Abstract—Extrinsic information transfer (EXIT) charts are a tool for predicting the convergence behavior of iterative processors for a variety of communication problems. A model is introduced that applies to decoding problems, including the iterative decoding of parallel concatenated (turbo) codes, serially concatenated codes, low-density parity-check (LDPC) codes, and repeat–accumulate (RA) codes. EXIT functions are defined using the model, and several properties of such functions are proved for erasure channels. One property expresses the area under an EXIT function in terms of a conditional entropy. A useful consequence of this result is that the design of capacity-approaching codes reduces to a curve-fitting problem for all the aforementioned codes. A second property relates the EXIT function of a code to its Helleseth–Kløve–Levenshtein information functions, and thereby to the support weights of its subcodes. The relation is via a refinement of information functions called split information functions, and via a refinement of support weights called split support weights. Split information functions are used to prove a third property that relates the EXIT function of a linear code to the EXIT function of its dual. Index Terms—Concatenated codes, duality, error-correction coding, iterative decoding, mutual information.

We study the convergence behavior of iterative decoding of a serially concatenated code. We rederive a existing analysis technique called EXIT chart [15] and show that for certain decoders the construction of an EXIT chart simplifies tremendously. The findings are extended such that simple irregular codes can be constructed, which can be used to improve the converence of the iterative decoding algorithm significantly. An efficient and optimal optimization algorithm is presented. Finally, some results on thresholds on the decoding convergence are outlined.

We optimize the random-like ensemble of Irregular Repeat Accumulate (IRA) codes for binary-input symmetric channels in the large blocklength limit. Our optimization technique is based on approximating the Evolution of the Densities (DE) of the messages exchanged by the Belief-Propagation (BP) message-passing decoder by a one-dimensional dynamical system. In this way, the code ensemble optimization can be solved by linear programming. We propose four such DE approximation methods, and compare the performance of the obtained code ensembles over the binary symmetric channel (BSC) and the binaryantipodal input additive white Gaussian channel (BIAWGNC). Our results clearly identify the best among the proposed methods and show that the IRA codes obtained by these methods are competitive with respect to the best-known irregular Low-Density Parity-Check codes (LDPC). In view of this and the very simple encoding structure of IRA codes, they emerge as attractive design choices.

We study the convergence behavior of iterative decoding for a number of serially concatenated systems, such as a serially concatenated code, coded data transmission over an inter-symbol interference channel, bit-interleaved coded modulation, or trellis-coded modulation. We rederive an existing analysis technique called EXIT chart, simplify its construction, and construct simple irregular codes to improve the convergence of iterative decoding. An efficient and optimal optimization algorithm yields systems, which approach information theoretic limits very closely. However, these systems exhibit their performance only for very long block lengths. To overcome this problem, we optimize the decoding convergence after a fixed, finite amount of iterations yielding systems, which perform very well for short block lengths, too. As an example, optimal system configurations for communication over an additive white Gaussian noise channel are presented.

The symbol-wise mutual information between the binary inputs of a channel encoder and the soft-outputs of a LogAPP decoder, i.e., the a-posteriori log-likelihood ratios (LLRs), is analyzed. This mutual information can be expressed as the expectation of a function of solely the absolute values of the a-posteriori LLRs. This result provides a simple and elegant method for computing the mutual information by simulation. As opposed to the conventional method, explicit measurements of histograms of the soft-outputs are not necessary. In fact, online estimation is possible, and bits having different statistical properties need not be treated separately. As a direct application, the computation of extrinsic information transfer (EXIT) charts is considered.

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This overview talk shows that the so-called turbo codes(decoders) entail a much broader principle. It discusses how the feedback of extrinsic information which we call the turbo principle can be used in many mobile communications receivers to improve performance through iterative processing. As an analysis and design tool the EXIT charts of mutual information transfer are used.

We show how to use EXIT charts for convergence prediction of a threefold serially concatenated system. The corresponding chart has three dimensions and allows to appropriately select system parameters and to find an optimal schedule of decoding iterations between the three decoders of such a system. Convergence thresholds are obtained to determine the minimal signalto -noise ratios for which convergence is possible. It turns out that threefold concatenated systems do not achieve any additional performance gain compared to suitably designed twofold systems. We conclude that a threefold concatenation should be considered only when the decoders cannot be chosen freely.

Abstract — This survey paper provides fundamentals in the design of LDPC codes. To provide a target for the code designer, we first summarize the EXIT chart technique for determining (near-)optimal degree distributions for LDPC code ensembles. We also demonstrate the simplicity of representing codes by protographs and how this naturally leads to quasi-cyclic LDPC codes. The EXIT chart technique is then extended to the special case of protograph-based LDPC codes. Next, we present several design approaches for LDPC codes which incorporate one or more accumulators, including quasi-cyclic accumulatorbased codes. The second half the paper then surveys several algebraic LDPC code design techniques. First, codes based on finite geometries are discussed and then codes whose designs are based on Reed-Solomon codes are covered. The algebraic designs lead to cyclic, quasi-cyclic, and structured codes. The masking technique for converting regular quasi-cyclic LDPC codes to irregular codes is also presented. Some of these results and codes have not been presented elsewhere. The paper focuses on the binary-input AWGN channel (BI-AWGNC). However, as discussed in the paper, good BI-AWGNC codes tend to be universally good across many channels. Alternatively, the reader may treat this paper as a starting point for extensions to more advanced channels. The paper concludes with a brief discussion of open problems. I.

We study the convergence behavior of iterative decoding of various serially concatenated systems such as a concatenated code, coded transmission over a channel introducing inter-symbol interference, bitinterleaved coded modulation, trellis coded modulation, a.s.o.