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A convergence proof for the turbo decoder as an instance of the gauss-seidel iteration

Many previous attempts at analyzing the convergence behavior of turbo and iterative decoding, such as EXIT style analysis (S. ten Brink, Oct. 2001) and density evolution (H. El Gamal and A. R. Hammons, Jr, Feb. 2001), ultimately appeal to results which become valid only when the block length grows rather large, while still other attempts, such as connections to factor graphs (F. R. Kshischang, et al., Feb. 2001) and belief propagation (R. J. McEliece, et al., Feb. 1998), have been largely unsuccessful at showing convergence due to loops in the turbo coding graph. The information geometric attempts (M. Moher and T. A. Gulliver, Nov. 1998), (T. Richardson, Jan. 2000), (S. Ikeda, et al., June 2004), (B. Muquest, et al., June 2002), and (J. Walsh, et al., March 2005), in turn have been inhibited by inability to efficiently describe intrinsic information extraction as an information projection. This paper recognizes turbo decoding as an instance of a Gauss-Seidel iteration on a particular nonlinear system of equations. This interpretation holds regardless of block length, and allows a connection to existing convergence results for nonlinear block Gauss Seidel iterations. We thus adapt existing convergence theory for the Gauss Seidel iteration to give sufficient conditions for the convergence of the turbo decoder that hold regardless of the block length