In this paper, we propose an analysis of the automorphism group of polar codes, with the aim of designing codes tailored forautomorphism ensemble(AE) decoding. Using a novel description of polar codes as monomial codes through negative monomials, we prove the equivalence between the notion ofdecreasing monomial codesand the universal partial order (UPO) framework for polar codes; this property is widely believed to hold true but a formal proof was missing. We further provide a rigorous mathematical connection between code word permutations and affine transformations, an important link to understand the considered automorphisms. Based on this mathematical formalisms, we analyze the algebraic properties of theaffine automorphisms groupof polar codes, providing a novel description of its structure. We classify automorphisms such that all automorphisms in the same class lead to the same result under permutation decoding, which gives rise to the concept ofredundantautomorphisms. Mathematically this is achieved by introducing equivalence classes of affine automorphisms under AE-based decoding. For practical application, we provide an algorithm to compute representatives for the equivalence classes, such that one automorphism from each equivalence class can be selected for use in AE decoding. A numerical analysis of the error correction performance of AE decoding of polar codes, based on equivalence classes, concludes the paper.

This paper proposes new polar code design principles for the low-latency automorphism ensemble (AE) decoding. Our proposal permits to design a polar code with the desired automorphism group (if possible) while assuring the decreasing monomial property. Moreover, we prove that some automorphisms are redundant under AE decoding, and we propose a new automorphisms classification based on equivalence classes. Finally, we propose an automorphism selection heuristic based on drawing only one element of each class; we show that this method enhances the block error rate (BLER) performance of short polar codes even with a limited number of automorphisms.

We propose a novel coupling technique for polar codes via a special kernel that enables efficient sliding window decoding. This feature allows to reduce the memory requirement of the decoder, an important possibility in wireless communication downlink scenarios. Our approach is based on the design of an ad-hoc kernel to be inserted in a multi-kernel polar code framework. Simulation results show that the proposed sliding window polar codes outperform polar codes transmitted in independent blocks at a negligible additional decoding overhead.

In this paper we deal with polar code automorphisms that are beneficial under low-latency automorphism ensemble (AE) decoding, and we propose polar code designs that have such automorphisms. Successive-cancellation (SC) decoding and thus SC-based AE decoding are invariant with respect to the only known polar code automorphisms, namely those of the lower-triangular affine (LTA) group. To overcome this problem, we provide methods to determine whether a given polar code has non-LTA automorphisms and to identify such automorphisms. Building on this, we design specific polar codes that admit automorphisms in the upper-diagonal linear (UTL) group, and thus render SC-based AE decoding effective. Demonstrated by examples, these new polar codes under AE decoding outperform conventional polar codes under SC list decoding in terms of error rate, while keeping the latency comparable to SC decoding. Moreover, state-of-the-art BP-based permutation decoding for polar codes is beaten by BP-based AE thanks to this design.

In this paper, we propose a new polar code construction by employing kernels of different sizes in the Kronecker product of the transformation matrix, thus generalizing the original construction by Arikan. These multi-kernel polar codes allow for more flexibility in terms of the code length and for various new design principles. Next to the common reliability design, we provide a design to maximize the minimal distance and a hybrid design combining reliability and distance properties. Numerical results demonstrate the advantage of multi-kernel polar codes under the new design principles compared to punctured and shortened Arikan polar codes.

In this paper we propose a novel frozen set design for polar codes and multi-kernel polar codes. We improve the existing hybrid distance-reliability design by minimizing the upper bound of the overall system error probability instead of minimizing its lower bound as previously proposed. This allows to better trade reliabilities of the input bits against distance properties of the code. We describe the new design approach, propose a greedy algorithm to limit the complexity of the code construction process, and evaluate its performance through numerical examples. In both MK polar codes and conventional polar codes, a substantial performance improvement is observed, matching the performance of CRC-aided polar codes under SCL without the need for a CRC.

The successive cancellation flip (SC-Flip) decoding algorithm arose as a valid low-complexity decoding algorithm for polar codes, however its decoding capabilities are still far away from list based decoders. In this paper, we propose an improved SC-Flip multiple error decoding framework based on error dependency, and specialize it for the two-error correction case. We propose to generate different lists of second error locations based on the index of the expected first errors. The inherent flexibility of this approach allows it to be modified for a desired trade-off between performance and complexity. Two second errors list construction approaches are presented, and are shown to yield gains over the original SC-Flip decoder at the same decoding complexity.

Product codes are widespread in optical communications, thanks to their high throughput and good error-correction performance. Systematic polar codes have been recently considered as component codes for product codes. In this paper, we present a novel construction for product polar codes based on non-systematic polar codes. We prove that the resulting product code is actually a polar code, having a frozen set that is dependent on the frozen sets of the component polar codes. We propose a low-complexity decoding algorithm exploiting the dual nature of the constructed code. Performance analysis and simulations show high decoding speed, that allows to construct long codes while maintaining low decoding latency. The resulting high throughput and good error-correction performance are appealing for optical communication systems and other systems where high throughput and low latency are required.

Research on polar codes has been constantly gaining attention over the last decade, by academia and industry alike, thanks to their capacity-achieving error-correction performance and low-complexity decoding algorithms. Recently, they have been selected as one of the coding schemes in the 5th generation wireless standard (5G). Over the years various polar code decoding algorithms, like SC-list (SCL), have been proposed to improve the mediocre performance of the successive cancellation (SC) decoding algorithm for finite code lengths; however, like SC, they suffer from long decoding latency. Fast decoding of polar codes tries to overcome this problem by identifying particular subcodes in the polar code and decoding them with efficient decoders. In this work, we introduce a generalized approach to fast decoding of polar codes to further reduce SC-based decoding latency. We propose three multi-node polar code subcodes whose identification patterns include most of the existing subcodes, extending them to SCL decoding, and allow to apply fast decoding to larger subsets of bits. Without any error-correction performance degradation, the proposed technique shows up to 23.6% and 29.2% decoding latency gain with respect to fast SC and SCL decoding algorithms, respectively, and up to 63.6% and 49.8% if a performance loss is accepted, whose amount depends on code and decoding algorithm parameters, along with the desired speedup.

In this paper, we propose a construction for multi-kernel polar codes based on the maximization of the minimum distance. Compared to the original construction based on density evolution, our new design shows particular advantages for short code lengths, where the polarization effect has less impact on the performance than the distances of the code. We introduce and compute the minimum-distance profile and provide a simple greedy algorithm for the code design. Compared to state-of-the-art punctured or shortened Arikan polar codes, multi-kernel polar codes with our new design show significantly improved error-rate performance.

In this work, we introduce β-expansion, a notion borrowed from number theory, as a theoretical framework to study fast construction of polar codes based on a recursive structure of universal partial order (UPO) and polarization weight (PW) algorithm. We show that polar codes can be recursively constructed from UPO by continuously solving several polynomial equations at each recursive step. From these polynomial equations, we can extract an interval for β, such that ranking the synthetic channels through a closed- form β-expansion preserves the property of nested frozen sets, which is a desired feature for low- complex construction. In an example of AWGN channels, we show that this interval for β converges to a constant close to 1.1892 when the code block-length trends to infinity. Both asymptotic analysis and simulation results validate our theoretical claims.

In this paper we propose a novel online caching method to perform the update phase in a distributed caching system, with a natural application to heterogeneous scenarios with cache-equipped small-cell base stations. We investigate the performance of our scheme, showing in particular that it results in a significant reduction of backhaul load, to the point of converging to the performance of the optimal offline placement scheme. While the optimal scheme is obtained at a high complexity cost, the performance of our solution is achieved in a low-complexity decentralized manner. Numerical simulations confirm that the novel online scheme adapts efficiently to varying networks parameters such as the file popularities, without the need of the usually time-consuming learning phase, which makes the proposal adapted for low-latency caching applications.

In this paper, we investigate a novel family of polar codes based on multi-kernel constructions, proving that this construction actually polarizes. To this end, we derive a new and more general proof of polarization, which gives sufficient conditions for kernels to polarize. Finally, we derive the convergence rate of the multi-kernel construction and relate it to the convergence rate of each of the constituent kernels.

Many security techniques working at the physical layer need a correct channel state information (CSI) at the transmitter, especially when devices are equipped with multiple antennas. Therefore such techniques are vulnerable to pilot contamination attacks (PCAs) by which an attacker aims at inducing false CSI. In this paper we provide a solution to some PCA methods, by letting two legitimate parties to compare their channel estimates. The comparison is made in order to minimize the information leakage on the channel to a possible attacker. By reasonable assumptions on both the channel knowledge by the attacker and the correlation properties of the attacker and legitimate channels we show the validity of our solution. An accurate analysis of possible attacks and countermeasures is provided, together with a numerical evaluation of the attainable secrecy outage probability when our solution is used in conjunction with beamforming for secret communications.

In this paper we investigate the problem of optimal cache placement under secrecy constraints in heterogeneous networks, where small-cell base stations are equipped with caches to reduce the overall backhaul load. For two models for eavesdropping attacks, we formally derive the necessary conditions for secrecy and we derive the corresponding achievable backhaul rate. In particular we formulate the optimal caching schemes with secrecy constraints as a convex optimization problem. We then thoroughly investigate the backhaul rate performance of the heterogeneous network with secrecy constraints using numerical simulations. We compare the system performance with and without secrecy constraints and we analyze the influence of the system parameters, such as the file popularity, size of the library files and the capabilities of the small-cell base stations, on the overall performance of our optimal caching strategy. Our results highlight the considerable impact of the secrecy requirements on the overall caching performance of the network.

In this paper, we study the problem of content placement for caching at the wireless edge with the goal to maximize the energy efficiency (EE) of heterogeneous wireless networks. In particular, we consider the minimization of two fundamental metrics: the expected backhaul rate and the energy consumption. We derive both metrics in closed-form expressions, and we solve the minimization problem as a convex optimization for each, highlighting the existence of a tradeoff between the two metrics. Further, we show the advantage of encoding the data using maximum-distance separable (MDS) codes over the alternative concept of file fragmentation, with respect to both backhaul rate and energy consumption. Then, we thoroughly study the performance of the optimal MDS-encoded caching scheme in terms of overall energy consumption for an important heterogeneous network scenario. We compare our optimal strategy to several other sub-optimal caching strategies, including the caching scheme, minimizing the backhaul rate, and we analyze the effects of the system parameters on the overall performance. Our analysis can be generalized to any network topology and to any small-cell base station capability. Our results show that the optimal placement of MDS-encoded content in caches at the wireless edge increases significantly the overall EE of the heterogeneous network. This demonstrates the importance of the edge caching strategy for energy-efficient network designs.

In this paper we investigate the problem of optimal MDS-encoded cache placement at the wireless edge to minimize the backhaul rate in heterogeneous networks. We derive the backhaul rate performance of any caching scheme based on file splitting and MDS encoding and we formulate the optimal caching scheme as a convex optimization problem. We then thoroughly investigate the performance of this optimal scheme for an important heterogeneous network scenario. We compare it to several other caching strategies and we analyze the influence of the system parameters, such as the popularity and size of the library files and the capabilities of the small-cell base stations, on the overall performance of our optimal caching strategy. Our results show that the careful placement of MDS-encoded content in caches at the wireless edge leads to a significant decrease of the load of the network backhaul and hence to a considerable performance enhancement of the network.

We analyze and optimize the secrecy performance of artificial noise (AN) in multi-input single-output wiretap channels with multiple antennas at the transmitter and a single antenna at the receiver and the eavesdropper. We consider two transmission schemes: 1) an on-off transmission scheme with a constant secrecy rate for all transmission periods, and 2) an adaptive transmission scheme with a varying secrecy rate during each transmission period. For the on-off transmission scheme, an easy-to-compute expression is derived for the hybrid outage probability, which allows us to evaluate the transmission outage probability and the secrecy outage probability. For the adaptive transmission scheme where transmission outage does not occur, we derive a closed-form expression for the secrecy outage probability. Using these expressions, we determine the optimal power allocation between the information signal and the AN signal and also determine the optimal secrecy rate such that the effective secrecy throughput is maximized for both transmission schemes. We show that the maximum effective secrecy throughput requires more power to be allocated to the AN signal when the quality of the transmitter-receiver channel or the transmitter-eavesdropper channel improves. We also show that both transmission schemes achieve a higher maximum effective secrecy throughput while incurring a lower secrecy outage probability than existing schemes.

We analyze and optimize the use of artificial noise (AN) for a predefined secrecy rate in wiretap channels with a multi-antenna transmitter, a single-antenna receiver, and a single-antenna eavesdropper. We derive a new closed-form expression for the secrecy outage probability that is independent of the channel realization. Based on this expression, we first optimize the power allocation between the information signal and the AN signal such that the secrecy outage probability is minimized. We then optimize jointly the power allocation and secrecy rate such that the secrecy throughput is maximized. As demonstrated by our analysis, the minimum secrecy outage probability requires more power to be allocated to the AN signal when the quality of the main channel quality or the eavesdropper's channel improves.

Luby transform (LT) codes are a class of rateless codes that automatically adapt their rate to the quality of the communication channel. In the original LT codes, fixed check-node degree distributions are used to combine variable nodes uniformly at random to extend the code graph and produce code bits. Here we propose a different approach: we design a sequence of rate-compatible degree distributions, and develop an algorithm that produces code bits in a manner such that the resulting degree distributions follow the designed sequence. Using this new design framework, we develop low-complexity LT codes suitable for time-varying noisy channels. Performance and complexity of the proposed LT codes are measured in terms of bit error rate and average number of edges per information and coded bit, respectively. Numerical examples illustrate the resulting trade-off between performance and complexity of the designed LT codes.

New insights into the role of artificial noise in securing communication in a Gaussian multi-antenna fading wiretap channel are presented. An appropriate secrecy-outage-based optimization framework is developed for the Multiple-Input Single-Output Single-Eavesdropper (MISOSE) case to measure the performance of artificial noise. It is assumed that only the legitimate receiver's instantaneous channel state information and the average statistics of the eavesdropper's channel are available at the transmitter. The optimization is based on maximizing the effective secret-message rate constrained by a given maximum secrecy outage criterion. Under this framework, a fundamental investigation is conducted into whether it is worthwhile for the transmitter to allocate any of its available power for artificial noise. By numerically solving the optimization problem, it is demonstrated that there are: (i) scenarios where artificial noise does indeed give significant gains in the secret-message rate, and (ii) scenarios where any amount of power allocation to artificial noise is wasteful in view of the overall performance.

This paper presents an investigation of employing transmit antenna selection for the MIMO wiretap channel. Instantaneous channel state information of the intended receiver and the eavesdropper are assumed to be available at the transmitter. We consider that the eavesdropper employs optimum combining, and investigate the two cases the intended receiver employs (a) optimum combining, (b) sub-optimal combining. We derive analytical expressions of the average secrecy capacities for these two cases in the high-SNR regime. We compare the performance of the proposed antenna selection scheme to a precoding scheme based on generalized singular value decomposition. Our investigations show that the average secrecy capacities increase with the number of antennas when the eavesdropper is at low SNR, but decrease when the eavesdropper is at high SNR.

This paper deals with the design of non-systematic irregular repeat-accumulate codes that are optimised for a finite number of decoding iterations. In many cases of practical interest, the usual approach for sparse-graph codes, which aims to approach capacity (with arbitrarily many decoder iterations), may not be suitable. This is of particular importance for systems with complexity or delay constraints. In this paper, we provide a design methodology which constrains the number of decoder iterations, as well as other measures of implementation complexity. Our approach uses extrinsic information transfer analysis, and our main contribution is the formulation of code optimisation problems which directly incorporate the number of iterations into the constraints. We focus on the single user binary erasure channel and the two-user binary adder channel, where this transfer analysis is exact. Generalisation to other sparse graph codes and other channels (under usual approximations) is straightforward.

This paper analyzes the impact of error propagation on the achievable throughput and throughput-delay tradeoff in wireless networks. It addresses the particular class of multihop routing schemes for parallel unicast that achieve a throughput scaling of Θ(n-1/2) per node in a network of n nodes. It is shown that in the finite-block-length case, necessitated by finite decoding memory at the nodes, the guaranteed per-node throughput in the network cannot scale better than o (n-r) per node for any r > 0. This bound on the guaranteed per-node throughput is tighter than the O (1/n) bound shown previously. Instead of focusing on the probability of error for each link, which is intractable, an approach of bounding mutual information is employed to show tight results on the achievable throughput and throughput-delay tradeoffs. It is shown that for multihop transmission protocols, error propagation leads to significant changes in the tradeoff between the throughput T(n) and the delay D(n), compared to previous results. The best known scaling behavior is only D(n) = Θ (n (log n) T(n)) under maximum throughput scaling, where the block length required scales as Ω (log n). When decoding memory at nodes is constrained to be O (log log n), the achievable tradeoff worsens to D(n) = Θ (n (log n)2T(n)).

A bound on the maximum information transmission rate through a cascade of Gaussian links is presented. The network model consists of a source node attempting to send a message drawn from a finite alphabet to a sink, through a cascade of Additive White Gaussian Noise links each having an input power constraint. Intermediate nodes are allowed to perform arbitrary encoding/decoding operations, but the block length and the encoding rate are fixed. The bound presented in this paper is fundamental and depends only on the design parameters namely, the network size, block length, transmission rate, and signal-to-noise ratio.

This paper presents an investigation of employing transmit antenna selection under secrecy constraints. The transmitter is equipped with multiple antennas while the intended receiver and the eavesdropper have a single antenna each. Instantaneous channel state information of the intended receiver and the eavesdropper are assumed to be available at the transmitter. We derive an analytical expression for the average secrecy rate, in terms of the number of available transmit antennas and the average signal-to-noise ratios at the intended receiver and the eavesdropper. We compare the performance of the antenna selection scheme, as revealed by our analysis, to a precoding scheme based on Generalized Singular Value Decomposition (GSVD) and characterize the loss in the secrecy capacity. In addition, we also derive an expression for the average secrecy capacity for the high SNR regime. From our investigation of the proposed system, we find that the increase of the average secrecy rate with the number of transmit antennas is sub-logarithmic.

This paper addresses the design of irregular repeat accumulate (IRA) codes for coded physical-layer network coding (PNC) for the binary-input Gaussian two-way relay channel, assuming perfect synchronization and equal received power at the relay. The design is based on a nontrivial extension of EXIT-chart based design. Specifically, we analyze the components of the IRA-PNC scheme and propose an approach to model the soft information exchanged between these components. Then, we develop upper and lower bounds on the extrinsic information transfer functions to characterize the iterative process of computing the network-coded information. Based on that, we construct optimized IRA codes to minimize the computation error at the relay. The optimized IRA-PNC has considerable performance improvement over the existing regular RA coded PNC. For a rate 3/4 code, as an example, we observed improvements of 2.6 dB, and the optimized IRA-PNC scheme is only about 1.7 dB away from the capacity upper bound of the Gaussian two-way relay channel.

This paper determines the range of feasible values of standard error exponents for binary-input memoryless symmetric channels of fixed capacity$C$and shows that extremes are attained by the binary symmetric and the binary erasure channel. The proof technique also provides analogous extremes for other quantities related to Gallager's$E_{0}$function, such as the cutoff rate, the Bhattacharyya parameter, and the channel dispersion.

This paper revisits earlier work on the achievable rate-region for the coded side-information problem. For specific source distributions we provide computable extreme rate points. As opposed to previous works, we present short and concise proofs and additional rate points below the time-sharing line of previously known rate points. Our results are based on a formulation as an optimization problem.

We investigate a channel-coded physical-layer network coding (CPNC) scheme for binary-input Gaussian two-way relay channels. In this scheme, the codewords of the two users are transmitted simultaneously. The relay computes and forwards a network-coded (NC) codeword without complete decoding of the two users' individual messages. We propose a new punctured codebook method to explicitly find the distance spectrum of the CPNC scheme. Based on that, we derive an asymptotically tight performance bound for the error probability. Our analysis shows that, compared to the single-user scenario, the CPNC scheme exhibits the same minimum Euclidean distance but an increased multiplicity of error events with minimum distance. At a high SNR, this leads to an SNR penalty of at most ln2 (in linear scale), for long channel codes of various rates. Our analytical results match well with the simulated performance.

We show that Gallager's random coding error exponent of an arbitrary binary-input memoryless symmetric channel is upper-bounded by that of the binary erasure channel and lower-bounded by that of the binary-symmetric channel of the same capacity. We apply the result to find the extremes of the channel dispersion for the aforementioned class of channels.

We investigate joint channel and physical layer network coding (CPNC) for Gaussian two-way relay channels. The two users' messages are encoded using the same binary linear code and are transmitted simultaneously with equal power. At the relay node, the network-coded message is recovered directly from the received signal sequence, and is then broadcast to the users. We propose a new methodology to explicitly find the distance spectrum of the coding scheme. Based on that, we analyze the error probability at the relay and derive an asymptotically tight performance bound (for high SNRs). We show that, with a general binary linear code, the CPNC scheme is subject to an asymptotic SNR loss of approximately ln 2 relative to the single-user case, regardless of the coding rate. Numerical results show that our analysis matches very well with the performance of the CPNC scheme.

New results on the achievable trade-off between the per-node throughput T(n) and the average delay D(n) in a static wireless network of n nodes are presented for physical link models. For links modeled by channels with additive white Gaussian noise with power-law attenuation, a trade-off of only D(n) = ⊖ (n (log n) T(n)) is guaranteed for T(n) = ⊖(n-½). This follows from showing that there is significant information loss in the network due to error propagation, unless the length of the channel code employed is sufficiently high. Constraining the block length to be bounded yields worse trade-offs: only ⊖(n (log n)2T(n)) is guaranteed for optimal throughput, provided there is rich fading diversity.

We consider the problem of communicating degraded message sets over multi-access channel with finite input and output alphabets that is controlled by an underlying i.i.d. state process. Inner and outer bounds are presented for the general case where encoders possess non-causal degraded state information. Two special cases of the general setup are then presented where the derived inner bound is shown to be the capacity region.

We formulate the maximum a posteriori (MAP) rule for the decode-and-forward transmission strategy operating with a noisy relay. From the MAP rule we derive an analytical bound on the error probability, taking into account decoding errors at the relay. We further determine a practical close-to-MAP decoding scheme based on a convenient error model for the decoding operation at the relay. This error model allows for a trellis representation of the code described jointly by the encoding process at the source and the re-encoding process at the relay. Numerical results demonstrate a close agreement between our analytical results and monte carlo simulations.

A novel adaptive binary decoding algorithm for LDPC codes is proposed, which reduces the decoding complexity while having a comparable or even better performance than corresponding non-adaptive alternatives. In each iteration the variable node decoders use the binary check node decoders multiple times; each single use is referred to as a sub-iteration. To process the sequences of binary messages in each iteration, the variable node decoders employ pre-computed look-up tables. These look-up tables as well as the number of sub-iterations per iteration are dynamically adapted during the decoding process based on the decoder state, represented by the mutual information between the current messages and the syndrome bits. The look-up tables and the number of sub-iterations per iteration are determined and optimized using density evolution. The performance and the complexity of the proposed adaptive decoding algorithm is exemplified by simulations.

In this paper, a theoretical framework of divergence minimization (DM) is applied to derive iterative receiver algorithms for coded CDMA systems. The DM receiver obtained performs joint channel estimation, multiuser decoding, and noise- covariance estimation. While its structure is similar to that of many ad-hoc receivers in the literature, the DM receiver is the result of applying a formal framework for optimization without further simplifications, namely the DM approach with a factorizable auxiliary model distribution. The well-known expectation- maximization (EM) algorithm and space-alternating generalized expectation-maximization (SAGE) algorithm are special cases of degenerate model distributions within the DM framework. Furthermore, many ad-hoc receiver structures from literature are shown to represent approximations of the proposed DM receiver. The DM receiver has four interesting properties that all result directly from applying the formal framework: (i) The covariances of all estimates involved are taken into account, (ii) The residual interference after interference cancellation is handled by the noise-covariance estimation as opposed to by LMMSE filters in other receivers, (iii) Posterior probabilities of the code symbols are employed rather than extrinsic probabilities, (iv) The iterative receiver is guaranteed to converge in divergence. The theoretical insights are illustrated by simulation results.

This paper deals with a novel design approach for a converging iterative receiver for coded CDMA that estimates both the channel coefficients and the transmitted symbols. The receiver design is based on the variational Bayesian space-alternating generalized expectation-maximization (VB-SAGE) method. Conceptually, the probability distribution of each user's code sequence and the probability distribution of all channel coefficients are updated in an iterative fashion. The obtained receiver, which we refer to as VB-SAGE receiver, performs iterative channel estimation, interference cancellation and single-user decoding. Even though the design is based on code sequence probabilities, only the soft decisions (mean values) and the variances of the code symbols are required for channel estimation and interference cancellation; moreover, the variances are functions of the soft decisions. These soft decisions are computed by the single-user decoders based on the outputs of the interference cancellation device. The iterative process of the VB-SAGE receiver is guaranteed to converge in the free energy. The VB-SAGE method proposed in this paper may also be used for other applications.

In this paper, the space alternating generalized expectation-maximization (SAGE) method is applied to derive receivers for coded CDMA systems. In the so-called SAGE receiver, hard decision code symbols are fed back from the single user decoders to the joint channel estimator and interference canceler. Using multivariate Gaussian approximations, the SAGE receiver is then modified such that soft values of code symbols are fed back from the single user decoders. We refer to this modified receiver as SAGE-based receiver. Compared to the SAGE receiver, the SAGE-based receiver considers the statistical properties of the code symbols, thus the channel estimation and the interference cancellation take symbol estimation errors into account. A performance improvement in terms of bit error rate (BER) and system load of the SAGE-based receiver compared to the SAGE receiver and other receivers is shown by means of Monte Carlo simulations.

This paper deals with the derivation and optimization of an iterative receiver architecture performing joint multiuser decoding and channel estimation. We consider an asynchronous multirate convolutional coded DS-CDMA system that communicates over quasi-static flat Rayleigh fading channels. The proposed receiver is derived within the space-alternating generalized expectation-maximization (SAGE) framework in connection with the noise-splitting approach. The used theoretical framework guarantees convergence of the receiver, as opposed to many other iterative receiver structures. Furthermore, the noise-splitting approach provides a set of noise-weighting coefficients that can be optimized under weak constraints. The inputs to the single-user decoders are linear combinations of two kinds of soft values with weights determined by the noise-weighting coefficients. These two kinds of soft values can be interpreted as a priori information and extrinsic information, respectively, if the channels are known. In the case of unknown channels, they are asymptotically a priori and asymptotically extrinsic, i.e., they become a priori and extrinsic when the length of the observed frame tends to infinity. In most cases, the optimum coefficients lead to extrinsic or asymptotically extrinsic values fed to the input of the single-user decoders. Monte Carlo simulations show that the proposed receiver is resistant to channel estimation errors and supports high system loads.

The degree distribution of low-density parity-check (LDPC) codes is optimized for systems that iterate over the receiver frontend, e.g., soft detector, demodulator, equalizer, etc., and the LDPC decoder. The overall extrinsic information transfer (EXIT) function of an iterative LDPC decoder is computed, based on the code's own EXIT chart, under the Gaussian assumption. While the optimization of the variable node distribution is a nonlinear problem, the optimization of the check node distribution is shown to be a linear problem. This fact is exploited to design codes where both the variable and the check node distributions are optimized, resulting in more robust constructions. The technique presented requires only knowledge of the measured EXIT function of the receiver frontend

An iterative receiver for joint multiuser-decoding and channel-estimation of coded CDMA is derived by applying the noise-splitting approach within the space alternating generalized expectation-maximization (SAGE) framework. We consider asynchronous single-rate DS/CDMA over flat Rayleigh fading channels. The resulting receiver structure comprises partial successive interference-cancellation (SIC), channel estimation, and soft-input/hard-output maximum likelihood sequence decoding (MLSD) for each user. Additionally, one obtains a set of noise-weighting coefficients that can be freely chosen within weak constraints. These coefficients determine the amount of feedback from the decoder output to the decoder input in the subsequent iteration, and thus, "how extrinsic" the decoder output values are. The noise-weighting coefficients strongly influence the system performance. Their optimization within the SAGE framework leads to extrinsic output values in most of the cases. The proposed receiver is evaluated by Monte-Carlo simulations, and it shows two major advantages: high load is supported and convergence is guaranteed.

Ideal coding schemes are defined as coding schemes that have the minimal symbol-error probability for a given code rate and a given communication channel. Ideal coding schemes with code-symbol decoding are analyzed with respect to their properties of processing mutual information. The wordwise mutual information, the symbol-wise mutual information and the extrinsic mutual information between code symbols and the corresponding decoder outputs are investigated. To analyze the extrinsic mutual information, the concept of information decomposition is introduced

Sparse intersymbol-interference (ISI) channels are encountered in a variety of high-data-rate communication systems. Such channels have a large channel memory length, but only a small number of significant channel coefficients. In this paper, trellis-based equalization of sparse ISI channels is revisited. Due to the large channel memory length, the complexity of maximum-likelihood detection, e.g., by means of the Viterbi algorithm (VA), is normally prohibitive. In the first part of the paper, a unified framework based on factor graphs is presented for complexity reduction without loss of optimality. In this new context, two known reduced-complexity algorithms for sparse ISI channels are recapitulated: The multi-trellis VA (M-VA) and the parallel-trellis VA (P-VA). It is shown that the M-VA, although claimed, does not lead to a reduced computational complexity. The P-VA, on the other hand, leads to a significant complexity reduction, but can only be applied for a certain class of sparse channels. In the second part of the paper, a unified approach is investigated to tackle general sparse channels: it is shown that the use of a linear filter at the receiver renders the application of standard reduced-state trellis-based equalizer algorithms feasible, without significant loss of optimality. Numerical results verify the efficiency of the proposed receiver structure

When the same data sequence is transmitted over two independent channels, or when a data sequence is transmitted twice but independently over the same channel, the independent observations can be combined at the receiver side. From an information-theory point of view, the overall mutual information between the data sequence and the received sequences represents a combination of the mutual information of the two channels. This concept is termed information combining. A lower bound and an upper bound on the combined information is presented, and it is proved that these bounds are tight. Furthermore, this principle is extended to the computation of extrinsic information on single code bits for a repetition code and for a single parity-check code of length three, respectively. For illustration of the concept and the bounds on information combining, two applications are considered. First, bounds on the information processing characteristic (IPC) of a parallel concatenated code are derived from its extrinsic information transfer (EXIT) chart. Second, bounds on the EXIT chart for an outer repetition code and for an outer single parity-check code of a serially concatenated coding scheme are computed.

Information combining is applied to an accumulator to obtain bounds on the extrinsic information transfer (EXIT) functions without relying on Gaussian or other approximations for the distribution of the a-priori messages.

A novel approach is presented for assessing the quality of transmission systems, comprising quantized source signals and APP source decoders, via Monte-Carlo simulation. A-posteriori probabilities are exploited in order to obtain an unbiased estimate of both the symbol error probability and the expected distortion for the transmission system; knowledge of the transmitted source signal is not necessary. Compared to the conventional method this blind quality estimation has a smaller estimation variance

By means of the mean reliability (defined as the mean of the absolute values of log-likelihood ratios), a new design of parallel concatenated "turbo" codes is proposed. This criterion allows us to describe the behavior of the constituent decoders and, furthermore, to predict the behavior of the iterative decoder for large block lengths. The mean reliability can also be used as a stopping criterion.