In this paper, analysis of linear time-variant systems is applied to incremental Delta-Sigma (I-DS) ADCs with periodic architectural reconfiguration in the frequency domain. The analysis will then be applied to the example of linear-exponential I-DS ADCs as a state-of-the-art dynamic reconfiguration technique. It is shown how a matched reconstruction filter of the reconfigured linear-exponential incremental Delta-Sigma modulator (I-DSM) can be mathematically derived. Using the calculated overall transfer functions of the linear-exponential I-DS ADC, accurate performance predictions can be given. The proposed method allows the accurate prediction of performances and gives insight and understanding of the performance improvements and trade-offs achieved by reconfiguration techniques in I-DS ADCs in general and the exponential phase in particular.

In any transmission scheme, security against eavesdroppers is of importance. Contrary to classical encryption, security directly at the physical layer can be achieved by applying suited coding and modulation schemes. In this paper, coded modulation schemes, i.e., the use of higher-order constellations, is studied for generating both security against a wiretapper and reliability for the legitimate user. Specifically, a multilevel coding (MLC) scheme is considered. The influence of the constellation, in particular its cardinality and its dimensionality, on the gained security is assessed. As LSB component code, a practical secure coding scheme based on punctured LDPC codes is used. The security level for the employed LDPC code ensembles within MLC constructions is examined with an adapted density evolution technique. Results from numerical simulations cover the theoretical considerations and match with the derived asymptotic behavior. The results are compared to the situation when straightforwardly applying a binary code according to the BICM approach.

Lattices are a popular field of study in mathematical research, but also in more practical areas like cryptology or multiple-input/multiple-output (MIMO) transmission. In mathematical theory, most often lattices over real numbers are considered. However, in communications, complex-valued processing is usually of interest. Besides, by the use of dual-polarized transmission as well as by the combination of two time slots or frequencies, four-dimensional (quaternion-valued) approaches become more and more important. Hence, to account for this fact, well-known lattice algorithms and related concepts are generalized in this work. To this end, a brief review of complex arithmetic, including the sets of Gaussian and Eisenstein integers, and an introduction to quaternion-valued numbers, including the sets of Lipschitz and Hurwitz integers, are given. On that basis, generalized variants of two important algorithms are derived: first, of the polynomial-time LLL algorithm, resulting in a reduced basis of a lattice by performing a special variant of the Euclidean algorithm defined for matrices, and second, of an algorithm to calculate the successive minima—the norms of the shortest independent vectors of a lattice—and its related lattice points. Generalized bounds for the quality of the particular results are established and the asymptotic complexities of the algorithms are assessed. These findings are extensively compared to conventional real-valued processing. It is shown that the generalized approaches outperform their real-valued counterparts in complexity and/or quality aspects. Moreover, the application of the generalized algorithms to MIMO communications is studied, particularly in the field of lattice-reduction-aided and integer-forcing equalization.

A well-known problem when using Physical Unclonable Functions for secret key generation and storage, is the instability of PUF responses due to environmental conditions like temperature variations. Using Ring Oscillator PUFs (RO-PUFs), a response bit is usually derived based on the comparison of a RO frequency with either a threshold or another RO frequency. Especially for multi-valued PUFs it is of importance to decrease RO frequency errors before digitization. Otherwise, these errors can result in a huge amount of bit-flips. To counteract environmental influence in the reproduction phase, we propose to map a frequency, which might be remeasured under a temperature condition different from initialization, closer to the initial frequency by using a method based on polynomial interpolation. This paper presents how such an approach can decrease errors in multi-valued responses and evaluates the error based on the used polynomial order and thus complexity.

The Hurwitz lattice provides the densest four-dimensional packing. This fact has motivated research on four-dimensional Hurwitz signal constellations for optical and wireless communications. This work presents a new algebraic construction of finite sets of Hurwitz integers that is inherently accompanied by a respective modulo operation. These signal constellations are investigated for transmission over the additive white Gaussian noise (AWGN) channel. It is shown that these signal constellations have a better constellation figure of merit and hence a better asymptotic performance over an AWGN channel when compared with conventional signal constellations with algebraic structure, e.g., two-dimensional Gaussian-integer constellations or four-dimensional Lipschitz-integer constellations. We introduce two concepts for set partitioning of the Hurwitz integers. The first method is useful to reduce the computational complexity of the symbol detection. This suboptimum detection approach achieves near-maximum-likelihood performance. In the second case, the partitioning exploits the algebraic structure of the Hurwitz signal constellations. We partition the Hurwitz integers into additive subgroups in a manner that the minimum Euclidean distance of each subgroup is larger than in the original set. This enables multilevel code constructions for the new signal constellations.

Physical unclonable functions (PUFs) produce responses by exploiting randomness that intrinsically occurs in integrated circuits due to uncontrollable variations in the manufacturing process of physical items. It is common practice that PUFs generate binary responses. Recently, it has been proposed to extract symbols from a higher-order alphabet in order to increase the length of the final response. In this paper, coding for this concept of multi-valued PUFs (MV-PUFs) is derived from the analogy to pulse-amplitude modulation in digital communications. To that end, based on ROPUF measurement data, we replace the classical binary symmetric channel model by a suited additive white Gaussian noise model. Consequently, the hard-input binary channel coding scheme is replaced by methods from coded modulation, utilizing the soft output. In addition, the functionality of helper data, which are required to stabilize noisy PUF responses, is transferred to the multi-valued case. By applying the designed methods to the available measurement data we eventually show that imagining the analog PUF output as$M$-ary amplitude-shift keying symbols observed over an AWGN channel, both the extracted entropy per response symbol and the reliability of the final key can be increased.

Vector approximate message passing is studied where vectorvalued diagonalization instead of a uniform one is employed. Thereby,individualvariancesare tracked within the algorithm instead of an average one. Straightforward application based on the expectation-consistent approximate inference framework does not give satisfactory results. The main reason for this is that the precision parameters may become negativeduring the iterations. In this contribution, improved versions for the update equation ("Onsager correction") are derived from basic estimation principles. Numerical simulations cover the superiority of the new variants.

Physical Unclonable Functions (PUFs) offer the possibility to produce unique fingerprints for integrated circuits. As raw PUF responses are affected by noise, some post-processing steps are necessary. We present a coding chain test framework for PUFs on Field Programmable Gate Arrays. The framework allows easy exchange, evaluation and comparison of different PUF implementations, coding algorithms and other chain modules. For a testing framework, the execution time of the evaluated algorithm is a bottleneck, since a huge amount of runs are supposed to be done. Hence, we additionally present a new type of Reed-Muller decoder hardware architecture using parallel modules to speed up the decoding process. The decoding time could be decreased by 95% in comparison to existing implementations at the cost of 41 times higher slice count.

Lattice-reduction-aided decision-feedback equalization (LRA DFE) and successive integer forcing are MIMO detection schemes which combine the equalization in a suited basis with the principle of successive interference cancellation (SIC). To this end, the reduction algorithm not only has to find a suited basis, but it should also provide an optimized detection order for SIC: the V-BLAST ordering, known to be optimal for conventional DFE. How these two tasks can be solved jointly has so far remained unclear in the literature. In this paper, we describe how the Lenstra-Lenstra-Lovász (LLL) reduction has to be adapted to achieve this aim. Moreover, we propose a weakened variant of the Hermite-Korkine-Zolotareff (HKZ) reduction that optimally solves both tasks jointly. Results obtained from numerical simulations complement the theoretical derivations.

Lattice-reduction-aided preequalization (LRA PE) is a powerful technique for interference handling on the multi-user multiple-input/multiple-output (MIMO) broadcast channel. However, recent advantages in the strongly related field of compute-and-forward and integer-forcing equalization have raised the question, if the factorization task present in LRA PE is really solved in an optimum way. In this paper, advanced factorization strategies are presented, significantly increasing the transmission performance. Specifically, the signal constellation and its related lattice as well as the factorization task/strategy are discussed. The impact of dropping the common unimodularity constraint in LRA PE is studied. Numerical simulations are given to show the effectiveness of all presented strategies.

We address the problem of increasing the sum rate in a multiple-access system from [1] for small number of users. We suggest an improved signal-code construction in which in case of a small number of users we give more resources to them. For the resulting multiple-access system a lower bound on the relative sum rate is derived. It is shown to be very close to the maximal value of relative sum rate in [1] even for small number of users. The bound is obtained for the case of decoding by exhaustive search. We also suggest reduced-complexity decoding and compare the maximal number of users in this case and in case of decoding by exhaustive search.

Information processing characteristics (IPC) provide a general framework for the analysis of a coding scheme. In this paper, we utilize IPCs to obtain performance estimates for the end-to-end coded channel for bit-interleaved coded modulation (BICM) using a given modulation format and coding scheme, i.e., its bit error rate and capacity. The proposed method enables to fully decouple the analysis of the coding scheme from the analysis of the higher-order modulation scheme and delivers very good performance estimates. Exemplarily, this is validated by means of numerical simulations for BICM using convolutional- and LDPC-coded amplitude-shift keying.

Since optimum diversity behavior can be achieved, equalization based on lattice basis reduction is of special interest for transmission over MIMO channels. Although the per-symbol equalization complexity is very low, the initial calculation of the required matrices still imposes an enormous burden in arithmetic complexity. In this paper, we present an alternative strategy to calculate the lattice-reduction-aided MMSE DFE. Thereby, operations on an augmented matrix, usually proposed to obtain the desired results, are avoided. A suited modification of the “fast V-BLAST algorithm” proposed by Benesty et al. is given. By means of numerical simulations it is covered, that performance does not suffer but algorithmic complexity can almost be halved.

In this paper we present decision-feedback differential detection (DF-DD) schemes for autocorrelation-based detection in impulse-radio ultra-wideband (IR-UWB) systems, a signaling scheme regarded as a promising candidate in particular for low-complexity wireless sensor networks. To this end, we first discuss ideal noncoherent sequence estimation and approximations thereof based on block-wise multiple-symbol differential detection (MSDD) and the Viterbi algorithm (VA) from the perspective of tree-search/trellis decoding. Exploiting relations well-known from tree-search decoding, we are able to derive the novel decision-feedback differential detection (DF-DD) schemes. A comprehensive comparison with respect to performance and complexity of the presented schemes in a typical IR-UWB scenario reveals-along with novel insights in techniques for complexity reduction of the sphere decoder applied for MSDD- that sorted DF-DD achieves close-to-optimum performance at very low, and in particular constant receiver complexity.

Power efficiency of noncoherent receivers for impulse-radio ultra-wideband (IR-UWB) transmission systems can significantly be improved, on the one hand, by employing multiple-symbol differential detection (MSDD), and, on the other hand, by providing reliability information to the subsequent channel decoder. In this paper, we combine these two techniques. Incorporating the computation of the soft information into a single-tree-search sphere decoder (SD), the application of this soft-output MSDD in a typical IR-UWB system imposes only a moderate complexity increase at, however, improved performance over hard-output MSDD, and in particular, over conventional symbol-by-symbol noncoherent differential detection.

The high peak-to-average power ratio (PAR) of the transmit signal is one serious issue, which occurs in orthogonal frequency-division multiplexing (OFDM). Hence, the application of a PAR reduction algorithm, which controls the peak-power of the transmit signal, is indispensable. In this paper, we consider selected mapping (SLM), a very popular technique for PAR reduction and its recently proposed extension successive SLM. One drawback of (successive) SLM is that the transmission of side information is necessary, which is extraordinarily prone to transmission errors. In this paper, the explicit transmission of this side information is considered. In order to reduce the increase in bit error rates, due to an erroneous detection of the side information, its estimation at the receiver and its mapping to signals embedded into the OFDM frame is optimized. In this case the application of successive SLM is very advantageous compared to the original approach of SLM as the detection error rate can be reduced significantly. Moreover, with successive SLM almost no differences between explicit transmission and perfect knowledge of the side information in terms of bit error rates occur.

A successive scheme for PAR reduction in (MIMO/ SISO) OFDM is presented, where K (parallel/consecutive) OFDM frames are treated jointly. Employing Reed-Solomon codes further candidate OFDM frames are generated and assessed successively; the currently best K are selected for possible transmission. The procedure stops if (i) all K best frames stay below a given tolerable PAR limit, or (ii) the maximally allowed number of candidates is exhausted. Thereby complexity compared to other PAR reduction schemes can be reduced significantly. Analytical derivations show that for PAR limits in the region of the "critical PAR" value xicrit = log(D), with D being the number of carriers, the average number candidates is close to Euler's number e = 2.71828..., which is particularly low.

Selected mapping (SLM) is a popular scheme for peak power reduction in orthogonal frequency-division multiplexing (OFDM) systems. In this letter, the performance of various versions of SLM, among them ordinary and directed SLM, in single- and multi-antenna point-to-point OFDM systems is assessed. Analytic expressions for the distribution of the PAR are derived. Numerical results cover that significant gains over conventional SLM can be achieved by directed SLM.

New schemes for peak-to-average power ratio reduction in orthogonal frequency-division multiplexing (OFDM) systems are proposed. Reed-Solomon (RS) and simplex codes are employed to create a number of candidates, from which the best are selected. Thereby, in contrast to existing approaches, the codes are arranged over a number of OFDM frames rather than over the carriers, hence a combination of the principles of multiple signal representation with selection (as done in selected mapping) and the use of channel coding is present. In particular, in multiple-antenna transmission, the proposed schemes do not cause any additional delay, but due to the utilization of the dimension space, additional gains can be achieved. Moreover, the schemes are very flexible; due to the selection step, any criterion of optimality can be taken into account. Besides multiple-antenna transmission, packet transmission is briefly considered, which, moreover, covers the appealing similarities with incremental redundancy check schemes in automatic repeat request (ARQ) applications and with decoding of codes transmitted over the erasure channel. The performance of the schemes is (using some approximations) derived analytically and is covered by numerical results that are in very good agreement with the theory. Significant gains can be achieved with these very flexible and versatile methods.

A new scheme for peak-to-average power ratio reduction in OFDM systems is proposed which combines the principles of multiple signal representation with selection (as done in selected mapping) and the use of channel coding. Thereby, in contrast to existing approaches, the code is arranged over a number of OFDM frames rather than over the carriers. Given the initial OFDM frames all linear combinations thereof are generated; from that the best are selected. The corresponding generator matrix is identical to that of the Simplex code. Specifically, we consider multi-antenna transmission, where the proposed scheme does not cause any additional delay, but utilize the dimension space adequately. The scheme is very flexible; due to the selection step any criterion of optimality can be taken into account. Performance is derived analytically and is covered by numerical results which are in very good agreement with theory.

A new bit allocation algorithm for uncoded transmission over parallel, independent channels (e.g., multicarrier transmission) is proposed. In contrast to state-of-the-art loading algorithms it is based on bit-level capacities and aims at the maximization of the so-called parallel decoding capacity. The algorithm can be implemented very efficiently and requires significantly lower complexity than existing approaches. Numerical results show a performance in terms of the bit-error ratio comparable to established bit-loading algorithms.

Peak-to-average power ratio (PAR) reduction in OFDM using antenna arrays (MIMO OFDM) is considered. In particular, generalizations of selected mapping (SLM) recently proposed in literature, are studied, and a new version, we call itdirectedSLM(dSLM), is introduced. It is shown that, in contrast to the other schemes, dSLM utilizes the potential offered by MIMO transmission-the complementary distribution function of the PAR exhibits a steeper (increased by a factor equal to the number of transmit antennas) decay. This effect is similar to the diversity gain in error performance when using MIMO transmission. We show that dSLM is very flexible and not restricted to any modulation format or OFDM frame size. Moreover, a variant of dSLM which does not require any side information to recover data at the receiver is presented.

In this paper variants of selected mapping are investigated for different MIMO OFDM scenarios (point-to- point, multiple-access, and broadcast channel). Particular focus is on the broadcast scenario, where transmitter side precoding is mandatory which usually increases peak-to-average power ratio (PAR). The combination of (non-linear) precoding with a selected mapping technique leads to a very high computational complexity because it operates on all carriers. This paper introduces a new PAR reducing method which affects only a subset of carriers and therefore saves complexity. Because PAR reduction is achieved by optimizing the sorting in the precoding scheme, this technique is called selected sorting.

Mappings for square quadrature amplitude modulation (QAM) are investigated in terms of the achievable capacity in bit-interleaved coded modulation (BICM). In G. Caire et al. (1998) it is conjectured that Gray labelings maximize this capacity. However, Gray mappings are not unique and it can be shown that the conjecture cannot hold in general for QAM constellation sizes larger than 16. Based on the bit level capacities of mappings, structures determining capacity are analyzed. Construction guidelines leading to capacity maximizing Gray mappings are presented.

The concept of "information combing" is extended to multiple access schemes, where the receiver is equipped with multiple receive antennas. In particular, the connection of the respective rate region to that of the scalar MACs, which are present if the receive antennas are treated separately, is derived. This interpretation leads to new, expedient insights and the gains over "scalar combing", i.e., the synergy available in MIMO channels, are quantified from a new point of view. The theoretical results are illustrated by numerical examples

Several approaches have recently been proposed for the efficient optimum or approximate solution of the detection problem in multiple-input multiple-output transmission systems. These are, however, difficult to compare. In the present work we briefly summarize the most popular and promising of these approaches and offer a way to visualize the tradeoff between complexity of the detection and the achievable power efficiency using "complexity-power diagrams". We conclude that the so-called sphere decoder algorithm is very attractive in terms of average complexity, while for low and constant processing delay lattice reduction with subsequent simple linear or nonlinear detection is more favorable

Point-to-point transmission over multiple-input/multiple-output channels with intersymbol interference is considered. Two different equalization strategies - single-carrier transmission with spatial/temporal decision-feedback equalization and multicarrier transmission with V-BLAST in each carrier - and the respective requirements on channel coding are studied and compared. Via capacity arguments, it is shown that bit-inter-leaved coded modulation is not well suited for the situations at hand. This holds in particular if channel coding is used in combination with rate loading, i.e., non-uniform rate distribution over the carriers. Approaches to overcome this problem are discussed

Sorted spectral factorization of matrix polynomials is studied. Such type of factoring Hermitian matrix polynomials is the key step in calculating the optimum receive filter matrices in spatial/temporal decision-feedback equalization as well as the optimum transmit filter matrices in spatial/temporal Tomlinson-Harashima-type precoding schemes. Contrary to other approaches, we inherently consider asymptotic rather than finite-length results for transmission over MIMO channels with intersymbol interference. It is shown how the different types of factorizations can be transformed onto a prototype factorization task, which in turn can be solved by first performing an unsorted factorization and then determining the optimal processing order. An easy-to-use iterative algorithm for unsorted spectral factorization and the adjustment of the optimized order in DFE and precoding are explained. Numerical simulations cover the impact of sorted and unsorted spectral factorization on the performance of DFE and precoding schemes.

Algorithms for the detection problem in multiple-input multiple-output (MIMO) transmission systems are compared, and the tradeoff between detection complexity and the achieved power-efficiency is visualized using "power-complexity" diagrams.

A precoding scheme for multiuser broadcast communications is described, which fills the gap between the low-complexity Tomlinson-Harashima precoding and the sphere decoder-based system of Peel et al. Simulation results show that, replacing the closest-point search with the Babai approximation, the full diversity order supported by the channel is available to each user, as in the system of Peel et al., and unlike Tomlinson-Harashima precoding, which suffers some diversity penalty. The complexity of the scheme is similar to that of Tomlinson-Harashima precoding.

The application of powerful coding for transmission over multiple-input/multiple-output channels is discussed. The authors emphasize that as an immediate consequence of the mutual information chain rule, multilevel coding (MLC) constitutes the optimum coded modulation scheme. On the other hand, simple bit-interleaved coded modulation (BICM) is only a convenient alternative for the case of two transmit and one receive antennas when combined with orthogonal space-time block codes. Starting from MLC, the authors further propose a hybrid coded modulation scheme, which favorably combines the advantages of MLC and BICM.

In this paper, Tomlinson-Harashima Precoding for multiple-input/multiple-output systems including multiple-antenna and multiuser systems is studied. It is shown that nonlinear preequalization offers significant advantages over linear preequalization which increases average transmit power. Moreover, it outperforms decision-feedback equalization at the receiver side which is applicable if joint processing at the receiver side is possible, and which suffers from error propagation. A number of aspects of practical importance are studied. Loading, i.e., the optimum distribution of transmit power and rate is discussed in detail. It is shown that the capacity of the underlying MIMO channel can be utilized asymptotically by means of nonlinear precoding.

The paper presents an improved version of precoding, i.e., nonlinear pre-equalization, for situations with one central transmitter (e.g., base station) and a number of distributed receivers (e.g., mobile terminals). The new method is based on Tomlinson-Harashima precoding, applied to multiple-input-multiple-output (MIMO) channels, and inspired by the concept of lattice-reduction-aided detection in MIMO communication systems. We show that, by using the proposed technique, a considerable degree of diversity can be gained over linear pre-equalization, as well as conventional precoding.

In this paper, powerful coding techniques for differential space-time modulation (DSTM) over Rayleigh flat fading channels and noncoherent demodulation without channel state information at the receiver are investigated. In particular, multilevel coding (MLC), bit-interleaved coded modulation (BICM), and so-called hybrid coded modulation (HCM) are devised and compared using capacity arguments. For improved noncoherent reception multiple-symbol differential detection (MSDD) and a low-complexity version of MSDD are adapted to DSTM.

In this contribution we study transmission over MIMO channels, where channel state information available at the transmitter enables the use of precoding as weak as an optimization of rate and power distribution over the parallel subchannels in the transmission system (loading). It is shown that spatial loading provides substantial gains and a revised version of the V-BLAST algorithm is given to calculate the matrix filters required for Tomlinson-Harashima precoding, or equivalently decision-feedback equalization.

We consider the lattice-reduction-aided detection scheme for 2/spl times/2 channels recently proposed by H. Yao and G.W. Wornell (see Proc. IEEE Globecom, 2002). By using an equivalent real-valued substitute MIMO channel model, their lattice reduction algorithm can be replaced by the well-known LLL (Lenstra-Lenstra-Lovasz) algorithm (Lenstra, A.K. et al., 1982), which enables the application of the algorithm to MIMO systems with arbitrary numbers of dimensions. We show how lattice reduction can also be favourably applied in systems that use precoding and give simulation results that underline the usefulness of this approach.

In this paper, powerful coding techniques for differential space-time modulation (DSTM) over Rayleigh flat fading channels and noncoherent detection without channel state information at the receiver are investigated. In particular, multilevel coding, bit-interleaved coded modulation, and so-called hybrid coded modulation (HCM) are devised and compared. For improved noncoherent reception multiple-symbol differential detection (MSDD) is adapted to DSTM. In order to reduce the computational effort required for MSDD, a low-complexity version of MSDD is applied. Evaluating the ergodic channel capacity for the different schemes as appropriate performance measure, HCM with simplified MSDD is shown to offer a favorable tradeoff between complexity and achievable power efficiency. Simulation results employing turbo codes in properly designed HCM schemes confirm the predictions from information theory.

Precoding, i.e., nonlinear pre-equalization, at the transmitter side has been proved to be a very efficient strategy for channel equalization in single-carrier digital transmission schemes. It enables the application of coded modulation in a seamless fashion. A drawback of precoding is that the signal at the input of the decision device exhibits a huge dynamic range. Based on dynamics shaping, a combined precoding/shaping technique introduced in the paper by Fischer et al. (1995), a new shaping strategy is developed in this paper. This technique enables a flexible tradeoff among: (1) reduction of the average transmit power; (2) avoidance of peaks in the transmit signal in order to facilitate line driver implementation; and (3) restriction of the maximum amplitude at the receiver side to a prescribed value. Over a wide range, all three demands can be met simultaneously. As the scheme is fully compatible with Tomlinson-Harashima (1971, 1972) precoding, it can replace the precoder even in existing and standardized schemes. Simulation results for a typical digital subscriber line scenario show the achievable gains.

Precoding at the transmitter side has been proved to be a very efficient strategy for channel equalization in single-carrier digital transmission schemes. It enables the application of coded modulation in a seamless fashion. A drawback of precoding is that the signal at the input of the decision device exhibits a huge dynamic range. Based on dynamics shaping, a combined precoding/shaping technique introduced previously (see Fischer, R. et al., IEEE JSAC, p.1622-33, 1995), a new shaping strategy is developed. This technique enables a flexible trade-off between (i) reduction of the average transmit power, (ii) avoidance of peaks in the transmit signal in order to facilitate a line driver implementation, and (iii) restriction of the maximum amplitude at the receiver side to a prescribed value. Over a wide range, all three demands can be met simultaneously. As the scheme is fully compatible with Tomlinson-Harashima precoding, it can replace the precoder even in existing and standardized schemes. Simulation results for a typical DSL scenario show the achievable gains.

In order to calculate the transmit power in shell-mapping-based transmission schemes, the frequencies of the shells are required. In this correspondence, a simple but general method for the calculation of these frequencies is derived. The method has approximately the same complexity as the shell-mapping encoder. As an example, the method is shown in detail for the shell-mapping scheme specified for the international telephone-line V.34 modem standard. Moreover, a very simple approximation is given which is tight for large constellations.

This paper deals with 2/sup l/-ary transmission using multilevel coding (MLC) and multistage decoding (MSD). The known result that MLC and MSD suffice to approach capacity if the rates at each level are appropriately chosen is reviewed. Using multiuser information theory, it is shown that there is a large space of rate combinations such that MLC and full maximum-likelihood decoding (MLD) can approach capacity. It is noted that multilevel codes designed according to the traditional balanced distance rule tend to fall in the latter category and, therefore, require the huge complexity of MLD. The capacity rule, the balanced distances rules, and two other rules based on the random coding exponent and cutoff rate are compared and contrasted for practical design. Simulation results using multilevel binary turbo codes show that capacity can in fact be closely approached at high bandwidth efficiencies. Moreover, topics relevant in practical applications such as signal set labeling, dimensionality of the constituent constellation, and hard-decision decoding are emphasized. Bit interleaved coded modulation, proposed by Caire et al. (see ibid., vol.44, p.927-46, 1998), is reviewed in the context of MLC. Finally, the combination of signal shaping and coding is discussed. Significant shaping gains are achievable in practice only if these design rules are taken into account.

Precoding at the transmitter side is a practicable method for transmission over intersymbol interference channels. In contrast to decision-feedback equalization no error propagation occurs and coded modulation can be applied as for the intersymbol interference free channel. Tomlinson-Harashima (1971, 1972) precoding and flexible precoding are analyzed and compared. The dualities and differences are discussed. The focus of interest is the application of precoding to fast digital transmission over twisted pair lines, such as high-rate or asymmetric digital subscriber lines. It turns out that flexibility-which is not necessary in the specific application, digital subscriber lines-of flexible precoding is paid with a performance loss compared to Tomlinson-Harashima precoding.

A new combined precoding/shaping technique for fast digital transmission over twisted pair lines is proposed. Major advantages of this "dynamics shaping" are: Dynamics of the signal at the input of the decision device are reduced by a great amount. Thereby, A/D-conversion, adaptive equalization, and symbol timing are rather facilitated. A trade-off between signal dynamics at the transmitter output, decision device input and SNR-gain by noise whitening is offered. For dynamics limitation relevant in practice, gains up to 6 dB are achieved. Additionally, the transmitter can be fixed to a typical application because, in contrast to Tomlinson-Harashima or other precoding techniques, blind adaptive equalization is practicable to remove residual intersymbol interference in the case of a mismatch of precoding and actual cable characteristics. The residual SNR-loss is negligible in most applications. SNR-gains due to noise prediction, channel coding and signal shaping simply can be combined us...