Turbo Decoding as Constrained Optimization

The turbo decoder was not originally introduced as a solution to an optimization problem. This has made explaining just why the turbo decoder performs as well as it does very difficult. Many authors have attempted to explain both the performance and convergence of the decoder, with varied success. In this document we show that the turbo decoder admits an exact interpretation as an iterative method (nonlinear block Gauss Seidel iteration) attempting to find a solution to a particular intuitively pleasing constrained optimization problem. In particular the turbo decoder is trying to find the maximum likelihood solution under the false assumption that the input to the encoders were chosen independently of one another, subject to a constraint on the probability that the messages so chosen happened to be the same. We provide an exact analytical objective function, along with an exact analytical form of the constraint, and then show that the turbo decoder is an iterative method originally suggested by Gauss, which is trying to solve the optimization problem by solving a system of equations which are the necessary conditions of Lagrange with a Lagrange multiplier of −1.