Bit-Interleaved Coded Modulation: Low Complexity Decoding Enis Akay and Ender Ayanoglu Center for Pervasive Communications and Computing Department of Electrical Engineering and Computer Science The Henry Samueli School of Engineering University of California, Irvine Irvine, California 92697-2625 Email: eakay@uci.edu ayanoglu@uci.edu Abstract— It has been shown by Zehavi that the performance of coded modulation can be improved over a Rayleigh fading channel by bit-wise interleaving at the encoder output, and by using an appropriate soft-decision metric for a Viterbi decoder at the receiver. Caire et al presented the details of the theory behind bit-interleaved coded modulation (BICM). In this paper we show that for Gray encoded M-ary quadra- ture amplitude modulation (QAM) systems, the bit metrics of BICM can be further simplified. In QAM systems, the maximum likelihood (ML) detector for BICM uses the minimum distance between the received symbol and M/2 constellation points on the complex plane as soft-decision metrics. We show that soft- decision bit metrics for the ML decoder can be further simplified to the minimum distance between the received symbol and √ M/2 constellation points on the real line R 1 . This reduces the number of calculations needed for each bit metric sub- stantially, and therefore reduces the complexity of the decoder without compromising the performance. Simulation results for single carrier modulation (SCM), and multi-carrier modulation (MCM) systems over additive white Gaussian noise (AWGN) and Rayleigh fading channels agree with our findings. In addition, we tie this result to the decoding methods for bit interleaved convolutional code standards used in industry. I. I NTRODUCTION The increasing interest and importance of wireless com- munications over the past couple of decades have led the consideration of coded modulation [1] for fading channels. It is known that, even for fading channels, the probability of error can be decreased exponentially with average signal to noise ratio using optimal diversity. Naturally, at first, several approaches using Ungerboeck’s method of keeping coding combined with modulation are applied over fading channels, as summarized in [2]. These approaches considered the performance of a trellis coded system that is based on a symbol-by-symbol interleaver with a trellis code. The order of diversity for any coded system with a symbol interleaver is the minimum number of distinct symbols between codewords. Thus, diversity can only be increased by preventing parallel transitions and increasing the constraint length of the code. In 1989 Viterbi et al [3] introduced a different approach. They designed schemes to keep their basic engine an off- the-shelf Viterbi decoder. This resulted in leaving the joint decoder/demodulator for two joint entities. Zehavi [4] later realized that the code diversity, and there- fore the reliability of coded modulation over a Rayleigh channel, could be improved. Using bit-wise interleaving and an appropriate soft-decision bit metric 1 at a Viterbi decoder, Zehavi achieved to make the code diversity equal to the smallest number of distinct bits, rather than channel symbols, along any error event. This leads to a better coding gain over a fading channel when compared to TCM, [4]. Following Zehavi’s paper, Caire et al [5] presented the the- ory behind BICM. Their work illustrated tools to evaluate the performance of BICM with tight error probability bounds, and design guidelines. In Section II we present a brief overview of BICM, and refer the reader to [5] for details. In QAM systems, the ML detector for BICM uses the minimum distance between the received symbol and M/2 con- stellation points on the complex plane as soft-decision metrics. In Section III, we show that soft-decision bit metrics for the ML decoder can be further simplified to the minimum distance between the received symbol and √ M/2 constellation points on the real line R 1 . This reduces the number of calculations needed for each bit metric substantially, and therefore reduces the complexity of the decoder without compromising the performance. Simulation results supporting our findings for SCM and MCM over AWGN and Rayleigh channels are presented in Section IV. We finish our paper with a brief conclusion in Section V, where we summarize our findings. II. BIT-I NTERLEAVED CODED MODULATION (BICM) BICM can be obtained by using a bit interleaver, π, be- tween an encoder for a binary code C and an N -dimensional memoryless modulator over a signal set χ ⊆ C N of size |χ| = M =2 m with a binary labeling map μ : {0, 1} m → χ. During transmission, the code sequence c is interleaved by π, and then mapped onto signal sequence x ∈ χ. The signal sequence x is then transmitted over the channel. The bit interleaver can be modeled as π : k → (k ′ ,i) where k denotes the original ordering of the coded bits c k , k ′ denotes 1 Note the use of the metric in this paper follows convolutional coding nomenclature and is not in the strict mathematical sense.