We propose a trainable version of the Additive Bahl-Cocke-Jelinek-Raviv (A-BCJR) algorithm by interpreting it as a Recurrent Neural Network (RNN). By leveraging the smoothness and differentiability of the max* operator, the core component of the A-BCJR forward and backward recursions, we derive a backpropagation algorithm that enables end-to-end training from the network’s output. The resulting model, referred to as T-BCJR, comprises a linear layer that computes edge metrics from state and input metrics, followed by a nonlinear max* layer that marginalizes these metrics back to the state and output domains. We further derive the corresponding delta backpropagation recursions, which exhibit a structural symmetry with the original forward-backward predictive steps. Unlike prior approaches that only replaced memoryless metric units in BCJR or Viterbi algorithms with Neural Networks (NNs), T-BCJR enables full trainability of the entire recursive structure. We derive a trainable detector, termed T-Detector, by connecting the T-BCJR to the channel output via an additional convolutional linear layer that emulates a channel shortening filter. The T-Detector can be trained to operate as a modulation- and channel-agnostic detector, capable of performing channel shortening, Maximum Likelihood (ML) sequence detection, and the computation of symbol-level or bit-level Log-Likelihoods (LLs) for soft-input decoding. Moreover, it is compatible with iterative receiver schemes involving outer channel decoders. The predictive step of the T-Detector maintains the same Digital Signal Processing (DSP) complexity as a conventional model-based detector, with the added benefit of being trainable from a cost function defined on the generated LLs. Experimental results demonstrate that the T-Detector can be trained to match the performance of the corresponding model-based detector for static and slow-time varying channels with appropriate pilot densities.

A Recurrent Neural Network (RNN) detector is obtained by interfacing the RNN BCJR, introduced by the authors, with the channel output through a convolutional linear layer that mimics the presence of a shortening filter. The RNN detector can be trained to implement a channel and modulation-agnostic detector, including the functions of channel shortening, Maximum Likelihood (ML) sequence detection, and symbol or bit Log-Likelihood (LL) computation for the following soft input channel decoder. The RNN detector has a processing complexity that matches that of the corresponding classical receiver. In this paper we explore the effectiveness of its employment in static and time-selective scenarios.

We explore the concept of a digital phase-locked loop (PLL) of the first type, derived as an alternative solution to Kalman's estimation problem by employing the Tikhonov distribution rather than the traditional Gaussian model. The resulting Tikhonov PLL is a complex-valued nonlinear filter that is simple to implement and demonstrates interesting features in channels affected by strong phase noise. We present a comparative analysis including the classical PLL and Kalman filter to highlight the strengths of the Tikhonov PLL in such contexts.

We present a new version of the additive BCJR algorithm based on a recurrent neural network whose structure reflects an underlying trellis diagram. Starting from a matrix version of the equations of the additive BCJR algorithm, we derive the equivalent trainable recurrent neural network model, named Recurrent Neural Network (RNN) BCJR. The RNN BCJR consists of a linear layer to form the edge metrics from the state and input metrics, followed by a SOFTMAX/max* layer to marginalize the edge metrics back to the state and output spaces. We derive the recursions for delta propagation to train the two-layer mixing matrices from the output cost function. Unlike the previous approaches, the proposed RNN BCJR can completely replace the BCJR and is trainable from the cost functions of the outputs. The trained RNN BCJR achieves the same optimal performance as the BCJR when the model is known but at the same time can adapt itself to model mismatch, thus outperforming BCJR.

In this paper, we consider channels affected by Wiener phase noise and derive a detection algorithm that exploits oversampling at the receiver for tracking better the phase noise variations. The resulting algorithm is a multi-sample discrete-phase BCJR that outperforms receivers using a single sample per symbol and maximizes the achievable information rate over phase noise channels. Finally, we assess the suitability of the multi-sample discrete-phase BCJR for performing iterative detection and decoding with standard codes.

The object of the paper is the design of two-stage co-decoders (codecs) with one hard decoded stage to reduce the decoder complexity. It will be shown that properly designed schemes with that structure can yield performance close to the information-theoretical limits while decreasing the decoder complexity. We focus the attention on the design of schemes for the binary input AWGN channel (BAWGN) in a large SNR range. The structure of the designed codec includes two stage encoders and one inner “channel polarizing code”, whose aim is to turn an AWGN channel into a worse and a better channel. The rationale of this operation is that both “very bad channels” and “very good channels” admit simpler hard-input decoders. After showing that for the positive (high) SNR and negative (low) SNR regime the optimal codec has a different optimal structure, the focus is concentrated on the case of low SNR, where binary modulation is indeed the best one. For low SNR regimes, the best solution consists of a very low rate (possibly zero) encoder at first stage coupled with a hard-input decoder, followed by a capacity-achieving encoder associated to a soft-input iterative decoder. The rate splitting between the two stages allows to construct flexible and low complexity encoding schemes for very large SNR ranges.

Recent investigations show that high-rate telemetry systems, based on the CCSDS 131.2-B-1 standard and employed in Earth observation missions, do not fully exploit the available capacity. Thus, we propose and discuss possible ways to improve the data return of these systems that can be considered for a revision of the current standard. Some of them are represented by a revision and/or an extension of the currently adopted modulation and coding formats. The remaining techniques are related to an enhancement of the transceiver architecture. Different bandwidth allocation strategies are also considered. The benefits of the proposed techniques are assessed by means of computer simulations with reference to a realistic scenario where a real non-linear amplifier, real filters, and real impairments are considered.

We consider the problem of designing constellation sets for multistage systems for high spectral efficiencies. In particular we will focus on two-stage systems where the cascaded decoding stages are associated to three types of receivers with different complexities and performances. The optimal receiver (S) processes the symbol LLR and delivers the maximum possible throughput, the simpler BICM receiver (B) processes in parallel bit LLR derived from the observation, and finally the simplest hard receiver (H) takes an ML hard decision on the observation and processes the sequence of estimated bits with algebraic-like decoders. Optimal constellation sets achieving the maximum possible mutual information depend on the target receiver structure and thus requires separate optimization. Using a variation of the optimization algorithm based on simulated annealing previously introduced we provide optimal constellation sets with 16 to 256 point for four variants of the two-stage receiver offering different trade-offs between complexity and performance. We will show that BH receivers, where the first stage uses a BICM approach and the second stage the even simpler hard decision receiver can provide performances within 0.2 bits from the theoretical Shannon limit in a proper range of Signal to Noise Ratio (SNR) if optimal constellation sets are used.

We present bounds and a closed-form high-SNR expression for the capacity of multiple-antenna systems affected by Wiener phase noise. Our results are developed for the scenario where a single oscillator drives all the radio-frequency circuitries at each transceiver (common oscillator setup), the input signal is subject to a peak-power constraint, and the channel matrix is deterministic. This scenario is relevant for line-of-sight multiple-antenna microwave backhaul links with sufficiently small antenna spacing at the transceivers. For the 2×2 multiple-antenna case, for a Wiener phase-noise process with standard deviation equal to 6°, and at the medium/high SNR values at which microwave backhaul links operate, the upper bound reported in the paper exhibits a 3dB gap from a lower bound obtained using 64-QAM. Furthermore, in this SNR regime the closed-form high-SNR expression is shown to be accurate.

In this paper, we optimize the constellation sets to be used in communication systems affected by phase noise. The main objective is to find the constellation which maximizes the channel mutual information under given power constraints. For any given constellation, the average mutual information (AMI) and the pragmatic average mutual information (PAMI) of the channel are calculated approximately, assuming that both the additive noise and phase noise are memoryless. Then, a simulated annealing algorithm is used to optimize the constellation. When the objective function is the PAMI, the proposed algorithm jointly optimizes the constellation and the binary labeling. We focus on constellations with 8, 16, 64 and 256 signals. The performances of the optimized constellations are compared with conventional constellations showing considerable gains in all system scenarios. In particular, it is shown that the optimized constellations are much more robust with respect to the changes in the phase noise characteristics than the phase shift keying (PSK) modulation and quadrature amplitude modulation (QAM).

In this paper we optimize constellation sets to be used for channels affected by phase noise. The main objective is to maximize the achievable mutual information of the constellation under a given power constraint. The mutual information and pragmatic mutual information of a given constellation is calculated approximately assuming that both the channel and phase noise are white. Then a simulated annealing algorithm is used to jointly optimize the constellation and the binary labeling. The performance of optimized constellations is compared with conventional constellations showing considerable gains in all system scenarios.

In this paper, we derive a phase-noise estimator for MIMO systems affected by independent phase noise at each antenna. The proposed phase-noise estimator employs the expectation-maximization algorithm and can incorporate channel decoding. The main feature of the proposed estimator lies in the fact that it estimates separately phase-noise samples at the transmitter and at the receiver, instead of estimating, as it is typical in single-antenna case, the sum of the samples at the two sides. Simulation results show that the proposed estimator allows obtaining a very good performance at an affordable complexity.

In this paper we use a variation of simulated annealing algorithm for optimizing two-dimensional constellations with 32 signals. The main objective is to maximize the symmetric pragmatic capacity under the peak-power constraint. The method allows the joint optimization of constellation and binary labeling. We also investigate the performance of the optimized constellation over nonlinear satellite channel under additive white Gaussian noise. We consider the performance over systems with and without pre-distorters. In both cases the optimized constellations perform considerably better than the conventional Amplitude Phase Shift Keying (APSK) modulations, used in the current digital video broadcasting standard (DVB-S2) on satellite channels. Based on our optimized constellations, we also propose a new labeling for the 4+12+16-APSK constellation of the DVB-S2 standard which is Gray over all rings.

We consider a frequency-division-multiplexed satellite system where nonlinear distortions may originate from the presence of high power nonlinear devices and can cause significant performance degradations. The spectral efficiency is used as a performance measure to compare, from an information-theoretic point of view, different transmission strategies and modulation formats. More precisely, we will consider transmission schemes employing continuous phase modulations, which are robust to nonlinearities, and schemes based on linear modulations and employing a detector taking into account the nonlinear effects or more traditional techniques, such those based on predistortion of the nonlinear device.

We introduce a novel message passing (BP) algorithm, named Analog-Digital Belief Propagation (ADBP). The algorithm works with factor graph over linear models and uses messages in a particular class of parameterized Gaussian-like distributions by tracking their parameters. With respect to the regular Gaussian BP, this algorithm adds two operations to the factor graph, namely the wrapping and the discretization of variables. This addition allows to use ADBP to construct iterative decoders for mod-M ring encoders that have a complexity independent from the size of the alphabets, thus opening the possibility to construct efficient decoders for systems with unbounded spectral efficiencies. In this paper we derive the updating rules of ADBP and show some possible simplifications of them that make ADBP suitable for implementation in practical systems.

In this paper we introduce a variation of simulated annealing algorithm for optimizing two-dimensional constellations with finite number of signals when the objective function is the symmetric capacity. Our method also allows the joint optimization of constellation and binary labeling when the objective function is the pragmatic capacity. The algorithm can be applied with constraints on both the peak-power and the average and peak-power of the constellation. The proposed algorithm does not impose any a-priori geometrical structure on the constellation and labeling. Here we provide results only for the peak-power constraints. For the usual symmetric capacity, optimal constellations depend on the signal to noise ratio and match in several cases the constellations already found in literature (APSK). On the other hand, for the pragmatic capacity several novel labeling-constellation schemes are introduced, showing interesting differences between the two approaches.

The pragmatic approach to coded continuous-phase modulation (CPM) is proposed as a capacity-achieving low-complexity alternative to the serially concatenated CPM (SC-CPM) coding scheme. In this paper, we first perform a selection of the best spectrally efficient CPM modulations to be embedded into SC-CPM schemes. Then, we consider the pragmatic capacity (a.k.a. BICM capacity) of CPM modulations and optimize it through a careful design of the mapping between input bits and CPM waveforms. The so obtained schemes are cascaded with an outer serially concatenated convolutional code to form a pragmatic coded-modulation system. The resulting schemes exhibit performance very close to the CPM capacity without requiring iterations between the outer decoder and the CPM demodulator. As a result, the receiver exhibits reduced complexity and increased flexibility due to the separation of the demodulation and decoding functions.

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.

The paper extends the pragmatic approach to coded continuous-phase modulation (CPM). It first proposes an optimization of the CPM capacity for given complexity and bandwidth efficiency. Then, the best CPM schemes are embedded into a pragmatic coded-modulation scheme, whose pragmatic capacity (a.k.a. BICM capacity) is maximized through a careful design of the mapping between input bits and CPM waveforms. The so obtained CPM schemes show a pragmatic capacity very close to the CPM capacity. An outer binary serially-concatenated convolutional code, cascaded with the CPM modulator through an interleaver, yields performance close to the pragmatic (and so to the CPM) capacity without requiring iterations between outer code and CPM modulator.

A key idea in coding for the broadcast channel (BC) is binning, in which the transmitter encode information by selecting a codeword from an appropriate bin (the messages are thus the bin indexes). This selection is normally done by solving an appropriate (possibly difficult) combinatorial problem. Recently it has been shown that binning for the Blackwell channel -a particular BC- can be done by iterative schemes based on Survey Propagation (SP). This method uses decimation for SP and suffers a complexity of O(n2). In this paper we propose a new variation of the Belief Propagation (BP) algorithm, named Reinforced BP algorithm, that turns BP into a solver. Our simulations show that this new algorithm has complexity O(n log n). Using this new algorithm together with a non-linear coding scheme, we can efficiently achieve rates close to the border of the capacity region of the Blackwell channel.

The paper describes a few simple solutions for a two-antenna receiver diversity combining scheme applied to the mobile terminal of a WCDMA cellular system. All schemes assume that combination takes place right after the low noise amplifiers following the receiving antennas, thus avoiding the need for duplication of the receiver chain. Combining strategies include optimal combining, suboptimal combining and selection combining, in addition to a new, very simple scheme exploiting the turbo code diversity. Different schemes are compared by simulating them in a very realistic 3GPP scenario, with a receiver including Rake, turbo decoder and channel estimation. To check the obtained performance with information-theoretical limits, a mixed analytical-simulation technique is used to evaluate bounds to the outage probability. The paper also addresses the problem of the feedback control algorithms that govern the RF combination through some suitable baseband signal processing.

This tutorial paper gives an overview of the implementation aspects related to turbo decoders, where the term turbo generally refers to iterative decoders intended for parallel concatenated convolutional codes as well as for serial concatenated convolutional codes. We start by considering the general structure of iterative decoders and the main features of the soft-input soft-output algorithm that forms the heart of iterative decoders. Then, we show that very efficient parallel architectures are available for all types of turbo decoders allowing high-speed implementations. Other implementation aspects like quantization issues and stopping rules used in conjunction with buffering for increasing throughput are considered. Finally, we perform an evaluation of the complexities of the turbo decoders as a function of the main parameters of the code.

In this paper, the design of a fully analog iterative decoder for a serially concatenated convolutional code is presented. The decoder is reconfigurable in both block length and code rate. An interleaver size up to 2400 bit is considered. The decoder core implements a single SISO working on a window of the whole code trellis. It is then reused several times to decode the two constituent codes. The resulting decoder performs iterations, but it is fully analog. The extrinsic information exchanged in the decoding process is stored in an analog memory and permuted through a reconfigurable interleaver. Behavioral analysis of the decoder as well as precision and mismatch impact on performance are reported in the paper.

In this paper, we present an all-analog implementation of the rate-1/3, block length 40, UMTS turbo decoder. The prototype was designed and fabricated in a 0.35$mu$m CMOS technology and operates at 3.3 V. We also introduce a discrete-time first-order model for analog decoders which allows fast BER simulations, while taking into account circuit transient behavior and component mismatch. The model is applied to the rate-1/3 analog turbo decoder for UMTS defined in the 3GPP standard, and the discrete-time model predictions are compared with the decoder experimental performance and the transistor-level simulations. These results demonstrated that this model can be successfully used as a tool to both predict analog decoder performance and give design guidelines for complex decoders, for which circuit-level simulations are impractical.

In this paper, we present an all-analog implementation of the rate-1/3, block length 40, universal mobile telecommunications system (UMTS) turbo decoder. The prototype was designed and fabricated in 0.35$\mu$m complementary metal-oxide-semiconductor technology and operates at 3.3 V. We also introduce a discrete-time first-order model for analog decoders which allows fast bit-error rate simulations, while taking into account circuit transient behavior and component mismatch. The model is applied to the rate-1/3 analog turbo decoder for UMTS defined in the Third Generation Partnership Project standard, and the discrete-time model predictions are compared with the decoder experimental performance and the transistor-level simulations. These results demonstrated that this model can be successfully used as a tool to both predict analog decoder performance and give design guidelines for complex decoders, for which circuit-level simulations are impractical.

In this paper, we discuss the design and testing results of an analog 0.35 /spl mu/m CMOS turbo decoder for the rate-1/3, 40 bit UMTS turbo code. The prototype was successfully tested at nominal conditions (2 Mbit/s), with an overall power consumption of 10.3 mW at 3.3 V. The tested BER curve shows a limited performance loss (about 0.5 dB) with respect to that of the digital implementation. We also discuss a discrete-time model of the analog decoder which allows us to run BER simulations including circuit transient behavior and device mismatch in a very short time. Circuit-level simulations demonstrate the validity of our model. According to the discrete-time simulation, a significant contribution to the performance loss is due to device mismatch.

We provide a performance analysis of a new class of serial concatenated convolutional codes (SCCC) where the inner encoder can be punctured beyond the unitary rate. The puncturing of the inner encoder is not limited to inner coded bits, but extended to systematic bits. We derive analytical upper bounds to the error probability of this particular code structure and address suitable design guidelines for the inner code puncturing patterns. We show that the proportion of systematic and parity bits to be deleted strongly depends on the SNR region of interest. Furthermore, we show that puncturing of the inner code systematic bits should be interleaver dependent. Based on these considerations, we derive design guidelines to obtain well-performing rate-compatible SCCCs families. Throughout the paper, the performance of the proposed codes are compared with analytical bounds, and with the performance of PCCC and SCCC proposed in literature

We deal with the design of interleavers in a coded code-division multiple-access (CDMA) scenario, where at the receiver an iterative turbo-like structure to perform multiuser detection is employed. The choice of the interleavers affects both the maximum-likelihood (ML) performance and the impact of the suboptimality of the iterative receiver. First, heuristic criteria of goodness for a set of interleavers, each assigned to a given active user, are introduced and motivated. One of these criteria is based on the intersection between the equivalent codes seen after the interleavers for each user pair. The design rules are valid for any kind of channel code. In particular, when the channel code used by every user is a terminated convolutional code, a very simple design rule, in the subset of congruential interleavers, is specified. The suitability of an interleaver set to iterative decoding is also treated. The analysis leads to a design rule which is shown to have great importance on the performance of a turbo-like receiver. Numerical results assess the validity of the derived design rules by showing that, for iterative multiuser receivers and reasonable block lengths, the suitability to iterative decoding is more important than the performance optimization.

In this paper, after a description of the principles of iterative (turbo) multiuser detection for CDMA systems, a new structure, based on a linear user separator, is introduced and analyzed. The new user separator processes the matched filter outputs according to a minimum mean-square-error (MMSE) criterion for the first few iterations, while it bypasses the MMSE filter when the interferers' bits are known with some reliability. This new receiver is tested by simulation and compared with other receivers previously introduced in literature. Also, a way of studying and designing iterative multiuser receivers is given by adapting the method of EXtrinsic Information Transfer (EXIT) charts. With the EXIT charts, the receiver performance (in an information-theoretic sense) can be evaluated as it evolves along the iterations. Thus, the receiver threshold can be reliably estimated.

Partially systematic rate-compatible serial concatenated convolutional codes (SCCCs) are considered. These are generated from a classical rate 1/3 serial concatenated mother code. To obtain rate-compatible SCCCs, the puncturing is limited to inner coded bits, by puncturing both the inner code's parity and systematic bits. It is shown that the performance can be enhanced in the "waterfall" or in the "error-floor" region, by simply spreading the puncturing over the parity and the systematic bits without increasing encoding and decoding complexity.

A new design criterion to obtain well performing rate-compatible serial concatenated convolutional codes (SCCC) families is proposed. To obtain rate-compatible SCCCs, the puncturing is limited to inner coded bits. However, the puncturing is not restricted to inner parity bits, but extended also to inner systematic bits, thus obtaining high rate SCCCs (i.e., beyond the outer code rate). The considerations presented in S. Benedetto, et al., (1998) to find "best" component encoders for SCCCs construction are extended to find the "best" rate-compatible puncturing patterns for a given input decoding delay I. A rate-compatibility restriction to the puncturing rule is used, implying that all the code bits of a high-rate punctured code are used by the lower rate codes. The two main applications of this technique are its use in hybrid ARQ/FEC schemes and to achieve unequal error protection (UEP) of an information sequence.

This paper deals with the design of an optimum soft-input soft-output (SISO) a-posteriori receiver front-end for the downlink of direct-sequence W-CDMA cellular systems. The proposed structure yields the minimum bit error probability when used alone with hard decision at its output, and it evaluates the optimal metric (in the presence of the intersymbol interference (ISI) due to multipath propagation) to be provided to the turbo decoder. Performance of the new SISO equalizer are shown for both uncoded and coded systems in the standard framework of UMTS system.

In this paper, a concatenated coding scheme consisting of an outer algebraic block code performing hard decoding connected to an inner turbo code through a rectangular interleaver is analyzed. The performance of the concatenated code is evaluated through a semi-analytical approach, based on the expression of the word error probability at the output of the outer decoder as a function of the error statistics at the output of the turbo decoder estimated through simulation. The proposed technique yields results that depend on the size of the interleaver, and permits to obtain trade-offs between that size and the outer code characteristics.

The main problem of space-time trellis codes is constituted by their complexity, which grows exponentially with the number of transmit antennas. To avoid this shortcoming, one can think to a suboptimum decoder in which a simplified metric, together with a preliminary filtering step, is used. In this paper, we develop this idea and give some possible choices for the filter design. These different choices are compared with one another by means of analytical tools and simulations.

The design and test results of a three-metal, double-poly, 0.35 μm; CMOS analog turbo decoder for the rate-1/3, block length 40, UMTS turbo code, are presented. A discrete-time model of analog decoding networks is also presented. This model can be used as a tool to both predict chip performance in a short time and give design guidelines for complex decoders, for which circuit-level simulations are impractical.

In this letter, we propose and compare some design criteria for the search of good rate-compatible systematic turbo codes (RCPTC) families. The considerations presented by Benedetto et al. to find "best" component encoders for turbo-code construction are extended to find good rate-compatible puncturing patterns leading to codes with promising performances.

This correspondence deals with the design and decoding of high-rate convolutional codes. After proving that every (n,n-1) convolutional code can be reduced to a structure that concatenates a block encoder associated to the parallel edges with a convolutional encoder defining the trellis section, the results of an exhaustive search for the optimal (n,n-1) convolutional codes is presented through various tables of best high-rate codes. The search is also extended to find the "best" recursive systematic convolutional encoders to be used as component encoders of parallel concatenated "turbo" codes. A decoding algorithm working on the dual code is introduced (in both multiplicative and additive form), by showing that changing in a proper way the representation of the soft information passed between constituent decoders in the iterative decoding process, the soft-input soft-output (SISO) modules of the decoder based on the dual code become equal to those used for the original code. A new technique to terminate the code trellis that significantly reduces the rate loss induced by the addition of terminating bits is described. Finally, an inverse puncturing technique applied to the highest rate "mother" code to yield a sequence of almost optimal codes with decreasing rates is proposed. Simulation results applied to the case of parallel concatenated codes show the significant advantages of the newly found codes in terms of performance and decoding complexity.

For high-data-rate applications, the implementation of iterative turbo-like decoders requires the use of parallel architectures posing some collision-free constraints to the reading/writing process from/into the memory. This consideration applies to the two main classes of turbo-like codes, i.e., turbo codes and low-density parity-check (LDPC) codes. Contrary to the literature belief, we prove in this paper that there is no need for an ad hoc code design to meet the parallelism requirement, because, for any code and any choice of the scheduling of the reading/writing operations, there is a suitable mapping of the variables in the memory that grants a collision-free access. The proof is constructive, i.e., it gives an algorithm that obtains the desired collision-free mapping. The algorithm is applied to two simple examples, one for turbo codes and one for LDPC codes, to illustrate how the algorithm works.

Recently, we proposed a new design technique to construct high-rate convolutional codes based on a structure formed by a block encoder and a simpler convolutional encoder (Graell i Amat, A. et al., IEEE Commun.. Lett., vol.5, no.11, p.453-5, 2001). The search technique was based on the optimization of the output weight enumerating function of the code. We now prove that every (n,n-1) convolutional code can be reduced to this structure. Following this result and suitably modifying our earlier search algorithm, we have been able to obtain the best (n, n-1) convolutional encoders to be used in the design of turbo codes. In this case, the search is aimed at the optimization of the input-output weight enumerating function of the encoders. We also derive an inverse puncturing method that can be applied to these high-rate convolutional codes to obtain a sequence of the (almost) best convolutional encoders. With such a method, a whole family of good encoders with different rates is obtained using the same encoder-decoder, thus permitting a great versatility that can be exploited in practical implementations.

We propose and compare some good rate-compatible serial concatenated convolutional code (SCCC) families. To obtain rate-compatible SCCCs, the puncturing is limited to inner coded bits. However, and this is the novelty proposed, we do not limit the puncturing to inner parity bits only, but we extend it also to inner systematic bits, thus obtaining higher rate SCCCs (i.e., beyond the outer code rate). The two main applications of this technique are its use in hybrid ARQ/FEC schemes and to achieve unequal error protection (UEP) of an information sequence.

A new hard iterative decoding algorithm to decode the serial concatenation of two block codes separated by an interleaver is presented. The new scheme is based on the use of erasures as a sort of extrinsic information to be exchanged between the two decoders. Extensive simulations highlight the coding gain of the new algorithm with respect to standard hard iterative decoding.

We consider the application of concatenated codes with interleaver and iterative decoding to optical communication systems. We show how to obtain the optimum log-likelihood ratio to be provided to the soft decoder in the optical channel environment, and compare the performance of a decoder using it with the one employing a Gaussian approximation of the optical channel. Simulation results refer to practical turbo-product codes, and encompass the effect of quantization on the log-likelihood ratio. The results show that the Gaussian assumption in computing the log-likelihood ratio for the optical channel leads to significant losses.

The performance of turbo codes is evaluated through analysis and simulation over the Rice multiplicative fading channel. The analysis is conducted extending the results presented by A.M. Viterbi and A.J. Viterbi (see Proc. ISIT'98, p.29, 1998) to the Rice multiplicative fading channel case. Simulation results are obtained using an extended version of the improved Jakes' channel simulator, first presented by M.F. Pop and N.C. Beaulieu (see IEEE Trans. Commun., vol.49, p.699-708, 2001).

Optimized high-rate convolutional codes are considered as the outer encoder of a serially concatenated structure where the inner encoder is replaced by the magnetic recording channel. Simulation results of the iterative decoding algorithm for an equalized Lorentzian channel model and a more realistic model that includes data-dependent transition noise are presented. The effect of precoder on performance is also studied, and simulation results are supported by EXIT chart analysis. All results refer to a comparison of the optimized codes with previously proposed schemes employing punctured codes or non optimized unpunctured codes with tail-biting decoding. Both trellis termination and tail-biting termination of the high-rate codes are studied. To terminate the code trellis we use the method derived by Amat, Montorsi and Benedetto, which only requires /spl nu/ (the code memory) tail-biting bits. Simulation results confirm the ML analysis: owing to their better distance properties, the scheme based on the new codes outperform state-of-the-art magnetic recording schemes based on both punctured and non optimized high-rate codes. The cost of using an unpunctured code versus the punctured one in terms of increased decoding complexity is turned into an advantage by applying to the high-rate code the soft-input soft-output (SISO) algorithm working on its dual trellis.

In this paper, the achievable performance of a turbo coded system adapted to a block fading channel model is evaluated. This model is suitable for analyzing, for instance, frequency-hopped multiple access (FHMA) systems that operate in a mobile satellite environment characterized by frequency-nonselective slow Rician fading, provided that the spacing between carriers is larger than the coherence bandwidth, resulting in basically uncorrelated blocks. In such systems, coded information is transmitted over a small number of fading channels in order to achieve diversity. The best coded information allocations over a certain number of fading channels are evaluated. Bounds on the achievable performance due to coding are derived using information-theoretic techniques. Moreover, simulation results are obtained and compared with the theoretical ones.

This paper considers the use of the best high-rate k/(k+1) convolutional codes obtained using the new construction technique described by Graell i Amat, Montorsi and Benedetto (see IEEE Communications Letters, vol.5, no.11, p.453-55, 2001) in a concatenated scheme. Simulation results for an AWGN channel and for a realistic magnetic recording channel are reported. It is shown that these codes, endowed with a decoding algorithm working on the dual code, yield performance improvements over the best known high-rate punctured codes with the same rate and memory in terms of both bit error probability and computational decoding complexity. For both the AWGN channel and the magnetic recording channel the new codes significantly lower the error floor with respect to known turbo-like code structures.

This paper presents an all-analog iterative decoding network for an EPR4 magnetic recording system. A powerful serially concatenated architecture is considered, consisting of a simple outer code, an interleaver with reasonable size and a rate 1 EPR4 channel as inner code. The analog chip design is based on analog 0.18 /spl mu/m CMOS technology. Simulation results for both digital and analog implementations are shown. Practical implementation issues such as considerations of mismatch effects over performance are also discussed.

We propose and compare some design criteria for the search of good rate-compatible punctured turbo (RCPT) code families. The considerations presented by S. Benedetto et al. (see IEEE Trans. on Commun., vol.46, no.9, p.1101-5, 1998) to find "best" component encoders for turbo code construction are extended to find the "best" rate-compatible puncturing patterns for a given interleaver length k. This approach is shown to be the optimal one, both in the maximum-likelihood sense (using transfer function bounds) and in the iterative decoding sense (through simulation results). A rate-compatibility restriction to the puncturing rule is used, implying that all the code bits of a high-rate punctured code are used by the lower rate codes. The two main applications of this technique are its use in modified type II ARQ/FEC schemes and to achieve unequal error protection of an information sequence.

In this work, the performance of turbo codes over the Rice multiplicative fading channel is evaluated through analysis and simulation. The analysis is conducted through improved upper bounds on bit error probabilities in conjunction with both the fully interleaved channel and the correlated Rice slow fading channel. The analytical expression of the improved bound is obtained by extending the results of Viterbi and Viterbi (see Proc. IEEE International Symposium on Information Theory, p.29, 1998) to the Rice multiplicative fading channel case. Simulation results are obtained and presented also for intermediate fading rates, ranging between the two extreme multipath fading rates considered in the analysis. Simulations are performed using the improved Jakes' channel simulator to simulate Rayleigh fading channels, and extended to adequately model the multipath Rice fading propagation environment.

We address several aspects of high-rate convolutional codes. Some results from an exhaustive search for codes optimized with respect to their distance spectrum, and encoders optimized with respect to their input-output weight enumerating function are presented. An additive version of the dual-SISO algorithm suitable to decode such codes with limited complexity is described, together with simulation results for both stand-alone and concatenated codes showing the codes performance improvement.

The well-known uniform error property for signal constellations and codes is extended to encompass information bits. We introduce a class of binary labelings for signal constellations, called bit geometrically uniform (BGU) labelings, for which the uniform bit error property holds, i.e., the bit error probability does not depend on the transmitted signal. Strong connections between the symmetries of constellations and binary Hamming spaces are involved. For block-coded modulation (BCM) and trellis-coded modulation (TCM) Euclidean-space codes, BGU encoders are introduced and studied. The properties of BGU encoders prove quite useful for the analysis and design of codes aimed at minimizing the bit, rather than symbol, error probability. Applications to the analysis and the design of serially concatenated trellis codes are presented, together with a case study which realizes a spectral efficiency of 2 b/s/Hz.

We compare the convergence thresholds of parallel and serially concatenated codes with interleaver that employ iterative decoding algorithms working at bit or symbol level. The analysis technique is an extension to that of the symbol-level decoder presented by S. ten Brink (see Electron. Lett., vol.35, no.10, p.806-8, 1999), and is valid for very large interleavers. The comparison shows that working at symbol level always yields a lower convergence threshold. In the serial concatenation, the advantage is quite significant.

We present a thorough comparison among three alternative solutions for the transmission of medium-high speed data over mobile telecommunication systems with high quality of service. They are based on concatenated codes with interleavers, the first two using the parallel concatenation known as turbo code, and the third the serial concatenation recently proposed. The comparison between the two alternatives has been based on the computation of the free distance of the resulting codes, and on extensive simulations over additive Gaussian noise (AWGN) and correlated multipath fading channels.

A new suboptimal turbo receiver for coded CDMA is presented. The user-separating front-end of this receiver has been thought of as the extension of linear multiuser detectors to an iterative structure. The complexity of this receiver is shown to be O(K), where K is the number of users. Simulations show that, for sufficiently high signal-to-noise ratios, the linear-US receiver tends to single-user performance.

We discuss the effects of quantization on the performance of the iterative decoding algorithm of concatenated codes with interleavers. Quantization refers here to the log-likelihood ratios coming from the soft demodulator and to the extrinsic information passed from one stage of the decoder to the next. We discuss the cases of a single soft-input soft-output (SISO) module, in its additive log-likelihood version (L-SISO), performing sequentially all iterations (an implementation solution coping with medium-low data rate as compared with the hardware clock), and that of a pipelined structure in which a dedicated hardware is in charge of each SISO operation (an implementation suitable for high data rates). We give design rules in both cases, and show that a suitable rescaling of the extrinsic information yields almost ideal performance with the same number of bits (five) representing both log-likelihood ratios and extrinsic information at any decoder stage.

In this paper, the basic theory of interleavers is revisited in a semi-tutorial manner, and extended to encompass noncausal interleavers. The parameters that characterize the interleaver behavior (like delay, latency, and period) are clearly defined. The input-output interleaver code is introduced and its complexity studied. Connections among various interleaver parameters are explored. The classes of convolutional and block interleavers are considered, and their practical implementation discussed. The trellis complexity of turbo codes is tied to the complexity of the constituent interleaver. A procedure of complexity reduction by coordinate permutation is also presented, together with some examples of its application.

Implementation rules for iterative decoders of concatenated codes with interleavers are proposed based on a study of the quantization effects on the performance. We consider both cases of a single soft-input soft-output (SISO) module performing sequentially all iterations and of a pipelined structure in which a dedicated hardware is in charge of each SISO operation. In the last case, we show that a suitable rescaling of the extrinsic informations yields almost ideal performance with the same number of bits (5) representing both LLRs and extrinsic information at any decoder stage.

As a powerful coding technique, turbo codes are a prime candidate for improving the reliability of communication over wireless channels, where fading is the main impairment. However, to date, only limited attention has been given to the performance of turbo codes on fading channels, and the influence of multipath fading has been fairly accurately investigated as far as the Rayleigh fading distribution is concerned only. Therefore, it is the purpose of this work to conduct a study of turbo codes performance considering the more general Rice fading case: the study is performed in conjunction with both the fully-interleaved channel and the correlated Rice slow-fading channel.

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.

Recursive systematic convolutional encoders have been shown to play a crucial role in the design of turbo codes. We recall some properties of binary convolutional encoders and apply them to a search for good constituent convolutional codes of turbo codes. Tables of the "best" recursive systematic convolutional encoders found are presented for various rates, together with the average bit-error probability performances of some turbo codes using them.

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.

A serially concatenated code with interleaver consists of the cascade of an outer encoder, an interleaver permuting the outer codeword bits, and an inner encoder whose input words are the permuted outer codewords. In this paper we derive design guidelines for the outer and inner codes that maximize the interleaver gain and the asymptotic slope of the error probability curves.

A new decoding algorithm for geometrically uniform trellis codes is presented. The group structure of the codes is exploited in order to improve the decoding process. Analytical bounds to the algorithm performance and to its computational complexity are derived. The algorithm complexity does not depend on the number of states of the trellis describing the code. Extensive simulations yield results on the algorithm performance and complexity, and permit a comparison with the Viterbi algorithm and the sequential Fano algorithm.

A parallel concatenated convolutional coding scheme consists of two constituent systematic: convolutional encoders linked by an interleaver. The information bits at the input of the first encoder are scrambled by the interleaver before entering the second encoder. The codewords of the parallel concatenated code consist of the information bits followed by the parity check bits of both encoders. Parallel concatenated codes (turbo codes), decoded through an iterative decoding algorithm of relatively low complexity, have been shown to yield remarkable coding gains close to theoretical limits. We characterize the separate contributions that the interleaver length and constituent codes give to the overall performance of the parallel concatenated code, and present some guidelines for the optimal design of the constituent convolutional codes.

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 previously 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 shed some light on some crucial questions, which have been floating around in the communications community since the proposal of turbo codes.

A description of the algorithms to evaluate the main parameters determining the performance of a trellis-coded modulation (TCM) scheme is presented. TCM schemes are divided into classes that have an increasing degree of symmetry, so as to properly match the various algorithms to each class. The algorithms are compared in terms of computational complexity and tested on a set of multidimensional PSK codes.>

The theory of geometrically uniform signal sets and codes over groups is applied to the case of L/spl times/MPSK constellations. Conditions for rotational invariance of group codes are discussed. The tables of geometrically uniform partitions found in Benedetto et al. (1993) are used to construct good geometrically uniform trellis codes over nonbinary Abelian groups. The present authors consider L/spl times/4PSK and L/spl times/8PSK constellations used to transmit information rates of 1 and 2 bit/two dimensions, respectively; and present tables of good codes over generating groups (Z4)/sup L/ and (Z8)/sup L/,for L ranging from 1 to 4. In most cases, they improve the tables of codes known so far. Moreover, the geometrical uniformity of codes allows a very easy performance evaluation, so that the authors also present a complete set of curves of error event probability for the obtained codes.>

The theory of geometrically uniform trellis codes is applied to the case of multidimensional PSK (phase shift keying) constellations. The symmetry group of an L*MPSK (M-ary PSK) constellation is completely characterized. Conditions for rotational invariance of geometrically uniform partitions of a signal constellation are given. Through suitable algorithms, geometrically uniform partitions of L*MPSK (M=4,8,16 and L=1,2,3,4) constellations are found, which present good characteristics in terms of the set of distances at a given partition level, the maximum obtainable rotational invariance, and the isomorphism of the quotient group associated with the partition. These partitions are used as starting points in a search for good geometrically uniform trellis codes based on binary convolutional codes.>