A factor graph is a bipartite graph that expresses how a "global" function of many variables factors into a product of "local" functions. Factor graphs subsume many other graphical models including Bayesian networks, Markov random fields, and Tanner graphs. Following one simple computational rule, the sum-product algorithm operates in factor graphs to compute---either exactly or approximately---various marginal functions by distributed message-passing in the graph. A wide variety of algorithms developed in artificial intelligence, signal processing, and digital communications can be derived as specific instances of the sum-product algorithm, including the forward/backward algorithm, the Viterbi algorithm, the iterative "turbo" decoding algorithm, Pearl's belief propagation algorithm for Bayesian networks, the Kalman filter, and certain fast Fourier transform algorithms.

Abstract—We present a unified graphical model framework for describing compound codes and deriving iterative decoding algorithms. After reviewing a variety of graphical models (Markov random fields, Tanner graphs, and Bayesian networks), we derive a general distributed marginalization algorithm for functions described by factor graphs. From this general algorithm, Pearl’s belief propagation algorithm is easily derived as a special case. We point out that recently developed iterative decoding algorithms for various codes, including “turbo decoding ” of parallelconcatenated convolutional codes, may be viewed as probability propagation in a graphical model of the code. We focus on Bayesian network descriptions of codes, which give a natural input/state/output/channel description of a code and channel, and we indicate how iterative decoders can be developed for parallel- and serially-concatenated coding systems, product codes, and low-density parity-check codes. I.

A code-division multiple-access system with channel coding may be viewed as a serially-concatenated coded system. In this paper we propose a low complexity method for decoding the resulting inner code (due to the spreading sequence), which allows iterative (turbo) decoding of the serially-concatenated code pair. The per-bit complexity of the proposed decoder increases only linearly with the number of users. Performance within

This paper introduces an iterative multiuser receiver for direct sequence code-division multiple access (DS-CDMA) with forward error control (FEC) coding. The receiver is derived from the maximum a posteriori (MAP) criterion for the joint received signal, but uses only single-user decoders. Iterations of the system are used to improve performance, with dramatic effects. Single-user turbo code decoders are utilized as the FEC system and a complexity study is presented. Simulation results show that the performance approaches single-user performance even for moderate signal-to-noise ratios.

We view the asynchronous random code-division multiple-access (CDMA) channel as a time-varying convolutional code. We study the case where the users encode their data, and, therefore, the single user transmitters and the CDMA channel appear as the concatenation of two coding systems. At the receiver we employ serial Turbo decoding strategies. Unlike conventional Turbo codes where both the inner and outer code may be selected, in our case, the inner code is due to the CDMA channel which we assume to be random. Nevertheless, the decoding system resembles the decoder of a serial Turbo code and single-user performance is obtained even for numbers of users approaching the spreading code length. Index Terms---Code-division multiple access, iterative methods, random codes, Turbo codes. I. INTRODUCTION T HE OUTPUT of a code-division multiple-access (CDMA) channel is a linear transformation of the input. In the case where the channel output noise is whitened, using the noise whitening-match...

In this paper, we present a unified graphical model framework for describing codes and deriving iterative decoding algorithms. We illustrate how the following systems can be described using graphical models: turbo-codes, serially-concatenated convolutional codes, frame-oriented turbo-codes, low-density parity-check codes, product codes, and convolutional codes on channels with memory. Recently proposed iterative decoding algorithms (e.g., turbo-decoding) can be viewed as a simple message passing procedure on the graphical model. This framework provides the means to derive new iterative decoders for new codes quite easily.

This paper discusses a code-division multiple access (CDMA) iterative multi-user receiver with forward error control (FEC) decoding. The maximum a-posteriori probability (MAP) criteria is used to derive the receiver. The decoding is done using a turbo-code decoder with modifications which are discussed. Iterations of the system are used to attain large performance improvements over conventional systems. 1. INTRODUCTION In mobile communications, direct-sequence codedivision multiple-access (DS-CDMA) systems, are now becoming available on the market. Unless orthogonal spreading codes are used the performance of these systems has been limited by their inherent multiple access interference. In [1] we demonstrated with random spreading codes, multiuser performance close to single user performance. This motivates us to use more powerful codes to see if we still get close to single user performance. In this paper we investigate an iterative DS/CDMA multi-user receiver, primarily intended fo...

This is a tutorial paper meant to introduce the reader to the new concept of turbo codes. This is a new and very powerful error correction technique which outperforms all previous known coding schemes. It can be used in any communication system where a significant power saving is required or the operating signal--to--noise ratio (SNR) is very low. Deep space communications, mobile satellite/cellular communications, microwave links, paging, etc., are some of the possible applications of this revolutionary coding technique. Part I of the paper discussed the history of turbo codes, why they are different from traditional convolutional/block codes, the turbo encoder structures and issues related to the interleaver design. Part II addresses the decoder architecture, the achievable performance for turbo codes for a wide range of coding rates and modulation techniques and discusses delay and implementation issues. 2 1 Introduction The optimum decoding of turbo codes is the maximum likeli...

This paper discusses the implementation of a very powerful but computationally efficient, iterative multi-user detector, intended for use as the basestation physical layer receiver for Wideband CDMA. This multiuser detector is unique in that it is fully compliant with the release '99 3GPP standard, a result that has not been shown before. The transmitter chain is discussed before the channel model used is introduced. The receiver design approach is then discussed. Special consideration is given to show how channel estimation is integrated into the receiver. We show single cell performance results that indicate a capacity increase of approximately three times over the "conventional" receiver implementation is possible (with and without antenna arrays). Results are then shown using a cell dimensioning simulation tool to indicate the overall gain in terms of increase in cell size. Here we can show that a gain of approximately 50% of the cell size is possible with such a multi-user detection approach. The dimensioning tool is also used to show the increase in capacity for a fixed cell size. Here we can show that a gain in the order of 330% is possible, compared to a conventional receiver approach.

We introduce a class of iteratively decodable trellis-constrained codes as a generalization of turbocodes, low-density parity-check codes, serially-concatenated convolutional codes, and product codes. In a trellis-constrained code, multiple trellises interact to define the allowed set of codewords. As a result of these interactions, the minimum-complexity single trellis for the code can have a state space that grows exponentially with block length. However, as with turbocodes and low-density parity-check codes, a decoder can approximate bit-wise maximum a posteriori decoding by using the sum-product algorithm on the factor graph that describes the code. We present two new families of codes, homogenous trellis-constrained codes and ring-connected trellis-constrained codes, and give results that show these codes perform in the same regime as do turbo-codes and low-density parity-check codes.