Iterative Decoding of Concatenated Codes: A Tutorial

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.