IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 5, MAY 2009 1351 Mapping Rearrangement for HARQ Based on Low-Order Modulation Leszek Szczecinski, Senior Member, IEEE, Andres Ceron, and Rodolfo Feick, Senior Member, IEEE Abstract—In this paper we consider hybrid automatic repeat request transmission based on binary phase shift keying mod- ulation. Our objective is to improve the performance of the retransmissions, keeping at the same time the complexity and the performance of the first transmission unaltered. We conclude that the so-called mapping rearrangement (MR) may considerably improve the performance if multi-dimensional modulation is applied. We evaluate the theoretical limits of bit-interleaved coded modulation with MR, verify the functioning of the practical coding scheme, and propose to improve the performance via it- erative detection-decoding (BICM-ID) which brings considerable gains. In particular, for high coding rates, a 4 dB SNR gain over transmission without MR may be achieved. Index Terms—Mapping rearrangement, constellation rear- rangement, mapping diversity, binary phase shift keying, au- tomatic repeat request, iterative detection, bit-interleaved coded modulation, BICM, ARQ, BICM-ID, BPSK, QPSK. I. I NTRODUCTION I N this work we propose to improve the performance of a hybrid automatic repeat request (HARQ) via the so-called mapping rearrangement (MR) when low-order modulations such as binary/quaternary phase shift-keying (BPSK/QPSK) are used. We show that in order to obtain MR gains, multi- dimensional (M-D) mappings should be used. We also pro- pose a simple coded modulation scheme capable of closely approaching the theoretical performance limits. This work targets improving the HARQ transmission [1][2], where the information bits are coded at the transmitter and eventual detection errors are handled via retransmissions. We are interested in HARQ where the same coded bits are sent and only their mapping onto the symbols changes throughout the transmissions; such a scenario may be classified as a HARQ type III (HARQ-III) [2]. It is known that appropriately designed mappings that vary throughout the transmissions offer performance gains if high-order modulation, e.g., 16- quadrature amplitude modulation (16-QAM), or 16-PSK is used, provided a joint detection of all transmitted packets, Paper approved by L. Rasmussen, the Editor for Electrical and Computer Engineering of the IEEE Communications Society. Manuscript received October 3, 2007; revised February 29, 2008 and April 4, 2008. L. Szczecinski is with the Institut National de la Recherche Scientifique, INRS-EMT, 800, Gauchetiere W. Suite 6900 Montreal, H5A 1K6, Canada (e-mail: leszek@emt.inrs.ca). A. Ceron was and R. Feick is with the Department of Electronics Engi- neering, Universidad Técnica Federico Santa María, Avenida Espana 1680, Valparaiso, Chile (e-mail: aceron@elo.utfsm.cl, feick@elo.utfsm.cl). This work was supported by NSERC, Canada (under research grant 249704- 02), by Fundacion Andes, Chile, and by Conicyt, Chile (under project PBCT- ACT-11/2004). It was presented at the IEEE Global Telecommunications Conference 2008, New Orleans, USA, Nov. 30-Dec. 4, 2008. Digital Object Identifier 10.1109/TCOMM.2009.05.070509 known as packet combining [3] [2], is carried out at the receiver. The gains of such a mapping rearrangement (MR) were initially shown in [4] and [5] and are also known as constella- tion rearrangement [6] or mapping diversity [7]. MR recently received more attention: new mappings were designed to minimize the bounds on uncoded bit error rate (MBER) after packet combining [7] or – to maximize the minimum squared Euclidean distance (MMSED) between the constellation points [8]. A form of MR was also included in the high speed downlink packet access (HSDPA) third-generation standard [9, Sec. 4.5.7][10][6] and the theoretical performance limits of MR for 16-QAM were analyzed in [11]. It is worth mentioning that the so-called incremental redundancy (IR) where each retransmission carries new parity bits is another option for HARQ. It requires the use of low-rate mother codes (and, when compared to MR, larger memory at the transmitter) to accommodate the possible rate decrease with retransmissions [12]. MR may thus be seen as a low-complexity alternative and/or complement to IR. In this work, we are interested in devising a MR mechanism for HARQ based on BPSK modulation which is much simpler to implement than high-order modulations (e.g., 16-QAM) known to provide MR gains [5][7]. In particular, the advantage of BPSK is its power efficiency (advantageous in mobile applications) and a robust time/frequency synchronization in low signal-to-noise ratio (SNR). We demonstrate that MR gains are attainable if we use multi-dimensional (M-D) modulation, i.e., when the bits are mapped into symbols constructed from temporally merged BPSK symbols. Although M-D modulation has already been used in the literature, our approach is novel through its applicative scope (HARQ) and a design based on desirable operational princi- ples for the resulting coding schemes, namely: • The proposed M-D MR comes into play only in retrans- missions so no constraints or cost is imposed on the first transmission. Consequently, we use the flexible bit- interleaved coded modulation (BICM), which is a natural choice for BPSK transmission. That is, we do not need to adjust the encoder’s structure to the M-D modulation as done in [13] to yield high spectral efficiency in the so-called trellis-coded modulation. • We propose M-D modulation to exploit the additional dimension offered by the retransmissions and to increase the capacity of the retransmission channel. That is, de- signing the mapping we actually change the constellation set which follows the implicit idea of MR in high-order 0090-6778/09$25.00 c  2009 IEEE