorithm compares with the iterative decoding of codes defined on graphs, in that it propagates in the trellises of the turbo code by removing transitions in the same way edges are removed in a bipartite graph under message-passing decoding. Performance comparison with LDPC codes for different coding rates is shown.
In this work, we develop a new iterative turbo receiver for LDPC-coded multi-antenna systems based on semidefinite relaxation (SDR). For a classical turbo receiver, forward error correction (FEC) code is only used at decoder. Nonetheless, by taking advantage of FEC code in the detection stage, our proposed SDR detector can output extrinsic information with much improved reliability to the decoder. We also propose a simplified SDR turbo receiver that solves only one SDR problem per codeword instead of solving multiple SDR problems in the iterative turbo processing. This scheme significantly red…
We review the recently introduced soft-aided bit-marking (SABM) algorithm and its suitability for product codes. Some aspects of the implementation of the SABM algorithm are discussed. The influence of suboptimal channel soft information is also analyzed.
This paper deals with two main issues regarding the short polar codes: the potential of FEC-assisted decoding and optimal code concatenation strategies under various design scenarios. Code concatenation and FEC-assisted decoding are presented systematically, assuming a packetized system. It is shown that FEC-assisted decoding can improve frame error rate of successive cancellation decoding arbitrarily, at the expense of some coding rate loss and decoding complexity linearly increasing with the number of codewords in the frame. This is compared with list decoding whose complexity grows linearly…
This paper is devoted to the finite-length analysis of turbo decoding over the binary erasure channel (BEC). The performance of iterative belief-propagation (BP) decoding of low-density parity-check (LDPC) codes over the BEC can be characterized in terms of stopping sets. We describe turbo decoding on the BEC which is simpler than turbo decoding on other channels. We then adapt the concept of stopping sets to turbo decoding and state an exact condition for decoding failure. Apply turbo decoding until the transmitted codeword has been recovered, or the decoder fails to progress further. Then th…
Guessing Random Additive Noise Decoding (GRAND) is a family of universal decoding algorithms suitable for decoding any moderate redundancy code of any length. We establish that, through the use of list decoding, soft-input variants of GRAND can replace the Chase algorithm as the component decoder in the turbo decoding of product codes. In addition to being able to decode arbitrary product codes, rather than just those with dedicated hard-input component code decoders, results show that ORBGRAND achieves a coding gain of up to 0.7dB over the Chase algorithm with same list size.
In this paper a new class of lattices called turbo lattices is introduced and established. We use the lattice Construction D to produce turbo lattices. This method needs a set of nested linear codes as its underlying structure. We benefit from turbo codes as our basis codes. Therefore, a set of nested turbo codes based on nested interleavers (block interleavers) and nested convolutional codes is built. To this end, we employ both tail-biting and zero-tail convolutional codes. Using these codes, along with construction D, turbo lattices are created. Several properties of Construction D lattices…
We present a novel algorithm that solves the turbo code LP decoding problem in a fininte number of steps by Euclidean distance minimizations, which in turn rely on repeated shortest path computations in the trellis graph representing the turbo code. Previous attempts to exploit the combinatorial graph structure only led to algorithms which are either of heuristic nature or do not guarantee finite convergence. A numerical study shows that our algorithm clearly beats the running time, up to a factor of 100, of generic commercial LP solvers for medium-sized codes, especially for high SNR values.
Iterative processing is widely adopted nowadays in modern wireless receivers for advanced channel codes like turbo and LDPC codes. Extension of this principle with an additional iterative feedback loop to the demapping function has proven to provide substantial error performance gain. However, the adoption of iterative demodulation with turbo decoding is constrained by the additional implied implementation complexity, heavily impacting latency and power consumption. In this paper, we analyze the convergence speed of these combined two iterative processes in order to determine the exact require…
The Progressive Edge Growth (PEG) algorithm is one of the most widely-used method for constructing finite length LDPC codes. In this paper we consider the PEG algorithm together with a scheduling distribution, which specifies the order in which edges are established in the graph. The goal is to find a scheduling distribution that yields "the best" performance in terms of decoding overhead, performance metric specific to erasure codes and widely used for upper-layer forward error correction (UL-FEC). We rigorously formulate this optimization problem, and we show that it can be addressed by usin…
This paper summarizes the design of a programmable processor with transport triggered architecture (TTA) for decoding LDPC and turbo codes. The processor architecture is designed in such a manner that it can be programmed for LDPC or turbo decoding for the purpose of internetworking and roaming between different networks. The standard trellis based maximum a posteriori (MAP) algorithm is used for turbo decoding. Unlike most other implementations, a supercode based sum-product algorithm is used for the check node message computation for LDPC decoding. This approach ensures the highest hardware …
In this paper we present a new Turbo analog error correcting coding scheme for real valued signals that are corrupted by impulsive noise. This Turbo code improves Donoho's deterministic construction by using a probabilistic approach. More specifically, our construction corrects more errors than the matrices of Donoho by allowing a vanishingly small probability of error (with the increase in block size). The problem of decoding the long block code is decoupled into two sets of parallel Linear Programming problems. This leads to a significant reduction in decoding complexity as compared to one-s…
In a recent paper [1] an improved analysis concerning the analysis of List Decoding was presented. The event that the correct codeword is excluded from the list is central. For the additive white Gaussian noise (AWGN) channel an important quantity is the in [1] called effective Euclidean distance. This was earlier considered in [2] under the name Vector Euclidean Distance, where also a simple mathematical expression for this quantity was easily derived for any list size. In [1], a geometrical analysis gives this when the list size is 1, 2 or 3.
In source coding, either with or without side information at the decoder, the ultimate performance can be achieved by means of random binning. Structured binning into cosets of performing channel codes has been successfully employed in practical applications. In this letter it is formally shown that various convolutional- and turbo-syndrome decoding algorithms proposed in literature lead in fact to the same estimate. An equivalent implementation is also delineated by directly tackling syndrome decoding as a maximum a posteriori probability problem and solving it by means of iterative message-p…
Turbo codes and CRC codes are usually decoded separately according to the serially concatenated inner codes and outer codes respectively. In this letter, we propose a hybrid decoding algorithm of turbo-CRC codes, where the outer codes, CRC codes, are not used for error detection but as an assistance to improve the error correction performance. Two independent iterative decoding and reliability based decoding are carried out in a hybrid schedule, which can efficiently decode the two different codes as an entire codeword. By introducing an efficient error detecting method based on normalized Euc…
We propose a new class of information-coupled (IC) Turbo codes to improve the transport block (TB) error rate performance for long-term evolution (LTE) systems, while keeping the hybrid automatic repeat request protocol and the Turbo decoder for each code block (CB) unchanged. In the proposed codes, every two consecutive CBs in a TB are coupled together by sharing a few common information bits. We propose a feed-forward and feed-back decoding scheme and a windowed (WD) decoding scheme for decoding the whole TB by exploiting the coupled information between CBs. Both decoding schemes achieve a c…
This paper presents TurboNet, a novel model-driven deep learning (DL) architecture for turbo decoding that combines DL with the traditional max-log-maximum a posteriori (MAP) algorithm. To design TurboNet, we unfold the original iterative structure for turbo decoding and replace each iteration by a deep neural network (DNN) decoding unit. In particular, the DNN decoding unit is obtained by parameterizing the max-log-MAP algorithm rather than replace the whole decoder with a black box fully connected DNN architecture. With the proposed architecture, the parameters can be efficiently learned fro…
This paper addresses the issue of phase noise in OFDM systems. Phase noise (PHN) is a transceiver impairment resulting from the non-idealities of the local oscillator. We present a case for designing a turbo receiver for systems corrupted by phase noise by taking a closer look at the effects of the common phase error (CPE). Using an approximate probabilistic framework called variational inference (VI), we develop a soft-in soft-out (SISO) algorithm that generates posterior bit-level soft estimates while taking into account the effect of phase noise. The algorithm also provides an estimate of t…
In this paper, we propose a turbo receiver for joint activity detection and data decoding in grant-free massive random access, which iterates between a detector and a belief propagation (BP)-based channel decoder. Specifically, responsible for user activity detection, channel estimation, and soft data symbol detection, the detector is developed based on a bilinear inference problem that exploits the common sparsity pattern in the received pilot and data signals. The bilinear generalized approximate message passing (BiG-AMP) algorithm is adopted to solve the problem using probabilities of the t…
For communication systems with heavy burst noise, an optimal Forward Error Correction (FEC) scheme is expected to have a large burst error correction capacity while simultaneously owning moderate random error correction capability. This letter presents a new FEC scheme based on multiple-symbol interleaved Reed-Solomon codes and an associated two-pass decoding algorithm. It is shown that the proposed multi-symbol interleaved coding scheme can achieve nearly twice as much as the burst error correction capability of conventional symbol-interleaved Reed-Solomon codes with the same code length and …
Partially information coupled turbo codes (PIC-TCs) is a class of spatially coupled turbo codes that can approach the BEC capacity while keeping the encoding and decoding architectures of the underlying component codes unchanged. However, PIC-TCs have significant rate loss compared to its component rate-1/3 turbo code, and the rate loss increases with the coupling ratio. To absorb the rate loss, in this paper, we propose the partially information coupled duo-binary turbo codes (PIC-dTCs). Given a rate-1/3 turbo code as the benchmark, we construct a duo-binary turbo code by introducing one extr…
Conventional turbo codes (CTCs) usually employ a block-oriented interleaving so that each block is separately encoded and decoded. As interleaving and de-interleaving are performed within a block, the message-passing process associated with an iterative decoder is limited to proceed within the corresponding range. This paper presents a new turbo coding scheme that uses a special interleaver structure and a multiple-round early termination test involving both sign check and a CRC code. The new interleaver structure is naturally suited for high speed parallel processing and the resulting coding …
Iterative probabilistic inference, popularly dubbed the soft-iterative paradigm, has found great use in a wide range of communication applications, including turbo decoding and turbo equalization. The classic approach of analyzing the iterative approach inevitably use the statistical and information-theoretical tools that bear ensemble-average flavors. This paper consider the per-block error rate performance, and analyzes it using nonlinear dynamical theory. By modeling the iterative processor as a nonlinear dynamical system, we report a universal "Z-crease phenomenon:" the zig-zag or up-and-d…
In this paper, we study a class of spatially coupled turbo codes, namely partially information- and partially parity-coupled turbo codes. This class of codes enjoy several advantages such as flexible code rate adjustment by varying the coupling ratio and the encoding and decoding architectures of the underlying component codes can remain unchanged. For this work, we first provide the construction methods for partially coupled turbo codes with coupling memory $m$ and study the corresponding graph models. We then derive the density evolution equations for the corresponding ensembles on the binar…
We study irregular binary turbo codes over non-ergodic block-fading channels. We first propose an extension of channel multiplexers initially designed for regular turbo codes. We then show that, using these multiplexers, irregular turbo codes that exhibit a small decoding threshold over the ergodic Gaussian-noise channel perform very close to the outage probability on block-fading channels, from both density evolution and finite-length perspectives.
Orthogonality is a much desired property for MIMO coding. It enables symbol-wise decoding, where the errors in other symbol estimates do not affect the result, thus providing an optimality that is worth pursuing. Another beneficial property is a low complexity soft decision decoder, which for orthogonal complex MIMO codes is known for two transmit (Tx) antennas i.e. for the Alamouti code. We propose novel soft decision decoders for the orthogonal complex MIMO codes on three and four Tx antennas and extend the old result of maximal ratio combining (MRC) to cover all orthogonal codes up to four …
We present an interleaving scheme that yields quasi-cyclic turbo codes. We prove that randomly chosen members of this family yield with probability almost 1 turbo codes with asymptotically optimum minimum distance, i.e. growing as a logarithm of the interleaver size. These interleavers are also very practical in terms of memory requirements and their decoding error probabilities for small block lengths compare favorably with previous interleaving schemes.
In this paper, iterative differential phase-shift keying (DPSK) demodulation and channel decoding scheme is investigated for the Joint Channel decoding and physical layer Network Coding (JCNC) approach in two-way relaying systems. The Bahl, Cocke, Jelinek, and Raviv (BCJR) algorithm for both coherent and noncoherent detection is derived for soft-in soft-out decoding of DPSK signalling over the two-user multiple-access channel with Rayleigh fading. Then, we propose a pragmatic approach with the JCNC scheme for iteratively exploiting the extrinsic information of the outer code. With coherent det…
Two classes of turbo codes over high-order finite fields are introduced. The codes are derived from a particular protograph sub-ensemble of the (dv=2,dc=3) low-density parity-check code ensemble. A first construction is derived as a parallel concatenation of two non-binary, time-variant accumulators. The second construction is based on the serial concatenation of a non-binary, time-variant differentiator and of a non-binary, time-variant accumulator, and provides a highly-structured flexible encoding scheme for (dv=2,dc=4) ensemble codes. A cycle graph representation is provided. The proposed …
In this paper, we present a belief propagation (BP) based algorithm for decoding non-orthogonal space-time block codes (STBC) from cyclic division algebras (CDA) having {\em large dimensions}. The proposed approach involves message passing on Markov random field (MRF) representation of the STBC MIMO system. Adoption of BP approach to decode non-orthogonal STBCs of large dimensions has not been reported so far. Our simulation results show that the proposed BP-based decoding achieves increasingly closer to SISO AWGN performance for increased number of dimensions. In addition, it also achieves ne…
It has been widely observed that there exists a fundamental trade-off between the minimum (Hamming) distance properties and the iterative decoding convergence behavior of turbo-like codes. While capacity achieving code ensembles typically are asymptotically bad in the sense that their minimum distance does not grow linearly with block length, and they therefore exhibit an error floor at moderate-to-high signal to noise ratios, asymptotically good codes usually converge further away from channel capacity. In this paper, we introduce the concept of tuned turbo codes, a family of asymptotically g…
We propose a decoder for Trellis-Constrained Codes, a super-class of Turbo- and LDPC codes. Inspired by amplitude amplification from quantum computing, we attempt to amplify the relative likelihood of the most likely codeword until it stands out from all other codewords.
In this paper we concentrate on rate-1/3 systematic parallel concatenated convolutional codes and their rate-1/2 punctured child codes. Assuming maximum-likelihood decoding over an additive white Gaussian channel, we demonstrate that a rate-1/2 non-systematic child code can exhibit a lower error floor than that of its rate-1/3 parent code, if a particular condition is met. However, assuming iterative decoding, convergence of the non-systematic code towards low bit-error rates is problematic. To alleviate this problem, we propose rate-1/2 partially-systematic codes that can still achieve a lowe…
This paper presents new FEC codes for the erasure channel, LDPC-Band, that have been designed so as to optimize a hybrid iterative-Maximum Likelihood (ML) decoding. Indeed, these codes feature simultaneously a sparse parity check matrix, which allows an efficient use of iterative LDPC decoding, and a generator matrix with a band structure, which allows fast ML decoding on the erasure channel. The combination of these two decoding algorithms leads to erasure codes achieving a very good trade-off between complexity and erasure correction capability.
Turbo code is a great achievement in the field of communication system. It can be created by connecting a turbo encoder and a decoder serially. A Turbo encoder is build with parallel concatenation of two simple convolutional codes. By varying the number of memory element (encoder configuration), code rate (1/2 or 1/3), block size of data and iteration, we can achieve better BER performance. Turbo code also consists of interleaver unit and its BER performance also depends on interleaver size. Turbo Decoder can be implemented using different algorithm, but Log -MAP decoding algorithm is less com…
Parameter recovering of channel codes is important in applications such as cognitive radio. The main task for that of a turbo code is to recover the interleaver. The existing optimal algorithm recovers interleaver parameters incrementally one by one. This algorithm continues till the end even if it has failed in the halfway. And there would be lots of wasted computation, as well as incorrectly recovered parameters that will badly deteriorate turbo decoding performance. To address such drawbacks, this paper proposes an early stopping method for the algorithm. Thresholds needed for the method ar…
In this paper we prove threshold saturation for spatially coupled turbo codes (SC-TCs) and braided convolutional codes (BCCs) over the binary erasure channel. We introduce a compact graph representation for the ensembles of SC-TC and BCC codes which simplifies their description and the analysis of the message passing decoding. We demonstrate that by few assumptions in the ensembles of these codes, it is possible to rewrite their vector recursions in a form which places these ensembles under the category of scalar admissible systems. This allows us to define potential functions and prove thresh…
Present-day communication systems routinely use codes that approach the channel capacity when coupled with a computationally efficient decoder. However, the decoder is typically designed for the Gaussian noise channel and is known to be sub-optimal for non-Gaussian noise distribution. Deep learning methods offer a new approach for designing decoders that can be trained and tailored for arbitrary channel statistics. We focus on Turbo codes and propose DeepTurbo, a novel deep learning based architecture for Turbo decoding. The standard Turbo decoder (Turbo) iteratively applies the Bahl-Cocke-Jel…
In this paper, we investigate in detail the performance of turbo codes in quasi-static fading channels both with and without antenna diversity. First, we develop a simple and accurate analytic technique to evaluate the performance of turbo codes in quasi-static fading channels. The proposed analytic technique relates the frame error rate of a turbo code to the iterative decoder convergence threshold, rather than to the turbo code distance spectrum. Subsequently, we compare the performance of various turbo codes in quasi-static fading channels. We show that, in contrast to the situation in the …
This paper provides simple lower bounds on the number of iterations which is required for successful message-passing decoding of some important families of graph-based code ensembles (including low-density parity-check codes and variations of repeat-accumulate codes). The transmission of the code ensembles is assumed to take place over a binary erasure channel, and the bounds refer to the asymptotic case where we let the block length tend to infinity. The simplicity of the bounds derived in this paper stems from the fact that they are easily evaluated and are expressed in terms of some basic p…
In this work, we consider pseudocodewords of (relaxed) linear programming (LP) decoding of 3-dimensional turbo codes (3D-TCs). We present a relaxed LP decoder for 3D-TCs, adapting the relaxed LP decoder for conventional turbo codes proposed by Feldman in his thesis. We show that the 3D-TC polytope is proper and $C$-symmetric, and make a connection to finite graph covers of the 3D-TC factor graph. This connection is used to show that the support set of any pseudocodeword is a stopping set of iterative decoding of 3D-TCs using maximum a posteriori constituent decoders on the binary erasure chann…
To meet the evolving data rate requirements of emerging wireless communication technologies, many parallel architectures have been proposed to implement high throughput turbo decoders. However, concurrent memory reading/writing in parallel turbo decoding architectures leads to severe memory conflict problem, which has become a major bottleneck for high throughput turbo decoders. In this paper, we propose a flexible and efficient VLSI architecture to solve the memory conflict problem for highly parallel turbo decoders targeting multi-standard 3G/4G wireless communication systems. To demonstrate…
Current optical coherent transponders technology is driving data rates towards 1 Tb/s/λand beyond. This trend requires both high-performance coded modulation schemes and efficient implementation of the forward-error-correction (FEC) decoder. A possible solution to this problem is combining advanced multidimensional modulation formats with low-complexity hybrid HD/SD FEC decoders. Following this rationale, in this paper we combine two recently introduced coded modulation techniques:the geometrically-shaped 4D-64 polarization ring-switched and the soft-aided bit-marking-scaled reliability decode…
In this article we focus on the problem of channel decoding in presence of a-priori information. In particular, assuming that the a-priori information reliability is not perfectly estimated at the receiver, we derive a novel analytical framework for evaluating the decoder's performance. It is derived the important result that a "good code", i.e., a code which allows to fully exploit the potential benefit of a-priori information, must associate information sequences with high Hamming weights to codewords with low Hamming weights. Basing on the proposed analysis, we analyze the performance of co…
In this paper, we investigate the performance of a class of spatially coupled codes, namely partially information coupled turbo codes (PIC-TCs) over the binary erasure channel (BEC). This class of codes enjoy flexible code rate adjustment by varying the coupling ratio. Moreover, the coupling method can be directly applied to any component codes without changing the encoding and decoding architectures of the underlying component codes. However, the theoretical performance of PIC-TCs has not been fully investigated. For this work, we consider the codes that have coupling memory $m$ and study the…
Semi-definite relaxation (SDR) detector has been demonstrated to be successful in approaching maximum likelihood (ML) performance while the time complexity is only polynomial. We propose a new receiver jointly utilizing the forward error correction (FEC) code information in the SDR detection process. Strengthened by the code constraints, the joint SDR detector provides soft information of much improved reliability to downstream decoder and therefore outperforms existing receivers with substantial gain.
As wireless communication systems look intently to compose the transition from voice communication to interactive Internet data, achieving higher bit rates becomes both increasingly desirable and challenging. Space-time coding (STC) is a communications technique for wireless systems that inhabit multiple transmit antennas and single or multiple receive antennas. Space-time codes make use of advantage of both the spatial diversity provided by multiple antennas and the temporal diversity available with time-varying fading. Space-time codes can be divided into block codes and trellis codes. Space…
