A Coding Theorem for Bit-Interleaved Coded Modulation

where SNR is the signal-to-noise ratio. For SNR ≪ 1 capacity can be achieved using binary linear codes and BPSK modulation, whereas for SNR ≫ 1 we need M -ary modulation with M > 2. A combination of M -ary modulation with M -ary linear codes cannot achieve Shannon capacity since the symbols occur with equal probability and a Gaussian input is required to achieve the capacity C; with M -ary linear codes we achieve the so-called uniform capacity C M = I(S;Y ), where S denotes the modulation symbols, and Y the channel output. If we want to restrict ourselves to binary codes we can combine 2-QAMmodulation with set-partitioning and trellis coding. The decoder for these Ungerboeck [1] codes is based on the Viterbi algorithm. Multi-level codes (Imai and Hirakawa [2]) can be considered as a generalization of Ungerboeck codes. Let us denote the bits in a modulation symbol by B1, . . . , BK . Starting by bit BK , a multi-stage decoder sequentially peels off the bit layers Bj, using the results of previous layers (Bj+1, . . . , BK). A code CK protects bit BK if the rate RK satisfies RK < I(BK ;Y ). Similarly, the following bit, BK−1, is successfully decoded if the rate RK−1 is chosen so that RK−1 < I(BK−1;Y,BK). In principle this method can achieve the uniform capacity C 2K (Huber and Wachsmann [3]), that is there are codes with rates Rj satisfying