On the Performance Gap between ML and Iterative Decoding of Finite-Length Turbo-Coded BICM in MIMO Systems

Abstract

As real-time applications typically require short length code words, this paper analyzes the minimum achievable code word error rate (CER) of the given finite length code words, namely, maximum likelihood (ML) decoding error probability. For coding schemes ML decoding is too complex, the key contribution of this paper is to analytically assess the performance gap between ML decoding and a practical decoding scheme. We analyze the combination of turbo codes and bit-interleaved coded-modulation (BICM) that is a spectrally efficient coding scheme adopted in 3GPP long term evolution (LTE). In single input single-output (SISO) systems, it was shown in the literature that turbo decoding delivers near-ML decoding performance. In this paper, we extend the analysis to a multi-input multi output (MIMO) system. In contrast to the SISO case, the turbo principle based iterative decoding scheme is subject to appreciable performance loss compared to ML decoding, even in good MIMO channel conditions. For this reason, we further analyze potential reasons and examine possible improvements. By means of simulation, it is shown that convergence to a non-ML code word rather than non-convergence is the key reason for the observed performance loss in good channel conditions.

Existing System

Interleaved Coded Modulation, Gaussian Approximation.

Proposed System

In this paper, we aim to estimate the ML decoding performance via upper and lower bounds because they possess a definite relation (i.e., higher and lower than) to the real ML decoding performance. Specifically, our contribution starts from the union bound, whose calculation involves sums of a great number of terms. Since a brute-force summation is infeasible, we propose to loosen it in a way that allows for a fast evaluation, provided that the signal-to-noise ratio (SNR) interval in which the union bound is informative is retained. Next, we apply Gallager’s first bounding technique together with the Chernoff bound. The obtained upper bound has a closed-form expression, and can be linked to the Duman-Salehi bounding technique originating from the Gallager’s second bounding technique. In order to validate the tightness of the upper bounds we subsequently include a lower bound that was also used. They jointly define a theoretic limit on the CER of ML decoding, serving as a baseline for practical decoding algorithms. In this paper, we evaluate the performance of the turbo principle based doubly iterative decoding algorithm. Different to the observation of turbo decoding in SISO systems, it unexpectedly fails to approach the ML performance even at high SNRs, therefore the practical iterative decoding schemes for turbo coded MIMO-BICM systems are not optimal. This result is the key contribution of this paper. By means of empirical studies, we notice that convergence to a non-ML code word rather than non-convergence is the key reason for the exhibited performance loss in good channel conditions. This implies the need of broadening the search space for an improved doubly iterative decoding performance.

Architecture

A Turbo BICM System