A Novel Multi-Dimensional Mapping of 8-PSK for BICM-ID Nghi H. Tran and Ha H. Nguyen Department of Electrical Engineering, University of Saskatchewan Saskatoon, SK, Canada S7N 5A9 Email: nghi.tran@usask.ca, ha.nguyen@usask.ca Abstract— Employing multi-dimensional constellation and mapping to improve the error performance of bit-interleaved coded modulation with iterative decoding (BICM-ID) has recently received a lot of attention, both in single-antenna and multiple- antenna systems. To date, except for the cases of BPSK and QPSK constellations, good multi-dimensional mappings have only been found by computer searching techniques. This paper introduces an explicit algorithm to construct a good multi-dimensional mapping of 8-PSK for improving the asymptotic performance of BICM-ID systems. By comparing the performance of the proposed mapping with an unachievable lower bound, it is conjectured that the proposed mapping is the globally optimal mapping. The superiority of the proposed mapping over the best conventional (two-dimensional) mapping and the multi- dimensional mapping found previously by computer search is also demonstrated. I. I NTRODUCTION Signal constellation and mapping play an important role in determining the bit error rate (BER) performance of a BICM-ID system. Good one- and two-dimensional mappings have been proposed for single-antenna systems [1]–[6] as well as multiple-antenna systems [7], [8]. More recently, considerable attention has been paid to the use of multi- dimensional (multi-D) mapping to further improve the error performance of BICM-ID. In particular, reference [9] studies hypercube mappings of QPSK constellation for BICM-ID in single-antenna systems, where it is shown that a significant coding gain can be achieved without any bandwidth nor power expansion. A parallel research work was also carried out in [10] but it only concentrates on an AWGN channel. The technique of multi-D mapping has also been extended to BICM-ID for multiple-antenna systems in [11]–[13]. It is observed in [13] that performance of the system with multi-D mapping can achieve near turbo-code performance with only a simple convolutional code. To the best of our knowledge, except [9] and [10], all research work related to the mapping design problem is only based on some searching techniques. For example, the brute- force computer searching can be carried out for the single- antenna systems that use a low-order signal constellation and the conventional two-dimensional (2-D) mapping [1]. In the case of multi-D mapping with a high-order constellation, exhaustive computer search to find the optimal mapping is impossible due to the huge complexity of the search. As an example, with a very simple single-antenna system employing 8-PSK, there are 64! = 1.27 × 10 89 possible 4-D mappings. For such a system, the binary switching algorithm (BSA) [14] can then be used. However, it should be emphasized that the BSA only gives locally optimal mappings. Furthermore, for a large constellation (such as high-order multi-D constellations), the BSA quickly becomes intractable and a more complicated searching inside a selected list must be applied [13]. Another searching technique, related to the quadratic assignment prob- lem (QAP), is also presented in [8]. This technique, however, has the same disadvantages with that of the BSA for larger constellations. The contribution of this paper is to introduce a novel design of multi-D mapping for BICM-ID that employs 8-PSK constellation. The proposed mapping exploits the symmetry of the 8-PSK constellation and the simple fact that an 8-PSK con- stellation can be decomposed into two QPSK constellations. In a 2-dimensional signal space, it can be easily verified that the proposed mapping is indeed the globally optimal mapping. For a higher dimension, unfortunately, whether the proposed mapping is the globally optimal mapping still remains to be answered. Nevertheless, by comparing the performance of the proposed mapping with an unachievable lower bound, it is shown that there is only a very small gap between the performance of the proposed mapping and the bound. It is therefore conjectured that the proposed mapping is actually the globally optimal mapping. II. SYSTEM MODEL ) ; ~ ( O c P ) ; ~ ( I c P ) ; ( O c P ) ; ( I c P ) ; ( O u P Fig. 1. Block diagram of a BICM-ID system. Fig. 1 shows the block diagram of a BICM-ID system. For This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings. 1-4244-0355-3/06/$20.00 (c) 2006 IEEE 5004