Frank R. Kschischang

The Factor Graph Approach to Model-Based Signal Processing

The message-passing approach to model-based signal processing is developed with a focus on Gaussian message passing in linear state-space models, which includes recursive least squares, linear minimum-mean-squared-error estimation, and Kalman filtering algorithms. Tabulated message computation rules for the building blocks of linear models allow us to compose a variety of such algorithms without additional derivations or computations. Beyond the Gaussian case, it is emphasized that the message-passing approach encourages us to mix and match different algorithmic techniques, which is exemplified by two different approaches - steepest descent and expectation maximization - to message passing through a multiplier node.

Coding for Errors and Erasures in Random Network Coding

The problem of error-control in random linear network coding is considered. A ldquononcoherentrdquo or ldquochannel obliviousrdquo model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic. Motivated by the property that linear network coding is vector-space preserving, information transmission is modeled as the injection into the network of a basis for a vector space V and the collection by the receiver of a basis for a vector space U. A metric on the projective geometry associated with the packet space is introduced, and it is shown that a minimum-distance decoder for this metric achieves correct decoding if the dimension of the space V capU is sufficiently large. If the dimension of each codeword is restricted to a fixed integer, the code forms a subset of a finite-field Grassmannian, or, equivalently, a subset of the vertices of the corresponding Grassmann graph. Sphere-packing and sphere-covering bounds as well as a generalization of the singleton bound are provided for such codes. Finally, a Reed-Solomon-like code construction, related to Gabidulins construction of maximum rank-distance codes, is described and a Sudan-style ldquolist-1rdquo minimum-distance decoding algorithm is provided.

On the trellis structure of block codes

The problem of minimizing the vertex count at a given time index in the trellis for a general (nonlinear) code is shown to be NP-complete. Examples are provided that show that (1) the minimal trellis for a nonlinear code may not be observable, i.e. some codewords may be represented by more than one path through the trellis and (2) minimizing the vertex count at one time index may be incompatible with minimizing the vertex count at another time index. A trellis produce is defined and used to construct trellises for sum codes. Minimal trellises for linear codes are obtained by forming the product of elementary trellises corresponding to the one-dimensional subcodes generated by atomic codewords. The structure of the resulting trellis is determined solely by the spans of the atomic codewords. A correspondence between minimal linear block code trellises and configurations of nonattacking rooks on a triangular chess board is established and used to show that the number of distinct minimal linear block code trellises is a Stirling number of the second kind. Various bounds on trellis size are reinterpreted in this context.

Optimal nonuniform signaling for Gaussian channels

Variable-rate data transmission schemes in which constellation points are selected according to a nonuniform probability distribution are studied. When the criterion is one of minimizing the average transmitted energy for a given average bit rate, the best possible distribution with which to select constellations points is a Maxwell-Boltzmann distribution. In principle, when constellation points are selected according to a Maxwell-Boltzmann distribution, the ultimate shaping gain ( pi e/6 or 1.53 dB) can be achieved in any dimension. Nonuniform signaling schemes can be designed by mapping simple variable-length prefix codes onto the constellation. Using the Huffman procedure, prefix codes can be designed that approach the optimal performance. These schemes provide a fixed-rate primary channel and a variable-rate secondary channel, and are easily incorporated into standard lattice-type coded modulation schemes. >

Power Reduction Techniques for LDPC Decoders

This paper investigates VLSI architectures for low-density parity-check (LDPC) decoders amenable to low-voltage and low-power operation. First, a highly-parallel decoder architecture with low routing overhead is described. Second, we propose an efficient method to detect early convergence of the iterative decoder and terminate the computations, thereby reducing dynamic power. We report on a bit-serial fully-parallel LDPC decoder fabricated in a 0.13-mum CMOS process and show how the above techniques affect the power consumption. With early termination, the prototype is capable of decoding with 10.4 pJ/bit/iteration, while performing within 3 dB of the Shannon limit at a BER of 10-5 and with 3.3 Gb/s total throughput. If operated from a 0.6 V supply, the energy consumption can be further reduced to 2.7 pJ/bit/iteration while maintaining a total throughput of 648 Mb/s, due to the highly-parallel architecture. To demonstrate the applicability of the proposed architecture for longer codes, we also report on a bit-serial fully-parallel decoder for the (2048, 1723) LDPC code in 10 GBase-T standard synthesized with a 90-nm CMOS library.

A Simplified Successive-Cancellation Decoder for Polar Codes

A modification is introduced of the successive-cancellation decoder for polar codes, in which local decoders for rate-one constituent codes are simplified. This modification reduces the decoding latency and algorithmic complexity of the conventional decoder, while preserving the bit and block error rate. Significant latency and complexity reductions are achieved over a wide range of code rates.

On Metrics for Error Correction in Network Coding

The problem of error correction in both coherent and noncoherent network coding is considered under an adversarial model. For coherent network coding, where knowledge of the network topology and network code is assumed at the source and destination nodes, the error correction capability of an (outer) code is succinctly described by the rank metric; as a consequence, it is shown that universal network error correcting codes achieving the Singleton bound can be easily constructed and efficiently decoded. For noncoherent network coding, where knowledge of the network topology and network code is not assumed, the error correction capability of a (subspace) code is given exactly by a new metric, called the injection metric, which is closely related to, but different than, the subspace metric of KOumltter and Kschischang. In particular, in the case of a non-constant-dimension code, the decoder associated with the injection metric is shown to correct more errors then a minimum-subspace-distance decoder. All of these results are based on a general approach to adversarial error correction, which could be useful for other adversarial channels beyond network coding.

On the Effective Weights of Pseudocodewords for Codes Defined on Graphs with Cycles

The behavior of an iterative decoding algorithm for a code defined on a graph with cycles and a given decoding schedule is characterized by a cycle-free computation tree. The pseudocodewords of such a tree are the words that satisfy all tree constraintsj pseudocodewords govern decoding performance. Wiberg [12] determined the effective weight of pseudocodewords for binary codewords on an AWGN channel. This paper extends Wiberg’s formula for AWGN channels to nonbinary codes, develops similar results for BSC and BEC channels, and gives upper and lower bounds on the effective weight. The 16-state tail-biting trellis of the Golay code [2] is used for exampIes. Although in this case no pseudocodeword is found with effective weight less than the minimum Hamming weight of the Golay code on an AWGN channel, it is shown by example that the minimum effective pseudocodeword weight can be less than the minimum codeword weight.

Capacity-achieving probability measure for conditionally Gaussian channels with bounded inputs

A conditionally Gaussian channel is a vector channel in which the channel output, given the channel input, has a Gaussian distribution with (well-behaved) input-dependent mean and covariance. We study the capacity-achieving probability measure for conditionally Gaussian channels subject to bounded-input constraints and average cost constraints. Many practical communication systems, including additive Gaussian noise channels, certain optical channels, fading channels, and interference channels fall within this framework. Subject to bounded-input constraint (and average cost constraints), we show that the channel capacity is achievable and we derive a necessary and sufficient condition for a probability measure to be capacity achieving. Under certain conditions, the capacity-achieving measure is proved to be discrete.

An Algebraic Approach to Physical-Layer Network Coding

The problem of designing physical-layer network coding (PNC) schemes via nested lattices is considered. Building on the compute-and-forward (C&F) relaying strategy of Nazer and Gastpar, who demonstrated its asymptotic gain using information-theoretic tools, an algebraic approach is taken to show its potential in practical, nonasymptotic, settings. A general framework is developed for studying nested-lattice-based PNC schemes-called lattice network coding (LNC) schemes for short-by making a direct connection between C&F and module theory. In particular, a generic LNC scheme is presented that makes no assumptions on the underlying nested lattice code. C&F is reinterpreted in this framework, and several generalized constructions of LNC schemes are given. The generic LNC scheme naturally leads to a linear network coding channel over modules, based on which noncoherent network coding can be achieved. Next, performance/complexity tradeoffs of LNC schemes are studied, with a particular focus on hypercube-shaped LNC schemes. The error probability of this class of LNC schemes is largely determined by the minimum intercoset distances of the underlying nested lattice code. Several illustrative hypercube-shaped LNC schemes are designed based on Constructions A and D, showing that nominal coding gains of 3 to 7.5 dB can be obtained with reasonable decoding complexity. Finally, the possibility of decoding multiple linear combinations is considered and related to the shortest independent vectors problem. A notion of dominant solutions is developed together with a suitable lattice-reduction-based algorithm.