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

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

Achievable Rates for Four-Dimensional Coded Modulation with a Bit-Wise Receiver

We study achievable rates for four-dimensional (4D) constellations for spectrally efficient optical systems based on a (suboptimal) bit-wise receiver. We show that PM-QPSK outperforms the best 4D constellation designed for uncoded transmission by approximately 1 dB. Numerical results using LDPC codes validate the analysis

On BICM receivers for TCM transmission

Recent results have shown that the performance of bit-interleaved coded modulation (BICM) using convolutional codes in nonfading channels can be significantly improved when the interleaver takes a trivial form (BICM-T), i.e., when it does not interleave the bits at all. In this paper, we give a formal explanation for these results and show that BICM-T is in fact the combination of a TCM transmitter and a BICM receiver. To predict the performance of BICM-T, a new type of distance spectrum for convolutional codes is introduced, analytical bounds based on this spectrum are developed, and asymptotic approximations are also presented. It is shown that the minimum distance of the code is not the relevant optimization criterion for BICM-T. Optimal convolutional codes for different constrain lengths are tabulated and asymptotic gains of about 2 dB are obtained. These gains are found to be the same as those obtained by Ungerboeck's one-dimensional trellis coded modulation (1D-TCM), and therefore, in nonfading channels, BICM-T is shown to be asymptotically as good as 1D-TCM.Comment: Submitted to the IEEE Transactions on Communication

Towards Fully Optimized BICM Transceivers

Bit-interleaved coded modulation (BICM) transceivers often use equally spaced constellations and a random interleaver. In this paper, we propose a new BICM design, which considers hierarchical (nonequally spaced) constellations, a bit-level multiplexer, and multiple interleavers. It is shown that this new scheme increases the degrees of freedom that can be exploited in order to improve its performance. Analytical bounds on the bit error rate (BER) of the system in terms of the constellation parameters and the multiplexing rules are developed for the additive white Gaussian Noise (AWGN) and Nakagami-$m$ fading channels. These bounds are then used to design the BICM transceiver. Numerical results show that, compared to conventional BICM designs, and for a target BER of $10^{-6}$, gains up to 3 dB in the AWGN channel are obtained. For fading channels, the gains depend on the fading parameter, and reach 2 dB for a target BER of $10^{-7}$ and $m=5$.Comment: Submitted to the IEEE Transactions on Communication

Performance Prediction of Nonbinary Forward Error Correction in Optical Transmission Experiments

In this paper, we compare different metrics to predict the error rate of optical systems based on nonbinary forward error correction (FEC). It is shown that the correct metric to predict the performance of coded modulation based on nonbinary FEC is the mutual information. The accuracy of the prediction is verified in a detailed example with multiple constellation formats, FEC overheads in both simulations and optical transmission experiments over a recirculating loop. It is shown that the employed FEC codes must be universal if performance prediction based on thresholds is used. A tutorial introduction into the computation of the threshold from optical transmission measurements is also given.Comment: submitted to IEEE/OSA Journal of Lightwave Technolog

Increasing Achievable Information Rates via Geometric Shaping

Achievable information rates are used as a metric to design novel modulation formats via geometric shaping. The proposed geometrically shaped 256-ary constellation achieves SNR gains of up to 1.18 dB.Comment: Additional references have been adde

General BER Expression for One-Dimensional Constellations

A novel general ready-to-use bit-error rate (BER) expression for one-dimensional constellations is developed. The BER analysis is performed for bit patterns that form a labeling. The number of patterns for equally spaced M-PAM constellations with different BER is analyzed.Comment: To appear in the Proceedings of the IEEE Global Communications Conference (GLOBECOM) 2012. Remark 3 modifie

On Achievable Rates for Long-Haul Fiber-Optic Communications

Lower bounds on mutual information (MI) of long-haul optical fiber systems for hard-decision and soft-decision decoding are studied. Ready-to-use expressions to calculate the MI are presented. Extensive numerical simulations are used to quantify how changes in the optical transmitter, receiver, and channel affect the achievable transmission rates of the system. Special emphasis is put to the use of different quadrature amplitude modulation formats, channel spacings, digital back-propagation schemes and probabilistic shaping. The advantages of using MI over the prevailing $Q$-factor as a figure of merit of coded optical systems are also highlighted.Comment: Hard decision mutual information analysis added, two typos correcte

Eight-dimensional Polarization-ring-switching Modulation Formats

We propose two 8-dimensional (8D) modulation formats (8D-2048PRS-T1 and 8D-2048PRS-T2) with a spectral efficiency of 5.5 bit/4D-sym, where the 8 dimensions are obtained from two time slots and two polarizations. Both formats provide a higher tolerance to nonlinearity by selecting symbols with nonidentical states of polarization (SOPs) in two time slots. The performance of these novel 8D modulation formats is assessed in terms of the effective signal-to-noise ratio (SNR) and normalized generalized mutual information. 8D-2048PRS-T1 is more suitable for high SNRs, while 8D-2048PRS-T2 is shown to be more tolerant to nonlinearities. A sensitivity improvement of at least 0.25 dB is demonstrated by maximizing normalized generalized mutual information (NGMI). For a long-haul nonlinear optical fiber transmission system, the benefit of mitigating the nonlinearity is demonstrated and a reach increase of 6.7% (560 km) over time-domain hybrid four-dimensional two-amplitude eight-phase shift keying (TDH-4D-2A8PSK) is observed