A Proof of Convergence of the MAP Turbo-Detector to the AWGN Case

In this paper, we consider a coded transmission over a frequency-selective channel. We propose to study analytically the convergence of the turbo-detector using a maximum a posteriori (MAP) equalizer and a MAP decoder. We show that the densities of the extrinsic log likelihood ratios (LLRs) exchanged during the iterations are e-symmetric and output-symmetric. Under the Gaussian approximation, this property allows to perform a one-dimensional (1-D) analysis of the turbo-detector. By deriving the analytical expressions of the extrinsic LLR distributions under the Gaussian approximation, we prove that the bit error rate (BER) performance of the turbo-detector converges to the BER performance of the coded additive white Gaussian noise (AWGN) channel at high signal to noise ratio (SNR), for any frequency-selective channel.