Recursive Convolutional Codes Research Articles

Efficient On-Off Keying Underwater Acoustic Communication for Seafloor Observation Networks

In the cableless seafloor observation networks (SONs), the links among network nodes rely on underwater acoustic communication (UAC). Due to the energy constraint and the high-reliability requirement of the cableless SONs, the noncoherent UAC has been a preferred choice, even though a noncoherent UAC scheme generally suffers from low spectral efficiency. In this paper, we propose a high-spectral-efficiency noncoherent UAC transmission scheme which is implemented as an orthogonal frequency-division multiplexing (OFDM) system adopting the on-off keying (OOK) modulation. To simultaneously achieve high performance at a low energy consumption, an irregular recursive convolutional code (IrCC) is employed and an accumulator (ACC) is introduced to achieve a modulation with memory at the transmitter side. The ACC enables a turbo iteration between the soft demapper called the ACC-OOK demapper and the soft decoder on the receiver side, and also reduces the decoding error floor. To account for the unknown signal-to-noise ratio (SNR), an iterative threshold estimation (ITE) algorithm is proposed to determine a proper decision threshold for the ACC-OOK demapper. The IrCC is designed to match the extrinsic information transfer (EXIT) curve of the ACC-OOK demapper, lowering the SNR threshold of the aforementioned turbo iteration. Simulations and experimental results verify the superiority of the proposed noncoherent UAC scheme over conventional ones.

A Serial Concatenation of Binary-Input Nonbinary-Output Convolutional Code and Recursive Convolutional Lattice Code

The recently proposed recursive convolutional lattice code (RCLC) can form a signal with pseudo-Gaussian constellations, and their parallel concatenation is shown to approach the Shannon limit. A practical limitation is that its input symbol is limited to $L^{2}$ -ary quadrature amplitude modulation (QAM), which has non-power-of-two constellation points when $L$ is chosen from the odd numbers. Therefore, encoding binary information by the RCLC is not straightforward. Furthermore, the information rate is limited to $\log _{2}~L$ bits per complex dimension due to their parallel concatenation. In this paper, we tackle these issues by introducing a serial concatenation of binary-input nonbinary-output convolutional code (CC) and the RCLC, where the outer CC outputs an $L$ -ary symbol that is matched to the input of the inner RCLC. We demonstrate that even with $L=3$ , the proposed approach can achieve 2 bits per complex dimension and still is able to approach the Shannon limit with lower decoding complexity compared with its parallel concatenation counterpart. As is demonstrated through theoretical analysis, the major practical drawback of the constellation generated by the RCLC is its Gaussian-like distribution, which has large peak-to-average power ratio. Therefore, we further introduce an approach to reduce the signal dynamic range for the proposed system. It is shown that a remarkable gain can be achieved in terms of capacity compared with the conventional QAM signals under the constraint of comparable power amplifier efficiency.

Capacity-Approaching Non-Binary Turbo Codes: A Hybrid Design Based on EXIT Charts and Union Bounds

In this paper, we introduce a novel design approach for capacity-approaching non-binary turbo codes. There are two important factors that impact the performance of turbo codes in general: 1) the convergence behavior of iterative decoding in the low signal-to-noise ratio (SNR) and 2) the error-floor effect in the high SNR. We thus design the non-binary turbo codes by means of the EXIT charts and truncated union bounds. We first reduce the search space of component recursive convolutional codes by the analysis based on the truncated union bounds in conjunction with the uniform interleaver, followed by its optimization through the EXIT chart analysis. The construction of the EXIT chart for non-binary turbo codes with fixed code coefficients is a non-trivial task by the fact that these messages have multiple parameters to identify. Therefore, we develop a new EXIT chart analysis for non-binary messages which does not rely on any specific message model. It is demonstrated through computer simulations that the well-designed non-binary turbo codes achieve a better performance than their binary counterparts as well as the conventional non-binary LDPC codes of the same field size. Furthermore, the code design is extended to high-order modulation, and our turbo codes designed for quadrature amplitude modulation are shown to outperform the conventional turbo trellis coded modulation schemes.

Blind Recognition of Recursive System Convolutional Codes

Recursive system convolutional codes (RSC codes) are the commonly used coding form for Turbo sub-encoders. For blind recognition problem of RSC codes, a novel method is proposed to recognize code parameters and generator polynomials. Firstly, code parameters are estimated by analyzing matrix constructed from the received codeword. Then, the mathematical model for recognition of generator polynomials is derived according to RSC codes’ characteristics, and the specific implementation steps are introduced. Finally, simulation results from Matlab show that the performance of the method is obviously more excellent than other methods, which indicates great significance to further research on Turbo codes.

Constrained Turbo Block Convolutional Codes for 100 G and Beyond Optical Transmissions

Constrained turbo block convolutional (CTBC) codes are developed for 100 G and beyond optical transmissions. The CTBC codes developed herein each fit within one optical transport network (OTN) frame. The CTBC codes involve a simple outer block code that is serially concatenated with a simple inner recursive convolutional code using a constrained interleaver that simultaneously delivers a high interleaver gain and a high minimum Hamming distance. Codes with 11.1 dB net coding gain (NCG) at 12.5% overhead (OH), 11.3 dB NCG at 15% OH, 11.6 dB NCG at 20% OH, and 11.9 dB NCG at 23.4% OH are reported. Compared with other codes that have been previously proposed for OTN applications, CTBC codes have much lower encoding/decoding complexity, improved NCG/OH tradeoffs, and avoid negative error floor effects.

Constrained Interleaving of Serially Concatenated Codes with Inner Recursive Codes

A novel constrained interleaving technique is proposed to improve serially concatenated codes (SCCs) with inner recursive convolutional codes (IRCCs). In this study, constrained interleavers are designed to achieve a minimum Hamming distance (MHD) for the SCC, d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SCC</sub> , between d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> and d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> while simultaneously maximizing the interleaver gain, where d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> and d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> are the MHD of the outer and inner codes respectively. Constrained interleavers can be constructed to achieve d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SCC</sub> =d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> while almost maintaining the interleaver gain of uniform interleaving. By imposing additional inter-row constraints, d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SCC</sub> of constrained interleaving is increased beyond d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> up to d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , however, at the expense of some interleaver gain. Numerical results demonstrate that constrained interleaving is an efficient way to construct SCCs with low error floors while achieving interleaver gain at relatively short interleaver sizes.

Analysis of recursive convolutional codes and turbo codes as sources with memory

The paper proposes a general method to analyze discrete sources with memory. Besides the classical entropy, we define new information measures for discrete sources with memory, similar to the information quantities specific to discrete channels. On the base of this method, we show for the first time that, as result of convolutional and turbo encoding, sources with memory are obtained. We apply this information analysis method for the general case of a recursive convolutional encoder of rate RCC=1/n0 and memory of order m, and for a turbo encoder of rate RTC=1/3, with two systematic recursive convolutional component encoders. Each component encoder has memory of order m, and is built based on the same primitive feedback polynomial. For the convolutional and turbo codes, the information quantities H(Y/S), H(S,Y), H(S/Y), H(Y), H(S) and I(S,Y) have been computed, where S and Y denote the set of states and the set of messages of the encoder, respectively. The analysis considered two cases: n0≤m+1 and n0>m+1. When n0=m+1, the mutual information I(S,Y) is maximum and equal to m, as is the entropy of the set of states. For turbo codes, the quantity I(S,Y) also depends on the input bit and on its probability.

A Turbo Coding Scheme for Channels with Synchronization Errors

We describe a turbo coding scheme with nonsystematic recursive convolutional codes for channels that exhibit insertions and deletions along with substitution errors. The scheme uses the MAP decoding algorithm for this channel and a special codeword arrangement to simplify the synchronization process. The decoder uses a special trellis structure to extract synchronization extrinsic information that is exchanged between the two constituent decoders. The proposed scheme is shown to provide improvement over earlier channel coding approaches for this channel model.

Outage Performance Analysis and Code Design for Three-Stage MMSE Turbo Equalization in Frequency-Selective Rayleigh Fading Channels

This paper analyzes the outage performance of soft-cancellation frequency-domain minimum-mean-square error (SC FD-MMSE) turbo equalization in a serially concatenated coded modulation scheme over frequency-selective Rayleigh fading channels with exponential delay-power profile. The convergence behavior of this iterative three-stage system is evaluated with the aid of correlation charts. Based on the convergence characteristic of the equalizer and the inner and outer channel decoders, we derive a simple closed-form approximation on the outage probability of the turbo receiver. In particular, using a union-bounding technique and a specific central limit theorem, it is shown that the outage probability can be well approximated by a sum of complementary Gaussian error functions. A design criterion based on the resultant analytical expression is then proposed to construct outer encoders that minimize the outage probability of three-stage turbo systems employing irregular recursive convolutional (IRC) codes. We provide examples of optimized IRC-based outer encoders and investigate by simulation the performance improvement obtained by optimized IRC-coded turbo equalization over turbo equalization using regular recursive convolutional codes. Numerical results of outage performance simulations in channels with different delay-power profiles are presented to verify the analysis.

An efficient variable-length code construction for iterative source-channel decoding

We present a novel variable-length code (VLC) construction which exhibits an inherent error correcting capability due to the exclusive presence of codewords with even Hamming weight. Besides error robustness, the proposed code construction features a similar codeword length distribution as Golomb-Rice codes, and therefore, in particular for sources with exponentially distributed symbols, has good source compression properties at the same time. We show that in a source channel coding framework with outer source encoding, inner channel encoding with a recursive convolutional code, and iterative decoding the proposed VLC construction can lead to significant performance improvements compared to fixed-length source encoding with optimized mappings. In particular, simulation results for the AWGN channel verify that for Gauss-Markov sources a performance close to the theoretical limit can be achieved.

Hybrid ARQ scheme based on recursive convolutional codes and turbo decoding

We propose a hybrid ARQ scheme using recursive convolutional codes, the turbo principle and combining diversity. While spending the same energy for transmission, the proposed scheme presents a better throughput for most of the SNR range and a smaller decoding complexity than another well-known method.

Design of Turbo-Coded Modulation for the AWGN Channel With Tikhonov Phase Error

We design 1-b/symbol/Hz parallel concatenated turbo-coded modulation (PCTCM) for the additive white Gaussian noise (AWGN) channel with Tikhonov phase error. Constituent recursive convolutional codes are optimized so that the turbo codes have low error floors and low convergence thresholds. The pairwise error probability based on the maximum-likelihood decoding metric is used to select codes with low error floors. We also present a Gaussian approximation method that accurately predicts convergence thresholds for PCTCM codes on the AWGN/Tikhonov channel. Simulation results show that the selected codes perform within 0.6 dB of constellation constrained capacity, and have no detectable error floor down to bit-error rates of 10/sup -6/.

High Rate Recursive Space-Time Trellis Code Designed for Serially Concatenated Space-Time Codes

Using several high rate recursive convolutional codes as the basic element and the trace criteria as the designing principle, a new kind of recursive space-time trellis code with more flexible and higher data rate is presented for the serially concatenated space-time code. When 2 b -ary modulation and N transmit antennas are used, the data rate of the new code can be arranged from b bps/Hz to Nb-1bps/Hz by modifying the number of recursive convolutional codes and the data rate of each code.

Design of recursive convolutional space-time codes with an arbitrary number of transmit antennas

A new class of recursive convolutional space-time codes (ReC-STC) with an arbitrary number of transmit antennas is designed by adopting several parallel two-state recursive systematic convolutional codes (RSCs). An intercross linear mapping rule that distributes the output bits of a RSC to the different positions of different transmitted signals is also described. The proposed ReC-STC can not only increase the data rate with the number of transmit antennas, but also perform well when it is used in a serially concatenated space-time code (SCSTC).

Iterative Source-Channel Decoding: Improved System Design Using EXIT Charts

The error robustness of digital communication systems using source and channel coding can be improved by iterative source-channel decoding (ISCD). The turbo-like evaluation of natural residual source redundancy and of artificial channel coding redundancy makes step-wise quality gains possible by several iterations. The maximum number of profitable iterations is predictable by an EXIT chart analysis. In this contribution, we exploit the EXIT chart representation to improve the error correcting/concealing capabilities of ISCD schemes. We propose new design guidelines to select appropriate bit mappings and to design the channel coding component. A parametric source coding scheme with some residual redundancy is assumed. Applying both innovations, the new EXIT-optimized index assignment as well as the appropriately designed recursive nonsystematic convolutional (RNSC) code allow to outperform known approaches to ISCD by far in the most relevant channel conditions.

High-Rate Recursive Convolutional Codes for Concatenated Channel Codes

This letter presents the results of the search for optimum punctured recursive convolutional codes (RCCs) of rate k/k+1, for k=2,...,8, suitable for concatenated channel codes whose constituent encoders are recursive, systematic convolutional codes. The mother codes that are punctured are rate-1/2 RCCs proposed for use in parallel and/or serial concatenation schemes. Extensive tables of systematic and nonsystematic puncturing patterns, optimized relative to various objective functions suitable for concatenated channel codes, are presented for several mother codes.

An Extensive Search for Good Punctured Rate-&amp;gt;tex&amp;lt;$ kover k+1$&amp;gt;/tex&amp;lt;Recursive Convolutional Codes for Serially Concatenated Convolutional Codes

In many practical applications requiring variable-rate coding and/or high-rate coding for spectral efficiency, there is a need to employ high-rate convolutional codes (CC), either by themselves or in a parallel or serially concatenated scheme. For such applications, in order to keep the trellis complexity of the code constant and to permit the use of a simplified decoder that can accommodate multiple rates, a mother CC is punctured to obtain codes with a variety of rates. This correspondence presents the results of extensive search for optimal puncturing patterns for recursive convolutional codes leading to codes of rate k/(k+1) (k an integer) to be used in serially concatenated convolutional codes (SCCC). The code optimization is in the sense of minimizing the required signal-to-noise ratio (SNR) for two target bit-error rate (BER) and two target frame-error rate (FER) values. We provide extensive sample simulation results for rate-k/(k+1) SCCC codes employing our optimized punctured CC.

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.

Serial Concatenation of Space-Time and Recursive Convolutional Codes

We propose a new serial concatenation scheme for space-time and recursive convolutional codes, in which a space-time code is used as the outer code and a single recursive convolutional code as the inner code. We discuss previously proposed serial concatenation schemes employing multiple inner codes and compare them with the new one. The proposed method and the previous one with joint decoding, both performing a combined decoding of the simultaneous output signals from multiple antennas, give a large performance gain over the separate decoding method. In decoding complexity, the new concatenation scheme has a lower complexity compared with the multiple encoding/joint decoding scheme due to the use of the single inner code. Simulation results for a communication system with two transmit and one receive antennas in a quasi-static Rayleigh fading channel show that the proposed scheme outperforms the previous schemes.

Iterative turbo decoder analysis based on density evolution

We track the density of extrinsic information in iterative turbo decoders by actual density evolution, and also approximate it by symmetric Gaussian density functions. The approximate model is verified by experimental measurements. We view the evolution of these density functions through an iterative decoder as a nonlinear dynamical system with feedback. Iterative decoding of turbo codes and of serially concatenated codes is analyzed by examining whether a signal-to-noise ratio (SNR) for the extrinsic information keeps growing with iterations. We define a for the iterative decoder, such that the turbo decoder will converge to the correct codeword if the noise figure is bounded by a number below zero dB. By decomposing the code's noise figure into individual curves of output SNR versus input SNR corresponding to the individual constituent codes, we gain many new insights into the performance of the iterative decoder for different constituents. Many mysteries of turbo codes are explained based on this analysis. For example, we show why certain codes converge better with iterative decoding than more powerful codes which are only suitable for maximum likelihood decoding. The roles of systematic bits and of recursive convolutional codes as constituents of turbo codes are crystallized. The analysis is generalized to serial concatenations of mixtures of complementary outer and inner constituent codes. Design examples are given to optimize mixture codes to achieve low iterative decoding thresholds on the signal-to-noise ratio of the channel.