Bit-Interleaved Coded Modulation

The principle of coding in the signal space follows directly from Shannon's analysis of waveform Gaussian channels subject to an input constraint. The early design of communication systems focused separately on modulation, namely signal design and detection, and error correcting codes, which deal with errors introduced at the demodulator of the underlying waveform channel. The correct perspective of signal- space coding, although never out of sight of information theorists, was brought back into the focus of coding theorists and system designers by Imai's and Ungerblock's pioneering works on coded modulation. More recently, powerful families of binary codes with a good tradeo ff between performance and decoding complexity have been (re-) discovered. Bit-Interleaved Coded Modulation (BICM) is a pragmatic approach combining the best out of both worlds: it takes advantage of the signal- space coding perspective, whilst allowing for the use of powerful families of binary codes with virtually any modulation format. BICM avoids the need for the complicated and somewhat less flexible design typical of coded modulation. As a matter of fact, most of today's systems that achieve high spectral e ffi ciency such as DSL, Wireless LANs, WiMax and evolutions thereof, as well as systems based on low spectral efficiency orthogonal modulation, feature BICM, making BICM the defacto general coding technique for waveform channels. The theoretical characterization of BICM is at the basis of e ffi cient coding design techniques and also of improved BICM decoders, e. g., those based on the belief propagation iterative algorithm and approximations thereof. In this text, we review the theoretical foundations of BICM under the uni fi ed framework of error exponents for mismatched decoding. This framework allows an accurate analysis without any particular assumptions on the length of the interleaver or independence between the multiple bits in a symbol. We further consider the sensitivity of the BICM capacity with respect to the signal-to-noise ratio (SNR), and obtain a wideband regime (or low-SNR regime) characterization. We review e ffi cient tools for the error probability analysis of BICM that go beyond the standard approach of considering in fi nite interleaving and take into consideration the dependency of the coded bit observations introduced by the modulation. We also present bounds that improve upon the union bound in the region beyond the cuto ff rate, and are essential to characterize the performance of modern randomlike codes used in concatenation with BICM. Finally, we turn our attention to BICM with iterative decoding, we review extrinsic information transfer charts, the area theorem and code design via curve fi tting. We conclude with an overview of some applications of BICM beyond the classical coherent Gaussian channel.