Iterative Decoding of Concatenated Codes: A Tutorial
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Iterative Decoding of Concatenated Codes: A Tutorial Phillip A. Regalia D´epartement Communications, Images et Traitement de l’ Information, Institut National des T´el´ecommunications, 91011 Evry Cedex, France Department of Electrical Engineering and Computer Science, Catholic University of America, Washington, DC 20064, USA Email: [email protected] Received 29 September 2003; Revised 1 June 2004 The turbo decoding algorithm of a decade ago constituted a milestone in error-correction coding for digital communications, and has inspired extensions to generalized receiver topologies, including turbo equalization, turbo synchronization, and turbo CDMA, among others. Despite an accrued understanding of iterative decoding over the years, the “turbo principle” remains elusive to master analytically, thereby inciting interest from researchers outside the communications domain. In this spirit, we develop a tutorial presentation of iterative decoding for parallel and serial concatenated codes, in terms hopefully accessible to a broader audience. We motivate iterative decoding as a computationally tractable attempt to approach maximum-likelihood decoding, and characterize fixed points in terms of a “consensus” property between constituent decoders. We review how the decoding algorithm for both parallel and serial concatenated codes coincides with an alternating projection algorithm, which allows one to identify conditions under which the algorithm indeed converges to a maximum-likelihood solution, in terms of particular likelihood functions factoring into the product of their marginals. The presentation emphasizes a common framework applicable to both parallel and serial concatenated codes. Keywords and phrases: iterative decoding, maximum-likelihood decoding, information geometry, belief propagation.
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INTRODUCTION
The advent of the turbo decoding algorithm for parallel concatenated codes a decade ago [1] ranks among the most significant breakthroughs in modern communications in the past half century: a coding and decoding procedure of reasonable computational complexity was finally at hand offering performance approaching the previously elusive Shannon limit, which predicts reliable communications for all channel capacity rates slightly in excess of the source entropy rate. The practical success of the iterative turbo decoding algorithm has inspired its adaptation to other code classes, notably serially concatenated codes [2, 3], and has rekindled interest [4, 5] in low-density parity-check codes [6], which give the definitive historical precedent in iterative decoding. The serial concatenated configuration holds particular interest for communication systems, since the “inner encoder” of such a configuration can be given more general interpretations, such as a “parasitic” encoder induced by a convolutional channel or by the spreading codes used in CDMA. The corresponding iterative decoding algorithm can then be extended into new arenas, giving rise to turbo equalThis is an open access article distributed under th
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