# Optimal Alphabets and Binary Labelings for BICM at Low SNR Journal article, 2011

Optimal binary labelings, input distributions, and input alphabets are analyzed for the so-called bit-interleaved coded modulation (BICM) capacity, paying special attention to the low signal-to-noise ratio (SNR) regime. For 8-ary pulse amplitude modulation (PAM) and for 0.75 bit/symbol, the folded binary code results in a higher capacity than the binary reflected gray code (BRGC) and the natural binary code (NBC). The 1 dB gap between the additive white Gaussian noise (AWGN) capacity and the BICM capacity with the BRGC can be almost completely removed if the input symbol distribution is properly selected. First-order asymptotics of the BICM capacity for arbitrary input alphabets and distributions, dimensions, mean, variance, and binary labeling are developed. These asymptotics are used to define first-order optimal (FOO) constellations for BICM, i.e. constellations that make BICM achieve the Shannon limit \$-1.59 \tr{dB}\$. It is shown that the \$\Eb/N_0\$ required for reliable transmission at asymptotically low rates in BICM can be as high as infinity, that for uniform input distributions and 8-PAM there are only 72 classes of binary labelings with a different first-order asymptotic behavior, and that this number is reduced to only 26 for 8-ary phase shift keying (PSK). A general answer to the question of FOO constellations for BICM is also given: using the Hadamard transform, it is found that for uniform input distributions, a constellation for BICM is FOO if and only if it is a linear projection of a hypercube. A constellation based on PAM or quadrature amplitude modulation input alphabets is FOO if and only if they are labeled by the NBC; if the constellation is based on PSK input alphabets instead, it can never be FOO if the input alphabet has more than four points, regardless of the labeling.

bit-interleaved coded modulation

natural binary code

Gray code

Average mutual information

channel capacity

pulse amplitude modulation (PAM)

phase shift keying (PSK)

binary labeling

Shannon limit

folded binary code

## Author

#### Erik Agrell

Chalmers, Signals and Systems, Kommunikationssystem, informationsteori och antenner, Communication Systems

Chalmers, Signals and Systems, Kommunikationssystem, informationsteori och antenner, Communication Systems

#### IEEE Transactions on Information Theory

0018-9448 (ISSN)

Vol. 57 10 6650-6672 6034708

Information and Communication Technology

#### Subject Categories

Telecommunications

#### DOI

10.1109/TIT.2011.2162179