IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 5, SEPTEMBER 2008 2815 BER Evaluation of BICM-ID via Bonferroni-Type Bounds Rolando Bettancourt, Leszek Szczecinski, Senior Member, IEEE, and Rodolfo Feick, Senior Member, IEEE Abstract—In this paper, we evaluate the bit error rate (BER) of bit-interleaved coded modulation (BICM) receivers that operate with a priori information obtained from the decoder during the so- called turbo iterations. We use Bonferroni-type lower and upper bounds to tightly estimate the BER performance in the range of signal-to-noise ratio (SNR) of interest. The results are shown to be useful for the convergence analysis of the iterative demapping via the so-called extrinsic information transfer (EXIT) charts and for the evaluation of the coded performance of BICM with iterative demapping (BICM-ID). Index Terms—BICM with iterative demapping (BICM-ID), bit-interleaved coded modulation (BICM), Bonferroni bounds, iterative demapping, turbo decoding, turbo demapping. I. I NTRODUCTION I N THIS PAPER, we propose a method for the evaluation of the bit error rate (BER) of bit-interleaved coded modu- lation (BICM) [1] receivers that operate with random a priori information. Such a scenario occurs in the context of BICM with iterative demapping (BICM-ID) [2], [3], where the soft- input soft-output (SISO) decoder and the demapper exchange information about the coded bits. The performance of BICM-ID strongly depends on the ap- propriate choice of the bits-to-symbols mapping, which has been subject to various propositions, e.g., [4]–[6]. These mostly heuristic approaches have also been complemented, e.g., in [7] and [8], by a formal algorithmic design of the mapping. In [7] and [8], the design criteria are based on the assump- tion of a sufficiently large number of iterations and a high operating signal-to-noise ratio (SNR) (which are necessary to ensure convergence), i.e., optimization of the error floor is targeted. Manuscript received January 23, 2007; revised July 5, 2007, August 17, 2007, October 11, 2007, and November 14, 2007. This work was supported in part by the Comisión Nacional de Investigación Científica y Tecnológica, Chile, under Programa Bicentenario de Ciencia y Tecnología Project ACT-11/2004, by Fundación Andes, Chile, and by the Natural Sciences and Engineering Research Council, Canada, under Grant 249704-02. The review of this paper was coordinated by Prof. J. Choi. R. Bettancourt was with the Department of Electronics Engineering, Univer- sidad Tecnica Federico Santa Maria, Valparaíso 110-V,Chile. He is now with VTR Globalcom, Santiago 3340, Chile (e-mail: rolando.bettancourt@vtr.cl). L. Szczecinski is with the Institut National de la Recherche Scientifique- Énergie, Matériaux et Télécommunications (INRS-EMT), University of Quebec, Montreal, QC H5A 1K6, Canada (e-mail: leszek@emt.inrs.ca). R. Feick is with the Department of Electronics Engineering, Universidad Tecnica Federico Santa Maria, Valparaíso 110-V, Chile (e-mail: feick@elo. utfsm.cl). Digital Object Identifier 10.1109/TVT.2007.913182 However, often, the so-called waterfall region matters more than obtaining a low error floor, particularly if retransmissions are available to take care of erroneously received packets. In such a case, optimizing the waterfall region (or finding mappings that guarantee the early convergence) translates into throughput gains. The EXIT analysis is probably the most popular method to evaluate the performance of iterative receivers [9], [10] and consists of tracking the evolution of the mutual information (MI) between L-values and coded bits throughout the iterative process. It yields the so-called extrinsic information transfer (EXIT) charts that are helpful to analyze convergence after a theoretically unbounded number of iterations and may also be used to approximate the performance when the number of iterations is limited [9], [11], [12]. Despite such appealing features, using EXIT charts for map- ping optimization is made difficult by the fact that simulations are necessary to obtain the EXIT functions. To circumvent this limitation, i.e., to avoid simulations, we propose to accurately approximate the EXIT functions. The proposed approach is based on a numerically efficient method that we developed for the evaluation of the BER at the demapper’s output. Under some simplifying assumptions, e.g., [9] and [11], the BER may be accurately related to the MI that is used in the EXIT analysis. The proposed method may be used to track MI throughout the iterations, producing results that are more accurate than those that result from BER tracking, as done in [10] and [13]. The main challenge is the evaluation of the demappers’ BER with random a priori L-values. For that purpose, the general BER-evaluation method proposed in [14] was already used in [15] via integration over the space of the random L-values. The complexity of such an approach grows as K log 2 M , with M being the constellation size and K—the number of integration nodes per dimension. To deal with this problem, we propose to apply the Bonferroni-type algorithmic bounds [16] on the BER. Their computational complexity is modest, and the advantage offered is that the effect of the a priori L-values is reduced to a single random variable, which is marginalized analytically. Numerical results indicate that the bounds tightly approach the bit error probabilities obtained via simulations. This paper is organized as follows. Section II introduces the notation and provides a quick overview of the principles of BICM-ID. In Section III, we make the derivations necessary to apply the algorithmic bounds to estimate the BER of the MAP receiver with random a priori information. These results are then applied in Section IV to obtain the EXIT functions of the demapper that are further used to estimate the coded BER. Concluding remarks are made in Section V. 0018-9545/$25.00 © 2008 IEEE