Mutual information transfer characteristics of soft in/soft out decoders are proposed as a tool to better understand the convergence behavior of iterative decoding schemes. The exchange of extrinsic information is visualized as a decoding trajectory in the extrinsic information transfer chart (EXIT chart). This allows the prediction of turbo cliff position and bit error rate after an arbitrary number of iterations. The influence of code memory, code polynomials as well as different constituent codes on the convergence behavior is studied for parallel concatenated codes. A code search based on the EXIT chart technique has been performed yielding new recursive systematic convolutional constituent codes exhibiting turbo cliffs at lower signal-to-noise ratios than attainable by previously known constituent codes.

/pdf/convergence-behavior-of-iteratively-decoded-parallel-4rfahetw9j.pdf

Convergence behavior of iteratively decoded parallel concatenated codes

Optical wireless communication (OWC) refers to transmission in unguided propagation media through the use of optical carriers, i.e., visible, infrared (IR), and ultraviolet (UV) bands. In this survey, we focus on outdoor terrestrial OWC links which operate in near IR band. These are widely referred to as free space optical (FSO) communication in the literature. FSO systems are used for high rate communication between two fixed points over distances up to several kilometers. In comparison to radio-frequency (RF) counterparts, FSO links have a very high optical bandwidth available, allowing much higher data rates. They are appealing for a wide range of applications such as metropolitan area network (MAN) extension, local area network (LAN)-to-LAN connectivity, fiber back-up, backhaul for wireless cellular networks, disaster recovery, high definition TV and medical image/video transmission, wireless video surveillance/monitoring, and quantum key distribution among others. Despite the major advantages of FSO technology and variety of its application areas, its widespread use has been hampered by its rather disappointing link reliability particularly in long ranges due to atmospheric turbulence-induced fading and sensitivity to weather conditions. In the last five years or so, there has been a surge of interest in FSO research to address these major technical challenges. Several innovative physical layer concepts, originally introduced in the context of RF systems, such as multiple-input multiple-output communication, cooperative diversity, and adaptive transmission have been recently explored for the design of next generation FSO systems. In this paper, we present an up-to-date survey on FSO communication systems. The first part describes FSO channel models and transmitter/receiver structures. In the second part, we provide details on information theoretical limits of FSO channels and algorithmic-level system design research activities to approach these limits. Specific topics include advances in modulation, channel coding, spatial/cooperative diversity techniques, adaptive transmission, and hybrid RF/FSO systems.

/pdf/survey-on-free-space-optical-communication-a-communication-239v68hfyw.pdf

Survey on Free Space Optical Communication: A Communication Theory Perspective

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.

/pdf/serial-concatenation-of-interleaved-codes-performance-12glcxdjs9.pdf

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

We study the turbo equalization approach to coded data transmission over channels with intersymbol interference. In the original system invented by Douillard et al. (1995), the data are protected by a convolutional code and the receiver consists of two trellis-based detectors, one for the channel (the equalizer) and one for the code (the decoder). It has been shown that iterating equalization and decoding tasks can yield tremendous improvements in bit error rate. We introduce new approaches to combining equalization based on linear filtering, with decoding.. Through simulation and analytical results, we show that the performance of the new approaches is similar to the trellis-based receiver, while providing large savings in computational complexity. Moreover, this paper provides an overview of the design alternatives for turbo equalization with given system parameters, such as the channel response or the signal-to-noise ratio.

http://www.ifp.illinois.edu/~singer/journalpapers/tuchler_2002a.pdf

Turbo equalization: principles and new results

Density evolution is an algorithm for computing the capacity of low-density parity-check (LDPC) codes under message-passing decoding. For memoryless binary-input continuous-output additive white Gaussian noise (AWGN) channels and sum-product decoders, we use a Gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm. We convert the infinite-dimensional problem of iteratively calculating message densities, which is needed to find the exact threshold, to a one-dimensional problem of updating the means of the Gaussian densities. This simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better, but also makes it easier to design good irregular LDPC codes for AWGN channels. For various regular LDPC codes we have examined, thresholds can be estimated within 0.1 dB of the exact value. For rates between 0.5 and 0.9, codes designed using the Gaussian approximation perform within 0.02 dB of the best performing codes found so far by using density evolution when the maximum variable degree is 10. We show that by using the Gaussian approximation, we can visualize the sum-product decoding algorithm. We also show that the optimization of degree distributions can be understood and done graphically using the visualization.

/pdf/analysis-of-sum-product-decoding-of-low-density-parity-check-2he3dyypdf.pdf

Analysis of sum-product decoding of low-density parity-check codes using a Gaussian approximation

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.

A soft-input soft-output APP module for iterative decoding of concatenated codes

In this article we apply transfer function bounding techniques to obtain upper bounds on the bit-error rate for maximum likelihood decoding of turbo codes constructed with random permutations. These techniques are applied to two turbo codes with constraint length 3 and later extended to other codes. The performance predicted by these bounds is compared with simulation results. The bounds are useful in estimating the 'error floor' that is difficult to measure by simulation, and they provide insight on how to lower this floor. More refined bounds are needed for accurate performance measures at lower signal-to-noise ratios.

/pdf/transfer-function-bounds-on-the-performance-of-turbo-codes-bifg084vyc.pdf

Transfer function bounds on the performance of turbo codes

In this article, we present two versions of a simplified maximum a posteriori decoding algorithm. The algorithms work in a sliding window form, like the Viterbi algorithm, and can thus be used to decode continuously transmitted sequences obtained by parallel concatenated codes, without requiring code trellis termination. A heuristic explanation is also given of how to embed the maximum a posteriori algorithms into the iterative decoding of parallel concatenated codes (turbo codes). The performances of the two algorithms are compared on the basis of a powerful rate 1/3 parallel concatenated code. Basic circuits to implement the simplified a posteriori decoding algorithm using lookup tables, and two further approximations (linear and threshold), with a very small penalty, to eliminate the need for lookup tables are proposed.

/pdf/soft-output-decoding-algorithms-in-iterative-decoding-of-3ch9keg086.pdf

Soft-Output Decoding Algorithms in Iterative Decoding of Turbo Codes

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.

Analysis, design, and iterative decoding of double serially concatenated codes with interleavers

We develop new, low complexity turbo codes suitable for bandwidth and power limited systems, for very low bit and word error rate requirements. Motivated by the structure of previously discovered low complexity codes such as repeat-accumulate (RA) codes with low density parity check matrix, we extend the structure to high-level modulation such as 8PSK, and 16QAM. The structure consists of a simple 4-state convolutional or short block code as an outer code, and a rate-1, 2 or 4-state inner code. Two design criteria are proposed: the maximum likelihood design criterion, for short to moderate block sizes, and an iterative decoding design criterion for very long block sizes.

F. Pollara

Papers

A soft-input soft-output APP module for iterative decoding of concatenated codes

Transfer function bounds on the performance of turbo codes

Soft-Output Decoding Algorithms in Iterative Decoding of Turbo Codes

Analysis, design, and iterative decoding of double serially concatenated codes with interleavers

Serial concatenated trellis coded modulation with rate-1 inner code