High-SNR Asymptotics of Mutual Information for Discrete Constellations with Applications to BICM

Asymptotic expressions of the mutual information between any discrete input and the corresponding output of the scalar additive white Gaussian noise channel are presented in the limit as the signal-to-noise ratio (SNR) tends to infinity. Asymptotic expressions of the symbol-error probability (SEP) and the minimum mean-square error (MMSE) achieved by estimating the channel input given the channel output are also developed. It is shown that for any input distribution, the conditional entropy of the channel input given the output, MMSE, and SEP have an asymptotic behavior proportional to the Gaussian Q-function. The argument of the Q-function depends only on the minimum Euclidean distance (MED) of the constellation and the SNR, and the proportionality constants are functions of the MED and the probabilities of the pairs of constellation points at MED. The developed expressions are then generalized to study the high-SNR behavior of the generalized mutual information (GMI) for bit-interleaved coded modulation (BICM). By means of these asymptotic expressions, the long-standing conjecture that Gray codes are the binary labelings that maximize the BICM-GMI at high SNR is proven. It is further shown that for any equally spaced constellation whose size is a power of two, there always exists an anti-Gray code giving the lowest BICM-GMI at high SNR.Research supported by the European Community’s Seventh’s Framework Programme (FP7/2007-2013) under grant agreements No. 271986 and No. 333680, by the Swedish Research Council, Sweden (under grants #621-2006-4872 and #621-2011-5950) and by the Ministerio de Economía y Competitividad of Spain (TEC2009-14504-C02-01, CSD2008-00010, and TEC2012-38800-C03-01)

On the BICM Capacity

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.Comment: Submitted to the IEEE Transactions on Information Theor

High-SNR Asymptotics of Mutual Information for Discrete Constellations

The high-signal-to-noise ratio (SNR) asymptotic behavior of the mutual information (MI) for discrete constellations over the scalar additive white Gaussian noise channel is studied. Exact asymptotic expressions for the MI for arbitrary one-dimensional constellations and input distributions are presented in the limit as the SNR tends to infinity. Using the relationship between the MI and the minimum mean-square error (MMSE), asymptotics of the MMSE are also developed. It is shown that for any input distribution, the MI and the MMSE have an asymptotic behavior proportional to a Gaussian Q-function, whose argument depends only on the minimum Euclidean distance of the constellation and the SNR. Closed-form expressions for the coefficients of these Q-functions are calculated

Bit-interleaved coded modulation in the wideband regime

The wideband regime of bit-interleaved coded modulation (BICM) in Gaussian channels is studied. The Taylor expansion of the coded modulation capacity for generic signal constellations at low signal-to-noise ratio (SNR) is derived and used to determine the corresponding expansion for the BICM capacity. Simple formulas for the minimum energy per bit and the wideband slope are given. BICM is found to be suboptimal in the sense that its minimum energy per bit can be larger than the corresponding value for coded modulation schemes. The minimum energy per bit using standard Gray mapping on M-PAM or M^2-QAM is given by a simple formula and shown to approach -0.34 dB as M increases. Using the low SNR expansion, a general trade-off between power and bandwidth in the wideband regime is used to show how a power loss can be traded off against a bandwidth gain.Comment: Submitted to IEEE Transactions on Information Theor

Signal Shaping for BICM at Low SNR

The mutual information of bit-interleaved coded modulation (BICM) systems, sometimes called the BICM capacity, is investigated at low signal-to-noise ratio (SNR), i.e., in the wideband regime. A new linear transform that depends on bits' probabilities is introduced. This transform is used to prove the asymptotical equivalence between certain BICM systems with uniform and nonuniform input distributions. Using known results for BICM systems with a uniform input distribution, we completely characterize the combinations of input alphabet, input distribution, and binary labeling that achieve the Shannon limit -1.59 dB. The main conclusion is that a BICM system achieves the Shannon limit at low SNR if and only if it can be represented as a zero-mean linear projection of a hypercube, which is the same condition as for uniform input distributions. Hence, probabilistic shaping offers no extra degrees of freedom to optimize the low-SNR mutual information of BICM systems, in addition to what is provided by geometrical shaping. These analytical conclusions are confirmed by numerical results, which also show that for a fixed input alphabet, probabilistic shaping of BICM can improve the mutual information in the low and medium SNR range over any coded modulation system with a uniform input distribution

Amplifier Limited Information Rates in High-Speed Optical Fiber Communication Systems

Due to the high transmission capacity, optical fiber systems have been extensively applied, as significant components, in the modern telecommunication infrastructure to meet the ever-increasing demand of data traffic. Optical amplifiers have been employed to amplify optical signals and to compensate for the transmission losses. They play a key role in relaying the signals in ultra-wideband optical fiber communication systems. However, the amplified spontaneous emission (ASE) noise will be introduced during such process, and it will degrade the performance of optical fiber systems and will pose constraints on the transmission information rates. The mutual information (MI) and the generalized mutual information (GMI) have been applied and investigated, as figures of merit, to evaluate the information rates in communication systems. The MI measures the highest achievable information rate (bits per symbol) that can be realized in a channel based on ideal symbol-wise encoder and decoder. The GMI, also known as the bit-interleaved coded modulation (BICM) capacity, indicates an upper bound on the number of bits per symbol that can be reliably transmitted through a channel based on the bit-wise decoding. Although the MI and the GMI are equal when the signal-to-noise ratio (SNR) tends to infinity, the MI is strictly higher than the GMI in any practical transmission scenarios. This discrepancy depends on the constellation cardinality and the binary labeling. In this work, we have investigated the impact of ASE noise on the MI and the GMI, and have developed corresponding analyses and estimations across different modulation formats, in linear optical fiber communication systems. Our work aims to explore the limit and requirements on optical amplifiers and to provide a comprehensive insight for the design of next-generation ultra-wideband optical fiber communication systems

A Simple Approximation for the Bit-Interleaved Coded Modulation Capacity

The generalized mutual information (GMI) is an achievable rate for bit-interleaved coded modulation (BICM) and is highly dependent on the binary labeling of the constellation. The BICM-GMI, sometimes called the BICM capacity, can be evaluated numerically. This approach, however, becomes impractical when the number of constellation points and/or the constellation dimensionality grows, or when many different labelings are considered. A simple approximation for the BICM-GMI based on the area theorem of the demapper's extrinsic information transfer (EXIT) function is proposed. Numerical results show the proposed approximation gives good estimates of the BICM-GMI for labelings with close to linear EXIT functions, which includes labelings of common interest, such as the natural binary code, binary reflected Gray code, etc. This approximation is used to optimize the binary labeling of the 32-APSK constellation defined in the DVB-S2 standard. Gains of approximately 0.15 dB are obtained

Amplifier limited information rates in high-speed optical fiber communication systems

Due to the high transmission capacity, optical fiber systems have been widely applied in the modern telecommunication infrastructure to meet the ever-increasing demand of data traffic. Optical amplifiers have been employed to amplify optical signals and to compensate for the transmission losses. They play a key role in relaying the signals in ultra-wideband optical fiber communication systems. However, the amplified spontaneous emission (ASE) noise will be introduced and will pose constraints on the transmission information rates. The mutual information (MI) and the generalized mutual information (GMI) have been applied to evaluate the information rates in communication systems. In this work, we have investigated the impact of ASE noise on the MI and the GMI, and developed corresponding analyses across different modulation formats. Our work aims to explore the limit and requirements on optical amplifiers in next-generation ultra-wideband optical fiber communication systems