Optimal Alphabets and Binary Labelings for BICM at Low SNR

Abstract

Abstract-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 dB. It is shown that the E b /N0 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 firstorder 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.

Conditions for a Monotonic Channel Capacity

“…In this case, the mutual information is governed by the mean value of the source distribution, according to [43,Th. 7].…”

On the equivalence of TCM encoders

“…Table I) can be generated by multiplying the matrices in R 2 and in T 2 . As a consequence of Theorems 1 and 2, the two TCM encoders [G, L] and [GT −1 , L R ] are equivalent for any G ∈ G k,m,ν and L ∈ L m , where L R and T are given by the factorization (7). In other words, all nonequivalent TCM encoders can be generated using one member of each modified Hadamard class only, and thus, a joint optimization over all G ∈ G k,m,ν and L ∈ L m can be reduced to an optimization over all G ∈ G k,m,ν and L ∈ R m with no loss in performance.…”

“…Of particular interest are the binary reflected Gray code (BRGC) [6] and the natural binary code (NBC) [7]. The BRGC is often used in BICM designs because it maximizes the BICM mutual information for medium and high signal-to-noise ratios [5,Sec.…”

Improved exponents and rates for bit-interleaved coded modulation

“…In short, BICM 1 is the classical BICM in [8] with equiprobable symbols. When the input probabilities are optimized for BICM 1 , we get the optimized BICM scheme in [11], [12]. In BICM 2 and BICM 3 , both mismatched receivers assume equiprobable Q X|Bj (x|b).…”