Improving the Performance of QPSK BICM-ID by Mapping on the Hypercube Nghi H. Tran and Ha H. Nguyen Department of Electrical Engineering University of Saskatchewan Saskatoon, SK, Canada S7N 5A9 hut461@engr.usask.ca, ha nguyen@engr.usask.ca Abstract— This paper introduces a new mapping of QPSK signals, viewed as a vertices of a multi-dimensional hypercube, to improve the performance of bit-interleaved coded modulation with iterative decoding (BICM-ID) over a Rayleigh fading chan- nel. The distance criterion to find the best mapping in terms of asymptotic performance is analytically derived. A general algorithm to construct the best mapping of a hypercube is then proposed. Numerical and simulation results show that the proposed mapping offers a significant coding gain over the conventional mappings of QPSK in a BICM-ID system. Such a coding gain is obtained without any bandwidth nor power expansion and with a small increase in the receiver complexity. I. I NTRODUCTION Bit-interleaved coded-modulation with iterative decoding (BICM-ID) is a spectral efficiency coded modulation technique for both additive white Gaussian noise (AWGN) and fading channels. It has been shown that for fixed signal constellation, interleaver and error control code, signal mapping plays an important role in determining the error performance of a BICM-ID system. A substantial amount of work has been carried out to address this mapping problem [1]–[3]. For example, reference [2] shows that, while Gray mapping of QPSK is the best for BICM (where no iteration is implemented between the channel decoder and the demodulator), it is the anti-Gray mapping that is preferred for BICM-ID 1 . Other research studies signal mappings of higher-order constellations in either one or two-dimensional signal spaces (such as PAM, PSK and QAM). By considering many symbols over a multiple symbol interval, a larger constellation is effectively created. Such a larger constellation in a higher dimensional space offers a more flexibility for mapping design. The benefit provided by multi-dimensional constellations has also been known in the design of trellis coded modulation. Motivated by the above observation, this paper studies sig- nal mapping in a multi-dimensional constellation to improve the performance of BICM-ID. More specifically, the system employing multi-dimensional hypercube built from QPSK is considered. By evaluating the upper bound for the bit error rate (BER) of the system, a distance criterion is derived to find the best mapping in terms of asymptotic performance. 1 Note that Gray and anti-Gray mappings are the only two mappings available for a QPSK constellation. Based on this distance criterion, an algorithm to construct the best mapping is then proposed. The paper is organized as follows. Section II briefly intro- duces the BICM-ID system employing hypercube constellation constructed from QPSK. In Section III, the upper bound on the BER performance of the system is then derived, from which a distance criterion is proposed to find the best mapping. The distance properties of a hypercube are studied in Section IV. Also presented in this section is the algorithm to construct the best mapping. Numerical and simulation results are provided in Section V to demonstrate the advantages of the proposed system. Finally, conclusions are draw in Section VI. II. SYSTEM MODEL BICM-ID systems are based on the concatenation of a con- volutional encoder, an interleaver and a M -ary modulator as shown in Fig. 1. This paper considers a BICM-ID system that Fig. 1. The block diagram of a BICM-ID system. employs QPSK modulation and a rate-1/2 convolutional code. This combination yields a spectral efficiency of 1 bit/s/Hz. Here, instead of mapping two coded bits to one QPSK symbol as in traditional systems, a group of 2n coded bits, n> 1, is mapped to n consecutive QPSK symbols. Since a QPSK constellation is built from two quadrature (orthogonal) carriers, there are 4 n distinct combinations of n QPSK symbols and they are the vertices of a 2n-dimensional hypercube (2n-cube). Obviously, using Gray or anti-Gray mapping for a QPSK constellation is just one special case of the above general 0-7803-8521-7/04/$20.00 (C) 2004 IEEE 1299 0-7803-8521-7/04/$20.00 © 2004 IEEE