Abstract- Iterative decoding of two-dimensional systematic convolutional codes has been termed “turbo ” (de)coding. Using log-likelihood algebra, we show that any decoder can he used which accepts soft inputs-including a priori values-and delivers soft outputs that can he split into three terms: the soft channel and a priori inputs, and the extrinsic value. The extrinsic value is used as an a priori value for the next iteration. Decoding algorithms in the log-likelihood domain are given not only for convolutional codes hut also for any linear binary systematic block code. The iteration is controlled by a stop criterion derived from cross entropy, which results in a minimal number of iterations. Optimal and suboptimal decoders with reduced complexity are presented. Simulation results show that very simple component codes are sufficient, block codes are appropriate for high rates and convolutional codes for lower rates less than 213. Any combination of block and convolutional component codes is possible. Several interleaving techniques are described. At a bit error rate (BER) of lo- * the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block/convolutional component codes, interleaver sizes less than 1000 and for three to six iterations. Index Terms- Concatenated codes, product codes, iterative decoding, “soft-inlsoft-out ” decoder, “turbo ” (de)coding.

In this paper, we present a general method for determining the capacity of low-density parity-check (LDPC) codes under message-passing decoding when used over any binary-input memoryless channel with discrete or continuous output alphabets. Transmitting at rates below this capacity, a randomly chosen element of the given ensemble will achieve an arbitrarily small target probability of error with a probability that approaches one exponentially fast in the length of the code. (By concatenating with an appropriate outer code one can achieve a probability of error that approaches zero exponentially fast in the length of the code with arbitrarily small loss in rate.) Conversely, transmitting at rates above this capacity the probability of error is bounded away from zero by a strictly positive constant which is independent of the length of the code and of the number of iterations performed. Our results are based on the observation that the concentration of the performance of the decoder around its average performance, as observed by Luby et al. [1] in the case of a binary-symmetric channel and a binary message-passing algorithm, is a general phenomenon. For the particularly important case of belief-propagation decoders, we provide an effective algorithm to determine the corresponding capacity to any desired degree of accuracy. The ideas presented in this paper are broadly applicable and extensions of the general method to low-density parity-check codes over larger alphabets, turbo codes, and other concatenated coding schemes are outlined.

A parallel concatenated coding scheme consists of two simple constituent systematic encoders linked by an interleaver. The input bits to the first encoder are scrambled by the interleaver before entering the second encoder. The codeword of the parallel concatenated code consists of the input bits to the first encoder followed by the parity check bits of both encoders. This construction can be generalized to any number of constituent codes. Parallel concatenated schemes employing two convolutional codes as constituent codes, in connection with an iterative decoding algorithm of complexity comparable to that of the constituent codes, have been recently shown to yield remarkable coding gains close to theoretical limits. They have been named, and are known as, "turbo codes". We propose a method to evaluate an upper bound to the bit error probability of a parallel concatenated coding scheme averaged over all interleavers of a given length. The analytical bounding technique is then used to s...

In this paper we present a general method for determining the capacity of message-passing decoders applied to low density parity check codes used over any binary-input memoryless channel with discrete or continuous output alphabets. We show that for almost all codes in a suitably defined ensemble, transmission at rates below this capacity results in error probabilities that approach zero exponentially fast in the length of the code, whereas for transmission at rates above the capacity the error probability stays bounded away from zero. Our results are based on the observation that the concentration of the performance of the decoder around its average performance, as observed by Luby et. al. [1] in the case of a binary symmetric channel and a binary message passing algorithm, is a general phenomenon. For the particularly important case of belief propagation decoders we provide an effective algorithm to determine the corresponding capacity to any desired degree of accuracy. The ideas pre...

Since the invention of \turbo codes" by Berrou et al. in 1993, the \turbo principle" has been adapted to several communication problems such as \turbo equalization", \turbo trellis coded modulation", and iterative multi user detection. In this paper we study the \turbo equalization" approach, which can be applied to coded data transmission over channels with intersymbol interference (ISI). In the original system invented by Douillard et al., the data is protected by a convolutional code and a receiver consisting of two trellis-based detectors are used, 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 (BER). We introduce new approaches to combining equalization based on linear ltering with the decoding. The result is a receiver that is capable of improving BER performance through iterations of equalization and decoding in a manner similar to turbo ...

Per-Survivor Processing (PSP) provides a general framework for the approximation of Maximum Likelihood Sequence Estimation (MLSE) algorithms whenever the presence of unknown quantities prevents the precise use of the classical Viterbi algorithm. This principle stems from the idea that data-aided estimation of unknown parameters may be embedded into the structure of the Viterbi algorithm itself. Among the numerous possible applications, we concentrate here on (a) adaptive MLSE, (b) simultaneous Trellis Coded Modulation (TCM) decoding and phase synchronization, (c) adaptive Reduced State Sequence Estimation (RSSE). As a matter of fact, PSP is interpretable as a generalization of decision feedback techniques of RSSE to decoding in the presence of unknown parameters. A number of algorithms for the simultaneous estimation of data sequence andunknown channel parameters are presented and compared with "conventional" techniques based on the use of tentative decisions. Results for uncoded modu...

For estimating the states or outputs of a Markov process, the symbol-by-symbol maximum a posteriori (MAP) algorithm is optimal. However, this algorithm, even in its recursive form, poses technical difficulties because of numerical representation problems, the necessity of non-linear functions and a high number of additions and multiplications. MAP like algorithms operating in the logarithmic domain presented in the past solve the numerical problem and reduce the computational complexity, but are suboptimal especially at low SNR (a common example is the Max-Log-MAP because of its use of the max function). A further simplification yields the soft-output Viterbi algorithm (SOVA). In this paper, we present a Log-MAP algorithm that avoids the approximations in the Max-Log-MAP algorithm and hence is equivalent to the true MAP, but without its major disadvantages. We compare the (Log-)MAP, Max-Log-MAP and SOVA from a theoretical point of view to illuminate their commonalities and differences. As a practical example forming the basis for simulations, we consider Turbo decoding, where recursive systematic convolutional component codes are decoded with the three algorithms, and we also demonstrate the practical suitability of the Log-MAP by including quantization effects. The SOVA is, at 10

this paper, we discuss such last-line-of-defense 0018--9219/99$10.00 1999 IEEE PROCEEDINGS OF THE IEEE, VOL. 87, NO. 10, OCTOBER 1999 1707 techniques that can be used to make low bit-rate video coders error resilient. We concentrate on techniques that use acknowledgment information provided by a feedback channel

Abstract—A number of important advances have been made in the area of joint equalization and decoding of data transmitted over intersymbol interference (ISI) channels. Turbo equalization is an iterative approach to this problem, in which a maximum a posteriori probability (MAP) equalizer and a MAP decoder exchange soft information in the form of prior probabilities over the transmitted symbols. A number of reduced-complexity methods for turbo equalization have recently been introduced in which MAP equalization is replaced with suboptimal, low-complexity approaches. In this paper, we explore a number of low-complexity soft-input/soft-output (SISO) equalization algorithms based on the minimum mean square error (MMSE) criterion. This includes the extension of existing approaches to general signal constellations and the derivation of a novel approach requiring less complexity than the MMSE-optimal solution. All approaches are qualitatively analyzed by observing the mean-square error averaged over a sequence of equalized data. We show that for the turbo equalization application, the MMSE-based SISO equalizers perform well compared with a MAP equalizer while providing a tremendous complexity reduction. Index Terms—Equalization, iterative decoding, low complexity, minimum mean square error. I.

Bit errors occur in wireless communication when interference or noise overcomes the coded and modulated transmission. Current wireless protocols may use forward error correction (FEC) to correct some small number of bit errors, but generally retransmit the whole packet if the FEC is insufficient. We observe that current wireless mesh network protocols retransmit a number of packets and that most of these retransmissions end up sending bits that have already been received multiple times, wasting network capacity. To overcome this inefficiency, we develop, implement, and evaluate a partial packet recovery (PPR) system. PPR incorporates two new ideas: (1) SoftPHY, an expanded physical layer (PHY) interface that provides PHY-independent hints to higher layers about the PHY’s confidence in each bit it decodes, and (2) a postamble scheme to recover data even when a packet preamble is corrupted and not decodable at the receiver. Finally, we present PP-ARQ, an asynchronous link-layer ARQ protocol built on PPR that allows a receiver to compactly encode a request for retransmission of only those bits in a packet that are likely in error. Our experimental results from a 31-node Zigbee (802.15.4) testbed that includes Telos motes with 2.4 GHz Chipcon radios and GNU Radio nodes implementing the 802.15.4 standard show that PP-ARQ increases end-to-end capacity by a factor of 2× under moderate load.