In layered communication networks there are only connections between intermediate nodes in adjacent layers . Applying network coding to such networks provides a number of benefits in theory as well as in practice. We propose alayering procedure to transform an arbitrary network into a layered structure. Furthermore, we derive a forward-backward duality for linear network codes, which can be seen as an analogon to the uplink-downlink duality in MIMO communication systems. I. I NTRODUCTION In [1] it was shown that communication between two nodes within a communication network is possible up to a rate that is equal to the minimum rate flowing through any possible cut between these two nodes—the mincutbetween them. This rate can be achieved by allowing intermediate nodes to code, i.e., to calculate functions of their incoming messages bef ore forwarding them. In [2] it was proved that it suffices to apply linear network coding(LNC), i.e., intermediate nodes just need to form linear combinations of their received messages from a finite fieldFq. If all operations are performed over a finite field of large enough sizeq, the factors at the intermediate nodes may even be drawn independently at random, which leads to a robust, decentralized, and capacity achieving approac h: random linear network coding(RLNC) [3], [4]. This paper studies network coding (NC) in layered networks, where intermediate nodes are arranged in layers and there exist only edges between nodes which are located in adjacent layers. We introduce a l yering procedure for establishing a layered structure in seemingly disparate an d unstructured network topologies. Applying NC to a layered network provides a number of benefits in theory for analysis as well as in practice. Moreover, we address the problem of bidirectional NCand derive aforward-backward duality . The paper is organized as follows: Sec. II gives a brief recapitulation and a classification of NC. In Sec. III we examine layered networks and introduce the layering procedure . Bidirectional NCis discussed in Sec. IV and some conclusions are drawn in Sec. V. II. B RIEF RECAPITULATION OF NETWORK CODING A. Problem Formulation We define a communication network as a directed, acyclic graphG = {N , E} with a set of nodesN and a set of edges E . The consideredmulticast scenarioconsists of a unique source nodeS ∈ N with n outgoing edges, and K destination nodesDk, k = 1, . . . ,K, with Nk ≥ n incoming edges. The This work was supported by DFG under grants FI 982/4-3 and HU 6 34/113, and by BMBF under grant 16 BP 12406. source transmitsn symbolsx1, . . . , xn ∈ Fq to each of the destination nodes Dk by injecting thesen symbols in parallel (one on each of its outgoing edges) into the network and each destination nodeDk tries to reconstruct all these symbols from its Nk receive symbolsyk,1, . . . , yk,Nk ∈ Fq. Nodes within the network are connected by edges i,j = (Ni,Nj) ∈ E . Each edge represents a noiseless 1communication link on which one symbol fromFq can be transmitted per usage. We further assume that each edge induces the same delay. 2 The in-degree d i and the out-degreed out i of a nodeNi is defined as the number of its incoming and outgoing edges, respectively. Coding at intermediate nodes is accomplished as follows: ea ch nodeNi collects the symbols from each of its d i incoming edges. Then, it computes possibly different functions of th ese symbols and transmits them on its d i outgoing edges. B. Classification of Network Coding Variants Essentially, there exist two distinct approaches to genera te outgoing messages at intermediate nodes. In the first one, which we denote as NC Variant I, each intermediate node calculates only a single function of its input symbols and transmits the resulting output symbol on all outgoing edges . This variant is applicable, e.g., in wireless networks, whe re intermediate nodes possess omnidirectional antennas, and thus, transmit a single signal. In NC Variant II intermediate node s computeindividual output symbols for their outgoing edges. This variant can be applied, e.g., in wired networks. In Fig. 1(a) an intermediate node Ni with d i incoming andd out i outgoing edges is depicted. The incoming and the outgoing symbols of node Ni are denoted asz i,δ, δ = 1, . . . , d in i , and z out i,ρ , ρ = 1, . . . , d i , respectively. The two NC variants are closely related to each other. This is specified in the following theo rem and is illustrated in Fig. 1. Theorem 1 A communication network employing NC Variant II can be transformed into an equivalent network which applies NC Variant I, by splitting up each intermediate node Ni with d i outgoing edges intod out i single output auxiliary nodes. These auxiliary nodes possess the same input edges as the original nodeNi. Proof: A Variant-II nodeNi is split up intod i auxiliary single output nodesNi,j , j = 1, . . . , d i , cf. Fig. 1(b). By repeating this procedure for all Variant-II nodes results i n an equivalent NC Variant I network. 1Since we do not treat error-correction coding for networks i n this paper, we restrict ourselves to the case of error-free NC. However, all statements contained in this paper are also applicable for noisy networ ks. 2If this is not the case, equal-delay edges can be achieved thr oug appropriate buffers at the intermediate nodes.

In layered communication networks there are only connections between intermediate nodes in adjacent layers. Applying network coding to such networks provides a number of benefits in theory as well as in practice. We propose a layering procedure to transform an arbitrary network into a layered structure. Furthermore, we derive a forward-backward duality for linear network codes, which can be seen as an analogon to the uplink-downlink duality in MIMO communication systems.

Random linear network coding (RLNC) is a method to maximise the information flow in a network by forming random linear combinations over a finite field Fq of the received information packets at each intermediate node. The network between one source node and one destination node acts as a linear map F(exp n)q ? F(exp N)q , which is represented by the network channel matrix. The connectivity within the network is assumed to be given, i.e. it is considered to be fixed but arbitrary and thus, the incidence matrix of the network is said to be known. The optimal decoding method is equivalent to solving a consistent system of linear equations over the respective finite field, e.g. by means of Gaussian elimination. Therefore, decoding is only successful if the respective square or tall network channel matrix has full column rank. Since the incidence matrix of the network is given, there is one degree of randomness less compared to the usual notion of random matrices. By exploiting similarities of RLNC with Luby transform (LT) coding, a method to establish rateless erasure resilience, which is also based on random matrices over finite fields, we derive an upper bound on the outage probability for RLNC with known incidence matrices.

Rateless codes, also known as digital fountain codes, are excellently suited for erasure correction in packet-switched communication networks. First applications are digital video broadcast or multicast over terrestrial networks or multimedia services in cellular networks. Such networks are usually prone to packet losses due to network congestions or unrecoverable bit errors within packets. The main attributes of rateless codes can be summarised as follows: • The transmitter is able to produce as many encoded packets as needed from a given source block consisting of k source packets. • The receiver is able to decode an exact copy of the entire source block from any subset of k(1 + εR) received (i.e. non-erased) encoded packets, where εR ≥ 0 is a small reception overhead. • No feedback channel is required for packet acknowledgements. In the literature, rateless codes are usually based on the simplifying design assumptions of input sequences of infinite length. The analysis and the characterisation of the so designed codes apply only to codes with very long input sequences and a corresponding latency. In contrast, this thesis focuses codes with finite (especially short to medium) lengths. These practical lengths enable applications that require a low transmission latency. In this context, various types of finite length LT codes and Raptor codes are investigated. The main contributions of this thesis are: • The derivation of analytical closed form expressions of the residual erasure probability under optimal decoding. • The derivation of tight upper and lower residual erasure bounds. • The generalisation of binary codes to higher order Galois fields. • The formulation of concrete design guidelines for highly efficient LT code ensembles with equal and unequal erasure protection. • New performance assessment tools for Raptor codes in terms of the so-called erasure weight and kernel weight profiles.

The erasure correction performance of Luby transform (LT) code ensembles over higher order Galois fields is analysed under optimal, \ie maximum likelihood (ML) erasure decoding. We provide the complete set of four bounds on the erasure probability after decoding on word as well as on symbol level. Especially the upper bounds are extremely close to the simulated residual erasure rates after decoding and can thus be used for code design instead of time-consuming simulations.

Finite length LT codes over higher order Galois fields F q for unequal error protection (UEP) are analysed under maximum likelihood (ML) decoding. We consider a biased sampling method to create the LT code graph. In contrast to a previous approach by Rahnavard et al., where a predetermined number of edges is created per importance class given a check node of degree d, our procedure allows to precisely adjust the desired class weights. Moreover, we provide upper and lower bounds on the symbol erasure probability for each importance class.

Low-density random linear fountain (LDRLF) codes are a type of LT (Luby transform) codes with optimum erasure correction under maximum likelihood (ML) decoding given a certain density or average check node degree. The upper bound on the residual symbol erasure rate is very tight and can be used for the design of LDRLF codes instead of performing time-consuming simulations. Using LDRLF codes for unequal error protection (UEP), the excellent erasure correction performance is maintained as well as the tightness of the upper bounds for each importance class. Since the UEP upper bounds may be complex to compute, we provide an extremely good approximation thereof which is well suited to design UEP LDRLF codes. Furthermore, we provide a heuristic criterion that has to be fulfilled in order to yield good approximations.

In this paper, two particular instances of LT codes with short message blocklength k and maximum likelihood (ML) decoding are investigated, i.e., random linear fountain (RLF) codes and (nearly) check-concentrated LT codes. Both show an almost equally good performance. The focus of this paper will be on RLF codes, a type of LT codes whose generator matrices are constructed from independent Bernoulli trials and have a binomial check node degree distribution. A new simple expression for an upper bound on the bit erasure probability under ML decoding is derived for RLF codes with density Δ = 0.5, i.e., with check node degree distribution Ω(x) = 2-k(1+x)k. It is shown that RLF codes with a minimum density far less than 0.5 are equally well suited to achieve a certain bit erasure probability for a given reception overhead. Furthermore, a characteristic term from a general upper bound on the bit erasure probability under ML decoding is identified that can be used to optimise check node degree distributions. Its implications on the performance of LT codes are qualitatively analysed.

Digital fountain codes over higher order Galois fields exhibit a better performance than their binary counter- parts under maximum likelyhood (ML) decoding when transmitted over a symbol erasure channel (SEC). Especially random linear fountain (RLF) codes exhibit an excellent performance, though at the expense of a high computational complexity for decoding due to their high density generator matrix. For practical applications, we propose RLF codes with a reduced density over higher order Galois fields. Although the reduction of the density results in an error floor at higher reception overheads, the level of this error floor can be well controlled by two parameters. For error floor levels that are tolerable in practical applications, a significant density reduction and thus a reduction of the computational complexity can be achieved. Furthermore, we derive a general upper bound on the symbol erasure rate for Luby Transform (LT) codes over Galois fields Fq of order q. Finally, we propose a method to enhance decoding of Fq-codes in the presence of bit erasures by using the binary images of the Fq-elements, such that not complete Fq-elements have to be discarded if their binary counterparts are impaired by bit erasures.

In some applications the transmission of discrete-time but continuous-amplitude (or multilevel) source symbols is required which might be more bandwidth efficient than conventional digital transmission. An appropriate method is to apply a source channel mapping (SCM) of M source symbols to N channel symbols. A geometrical approach for SCM has been introduced by Shannon and Kotel'nikov (Shannon-Kotel'nikov mappings). These systems are used to map M continuous-amplitude and discrete-time source symbols to N continuous-amplitude and discrete-time channel symbols without the intermediate step of a binary representation. These schemes are usually decoded using a maximum likelihood (ML) decoder which leads to optimum results in the mean square error sense for very good channels, but is suboptimal for noisy channels. In this paper the performance of an improved decoder, the minimum mean square error (MMSE) decoder is assessed. As a special case of a 1:2 expansion case (rate 1/2) Shannon-Kotel'nikov mapping, the Archimedes spiral is considered. The properties of the ML and MMSE decoder are examined and a graphical interpretation of the superior performance of the MMSE decoder is given. Furthermore, the robustness of the MMSE decoder w.r.t. an inaccurate estimation of the channel quality is determined. The concepts of the MMSE decoder which lead to a superior performance to the ML decoder can be generalized and applied to all Shannon-Kotel'nikov mappings.

In some applications the transmission of discrete-time but continuous-amplitude (or multilevel) source symbols is required which might be more bandwidth efficient than conventional digital transmission. An appropriate method is to apply a source channel mapping (SCM) of M source symbols to N channel symbols. A geometrical approach for SCM has been introduced by Shannon and Kotel'nikov (Shannon-Kotel'nikov mappings). These systems are used to map M continuous-amplitude and discrete-time source symbols to N continuous-amplitude and discrete-time channel symbols without the intermediate step of a binary representation. These schemes are usually decoded using a maximum likelihood (ML) decoder which leads to optimum results in the mean square error sense for very good channels, but is suboptimal for noisy channels. In this paper the performance of an improved decoder, the minimum mean square error (MMSE) decoder is assessed. As a special case of a 1:2 expansion case (rate 1/2) Shannon-Kotel'nikov mapping, the Archimedes spiral is considered. The properties of the ML and MMSE decoder are examined and a graphical interpretation of the superior performance of the MMSE decoder is given. Furthermore, the robustness of the MMSE decoder w.r.t. an inaccurate estimation of the channel quality is determined. The concepts of the MMSE decoder which lead to a superior performance to the ML decoder can be generalized and applied to all Shannon-Kotel'nikov mappings.

The rate distortion function is a widely used theoretical bound which describes the minimum mean square error (MMSE) distortion for a given number of quantization bits when quantizing a scalar random variable. An analytical solution for this function is only available for a small number of probability density functions (pdf), such as the Gaussian pdf. For arbitrary pdfs, the Blahut-Arimoto algorithm needs to be applied to iteratively estimate the rate distortion function. We propose a novel (semi-)analytical and non-iterative method to calculate the rate distortion function for sources with arbitrary pdfs. This method is based on the Guo-Shamai-Verdu¿ (GSV) theorem. Furthermore, it is possible to apply the proposed method for calculating the optimum performance theoretically attainable (OPTA) for arbitrarily distributed input symbols observed through an AWGN channel.

Linear analog block codes have been considered for transmission of discrete-time and continuous-amplitude data. In this paper, the fidelity measure parameter SNR (pSNR) at the receiver is derived for an arbitrary generator matrix P using an additive white Gaussian noise (AWGN) channel. In contrast to [1], it is shown that the performance of linear analog bock codes is dependent on the eigenvalues of the matrix P T P and not only on the dimensions of the matrix P. Surprisingly, the quality of the received values is independent of the code rate r , and e.g. a simple identity matrix has the optimal eigenvalues. Furthermore, the theoretical fidelity bound OPTA (Optimum Performance Theoretically Attainable) is used to assess the performance of a transmission system of continuous-amplitude data.

The FlexCode project is a joint research project of KTH Stockholm, RWTH Aachen University, Ericsson AB, Nokia Siemens Networks, and Orange/France Telecom under the umbrella of the sixth framework programme of the European Commission (http://www.flexcode.eu). In this paper we present the channel coding approach used in the FlexCode project. Furthermore, a brief introduction to the channel model utilized in FlexCode is given. The presented channel encoder enables iterative source-channel decoding at the receiver in order to achieve near-capacity transmission of the source coder parameters. The structure of the encoder enables to flexibly select the coding rate as well as the size of the input block. This joint source-channel coding approach is able to handle both considered types of quantization in the FlexCode project: constrained entropy and constrained resolution. On the other hand, the channel coding approach presented in this paper is able to achieve near Shannon-limit performance for arbitrary bit streams which is shown by a simulation example.

Turbo DeCodulation is the combination of iterative demodulation and iterative source-channel decoding in a multiple Turbo process. The receiver structures of bit-interleaved coded modulation with iterative decoding (BICM-ID) and iterative source-channel decoding (ISCD) are merged to one joint Turbo system, which we further enhance in this paper by using a low- density parity check (LDPC) code for channel coding, resulting in a third iterative loop. We propose to use a special LDPC code structure with short sub-codes, which can be implemented very effectively in parallel. The quadripartite Tanner graph of the Turbo DeCodulation is presented, showing that the processing of all receiver nodes could be parallelized. Simulation results including an EXIT chart analysis demonstrate the excellent capabilities of Turbo DeCodulation with its performance gain exceeding the combined gain of BICM-ID and ISCD.

We present a system for iterative source-channel decoding (ISCD) using irregular index assignments: The concept of irregular codes is applied to the index assignment of a scalar quantizer. The optimization performed in the EXIT chart enables near optimum transmission. The irregular index assignments are constructed by using high-rate block codes. This construction allows to use a very simple stopping criterion at the receiver and thereby to potentially reduce the number of iterations. We demonstrate the performance of this system by means of a simulation example over an AWGN channel with hard decision at the output. Furthermore we present simple yet effective measures for the case that no channel state information (e.g., instantaneous bit error rate) is available.

Iterative source-channel decoding (ISCD) aims at the exploitation of the time-variant residual redundancy of the source samples, e.g., source codec parameters, for error concealment and quality improvements. In most previous publications the receiver had perfect knowledge of the amount of residual redundancy. This assumption would require a reliable, i.e. highly redundant, transmission of side information. In contrast, in this paper we present a relatively simple scheme, yet efficient and robust, by which the residual redundancy at the receiver can be estimated accurately without any side information, and then can be exploited adaptively. We present the achievable performance gains in an ISCD system including the estimation of the residual source redundancy at the receiver for various scenarios. Several methods of different performance and computational complexity are proposed, with some of them even outperforming a system with perfect side information. The latter, not quite intuitive fact, is explained in the paper.

Due to complexity and delay constraints a usually significant amount of residual redundancy remains in the source samples after source coding. This residual redundancy can be exploited by iterative source-channel decoding for error concealment and quality improvements. One key design issue in joint source-channel (de-) coding is the index assignment. Besides conventional index assignments optimized index assignments have been developed, e.g., considering zeroth or first order a priori information of the source samples. However, in real-world scenarios it is unlikely that the amount of residual redundancy is constant over time and thus it may occur that the just deployed index assignment is suboptimal at times when the residual redundancy differs too much from the amount that it is optimized for. In this paper the performance of optimized index assignments is examined that consider first order a priori knowledge under such suboptimal conditions.

Due to complexity and delay constraints a usu- ally significant amount of residual redundancy remains in the source samples after source coding. This residual redun - dancy can be exploited by iterative source-channel decodin g for error concealment and quality improvements. One key design issue in joint source-channel (de-)coding is the ind ex assignment. Besides conventional index assignments opti- mized index assignments have been developed, e.g., consid- ering zeroth or first order a priori information of the source samples. However, in real-world scenarios it is unlikely th at the amount of residual redundancy is constant over time and thus it may occur that the just deployed index assignment is suboptimal at times the residual redundancy differs too much from the amount that it is optimized for. In this paper the performance of optimized index assignments is examined that consider first order a priori knowledge under such sub- optimal conditions.

In this paper a novel interpretation of encoding and decoding of recursive convolutional codes is presented. By means of Galois field arithmetic a code is separated into sub-codes with a single delay operator. One of these simple sub-codes is sufficient for encoding and decoding with the equivalent trellis diagram. This paper is not targeted at performance improvements but a new insights for the analysis of recursive convolutional codes resulting in novel, possibly less complex approaches for their implementation, e.g., on a chip.