Performance of hybrid turbo codes

The authors present the latest results on turbo codes which are built from a serial concatenation between a block code (BCH) and a recursive systematic convolutional code. The performance on a Gaussian channel with QPSK modulation is evaluated     

Turbo codes extended with outer BCH code

The `error floor' observed in several simulations with the turbo codes is verified by calculation of an upper bound to the bit error rate for the ensemble of all interleavers. Also an easy way to calculate the weight enumerator used in this bound is presented. An extended coding scheme is proposed including an outer BCH code correcting a few bit errors     

Serial concatenation of interleaved convolutional codes andM-ary continuous phase modulations

We consider the serial concatenation of a convolutional code and a continuous phase modulation separated by a random interleaver. It is shown that the continuous phase modulator has the same behavior as a recursive systematic convolutional code in serial turbo codes. The interleaver gain produced by the modulator and the random interleaving depends only on the minimum Hamming distance of the outer code. A soft-input soft-output detection algorithm is described and applied in an iterative joint detection-decoding scheme. Simulated performance over Gaussian and Rayleigh channels shows a dramatic gain over both uncoded modulation and classic detection cases.     

Convolutional double accumulate codes (or double turbo DPSK)

We consider the serial concatenation of a convolutional code with two differential encoders (or accumulate codes). Simulations of such a concatenated scheme using a rate 1/2 outer convolutional code (sometimes punctured to increase the overall rate) show that, at high rates, the bit error rate (BER) and frame error rate (FER) performance of these codes is significantly better than a concatenation with only one accumulate code and a parallel concatenated convolutional code (PCCC) of similar complexity. An intuitive explanation for the performance of these codes is also given     

A Coded Modulation Scheme for Deep-Space Optical Communications

A coded modulation scheme for deep-space optical communications is studied, which is a serial concatenation of an outer low-density parity-check (LDPC) code, an interleaver, a bit-accumulator, and pulse-position modulation (PPM). It is referred as LDPC-APPM. It is decoded with an iterative demodulator-decoder using standard turbo-decoding techniques. Simulation results show that the LDPC-APPM with an outer LDPC code of medium code length suffers a loss of about 0.7 dB at a bit-error rate of relative to the serially concatenated PPM (SCPPM) of similar parameters, which consists of the serial concatenation of an outer convolutional code, an interleaver, a bit-accumulator, and PPM; however, it has better code rate flexibility especially for high rates and simpler implementation structure than SCPPM. Further, it can achieve better performance than the LDPC-PPM of the same rate and length, which consists of the serial concatenation of an LDPC code and PPM.     

Iterative multiuser interference reduction: turbo CDMA

We view the asynchronous random code division multiple-access (CDMA) channel as a time-varying convolutional code. We study the case where the users encode their data, and, therefore, the single user transmitters and the CDMA channel appear as the concatenation of two coding systems. At the receiver we employ serial turbo decoding strategies. Unlike conventional turbo codes where both the inner and outer code may be selected, in our case, the inner code is due to the CDMA channel which we assume to be random. Nevertheless, the decoding system resembles the decoder of a serial turbo code and single-user performance is obtained even for numbers of users approaching the spreading code length     

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.     

On the frame-error rate of concatenated turbo codes

Turbo codes with long frame lengths are usually constructed using a randomly chosen interleaver. Statistically, this guarantees excellent bit-error rate (BER) performance but also generates a certain number of low weight codewords, resulting in the appearance of an error floor in the BER curve. Several methods, including using an outer code, have been proposed to improve the error floor region of the BER curve. We study the effect of an outer BCH code on the frame-error rate (FER) of turbo codes. We show that additional coding gain is possible not only in the error floor region but also in the waterfall region. Also, the outer code improves the iterative APP decoder by providing a stopping criterion and alleviating convergence problems. With this method, we obtain codes whose performance is within 0.6 dB of the sphere packing bound at an FER of 10^-6     

VLSI architectures for turbo codes

A great interest has been gained in recent years by a new error-correcting code technique, known as “turbo coding”, which has been proven to offer performance closer to the Shannon's limit than traditional concatenated codes. In this paper, several very large scale integration (VLSI) architectures suitable for turbo decoder implementation are proposed and compared in terms of complexity and performance; the impact on the VLSI complexity of system parameters like the state number, number of iterations, and code rate are evaluated for the different solutions. The results of this architectural study have then been exploited for the design of a specific decoder, implementing a serial concatenation scheme with 2/3 and 3/4 codes; the designed circuit occupies 35 mm², supports a 2 Mb/s data rate, and for a bit error probability of 10^-6, yields a coding gain larger than 7 dB, with ten iterations     

一种LDPC编码高阶调制系统的联合解调解码方法

该文用一种级联码模型描述了LDPC编码高阶调制系统。该级联码模型以LDPC码为外码,二-十进制转换码为内码,再加一个删余模块构成。基于这种级联码模型,该文给出了其联合校验方程和二分图,并提出了级联码置信度传播算法,实现了LDPC编码高阶调制系统的联合解调解码。仿真表明,该文提出的联合解调解码算法有效地改进了LDPC编码高阶调制系统的性能。     

On turbo encodedBicm

The superior performance of the binary turbo codes has stimulated vigorous efforts in generating bandwidth efficient modulation schemes adhering to these codes. Several approaches for the integration of turbo-coding and modulation have emerged in recent years but none seem to dominate. In the bit interleaved coded modulation (Bicm) scheme is used to achieve high bandwidth and power efficiency, while separating coding and modulation. As is now well known, theBicm scheme achieves capacity remarkably close to the constellation channel capacity. The turbo-Bicm scheme enjoys high coding diversity (well suited for fading channels), high flexibility as well as design and implementation simplicity, while maintaining good power efficiency. The system comprises one standard turbo code, an interleaver, a mapper and a modulator at the transmitter, corresponding to a demodulator, a de-interleaver and a turbo decoder at the receiver. A modified system, which improves on performance by incorporating the demodulation in the iterative decoding procedure, is investigated, and some performance gain is demonstrated, especially for low rate codes. Information theoretic arguments for the somewhat minor potential improvement in performance are detailed. The preferred mapper and interleaver for this system are considered. Extending previous works, for higher level modulations, we analyze a system including a convolutional code, an interleaver, a differential encoder (De), a mapper and a modulator at the transmitter. As for theBpsk modulation, the serial concatenation of a convolutional code withDe outperforms the single convolutional code. The serial concatenation withDe approach is analyzed also for a turbo code, where it is found to fail in achieving performance improvement. Several structures for the serial concatenation withDe are examined. These results are substantiated through the ‘spectral thinning’ phenomena of the weight distribution of the convolutional and turbocodes.     

Concatenated codes for fading channels based on recursive space-time trellis codes

We propose a class of codes which combine the principles of turbo coding and space-time trellis codes. It is first shown that several classes of space-time codes have an equivalent recursive realization. This fact is then exploited to design serial concatenated coding schemes with an outer code, interleaver, and an inner recursive space-time encoder. Two solutions are proposed in this paper - the use of convolutional outer codes aimed mainly to improve the power efficiency and the use of very high-rate outer codes to obtain significant improvement in power efficiency with a marginal decrease in spectral efficiency. We show that single parity check based turbo product codes are a good candidate for very high-rate outer codes. Finally, we propose an automatic repeat request scheme based on recursive realizations of space-time codes and show that the proposed scheme provides significant reduction in frame error rate.     

Iterative decoding of space-time differentially coded unitary matrix modulation

Noncoherent communication over the Rayleigh flat fading channel with multiple transmit and receive antennas is investigated. Codes achieving bit error rate (BER) lower than 10/sup -4/ at bit energy over the noise spectral density ratio (E/sub b//N/sub 0/) of 0.8 to 2.8 dB from the capacity limit were found with coding rates of 0.5 to 2.25 bits per channel use. The codes are serial concatenation of a turbo code and a unitary matrix differential modulation code. The receiver is based on a high-performance joint iterative decoding of the turbo code and the modulation code. Information-theoretic arguments are harnessed to form guidelines for code design and to evaluate performance of the iterative decoder.     

A new high rate code scheme with highly parallel and low complexity decoding algorithm

A new high rate code scheme is proposed in this paper. It consists of serial concatenated recursive systematic ordinary (nonpunctured) convolutional codes with only 8 states in the trellis of the corresponding reciprocal dual codes. With a low complexity and highly parallel decoding algorithm, over additive white Gaussian noise channels, the proposed codes can achieve good bit error rate (BER) performance comparable to that of turbo codes and low density parity check (LDPC) codes. At code rate R=16/17, the overall decoding complexity of the proposed code scheme is almost half that of the LDPC 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.     

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.     

A soft-input soft-output APP module for iterative decoding ofconcatenated codes

Concatenated coding schemes consist of the combination of two or more simple constituent encoders and interleavers. The parallel concatenation known as “turbo code” has been shown to yield remarkable coding gains close to theoretical limits, yet admitting a relatively simple iterative decoding technique. The recently proposed serial concatenation of interleaved codes may offer superior performance to that of turbo codes. In both coding schemes, the core of the iterative decoding structure is a soft-input soft-output (SISO) a posteriori probability (APP) module. In this letter, we describe the SISO APP module that updates the APP's corresponding to the input and the output bits, of a code, and show how to embed it into an iterative decoder for a new hybrid concatenation of three codes, to fully exploit the benefits of the proposed SISO APP module     

Concatenated code system design for storage channels

A number of papers have been published on the concatenation of an outer code with a partial response (PR) channel, where the outer code is a turbo code, a convolutional code, or a low-density parity-check code. This paper deals with the second case, assuming EPR4 and modified extended EPR4 (MEEPR4) partial response (PR) targets. The goals in this work include (1) the joint optimization of interleaver and precoder for a fixed outer convolutional code and PR target, (2) the choice of optimal code rate for both PR targets assuming a Lorentzian model, and (3) an assessment of the performance of these codes in the presence of thermal asperities. We introduce mathematical and algorithmic tools for accomplishing these goals and present simulation results that support our approach. Among the positive results is the ability to lower the well-known error rate floor of these concatenated schemes for arbitrary PR channels     

Optimizations of a Hardware Decoder for Deep-Space Optical Communications

The National Aeronautics and Space Administration has developed a capacity approaching modulation and coding scheme that comprises a serial concatenation of an inner accumulate pulse-position modulation (PPM) and an outer convolutional code [or serially concatenated PPM (SCPPM)] for deep-space optical communications. Decoding of this code uses the turbo principle. However, due to the nonbinary property of SCPPM, a straightforward application of classical turbo decoding is very inefficient. Here, we present various optimizations applicable in hardware implementation of the SCPPM decoder. More specifically, we feature a Super Gamma computation to efficiently handle parallel trellis edges, a pipeline-friendly ";maxstar top-2"; circuit that reduces the max-only approximation penalty, a low-latency cyclic redundancy check circuit for window-based decoders, and a high-speed algorithmic polynomial interleaver that leads to memory savings. Using the featured optimizations, we implement a 6.72 megabits-per-second (Mbps) SCPPM decoder on a single field-programmable gate array (FPGA). Compared to the current data rate of 256 kilobits per second from Mars, the SCPPM coded scheme represents a throughput increase of more than twenty-six fold. Extension to a 50-Mbps decoder on a board with multiple FPGAs follows naturally. We show through hardware simulations that the SCPPM coded system can operate within 1 dB of the Shannon capacity at nominal operating conditions.     

Turbo processing in transmit antenna diversity systems

We consider turbo-trellis-coded transmission over fading multiple-input-multiple-output (M1M0) channels with transmit diversity using space-time block codes. We give a new view on space-time block codes as a transformation of the fading MIMO channel towards a Gaussian single-input-single-output (siso) channel and provide analytical results on the BER of space-time block codes. Furthermore, we describe the concatenation of Turbo-TCM with a space-time block code and show that in addition to the transmit diversity substantial benefits can be obtained by turbo iterations as long as the channel is time-varying during transmission of a coded block or frequency hopping is applied. Finally, a double iterative scheme for turbo equalization and turbo decoding of the concatenation of Turbo-TCM and space-time block code in frequency-selective MIMO channels is described.