Multidimensional Voronoi constellations (VCs) have been shown to be more power-efficient than quadrature amplitude modulation (QAM) formats given the same uncoded bit error rate, and also have higher achievable information rates. However, a coded modulation scheme that sustains these gains after forward error correction (FEC) coding is still lacking. This paper designs coded modulation schemes with soft-decision FEC codes for VCs, including bit-interleaved coded modulation (BICM) and multilevel coded modulation (MLCM), together with three bit-to-integer mapping algorithms and log-likelihood ratio calculation algorithms. Simulation results show that VCs can achieve up to 1.84 dB signal-to-noise ratio (SNR) gains over QAM with BICM, and up to 0.99 dB SNR gains over QAM with MLCM for the additive white Gaussian noise channel at the bit error rate of$1.81\times 10^{-3}$, with a low decoding complexity.

The capacity of a discrete-time channel with correlated phase noises is investigated. In particular, the electro-optic frequency comb system is considered, where the phase noise of each subchannel is a combination of two independent Wiener phase-noise sources. Capacity upper and lower bounds are derived for this channel and are compared with lower bounds obtained by numerically evaluating the achievable information rates using quadrature amplitude modulation constellations. Capacity upper and lower bounds are provided for the high signal-to-noise ratio (SNR) regime. The multiplexing gain (pre-log) is shown to beM− 1, whereMrepresents the number of subchannels. A constant gap between the asymptotic upper and lower bounds is observed, which depends on the number of subchannelsM. For the specific case ofM= 2, capacity is characterized up to a term that vanishes as the SNR grows large.

In optical fiber communication, due to the random variation of the environment, the state of polarization (SOP) fluctuates randomly with time leading to distortion and performance degradation. The memory-less SOP fluctuations can be regarded as a two-by-two random unitary matrix. In this paper, for what we believe to be the first time, the capacity of the polarization drift channel under an average power constraint with imperfect channel knowledge is characterized. An achievable information rate (AIR) is derived when imperfect channel knowledge is available and is shown to be highly dependent on the channel estimation technique. It is also shown that a tighter lower bound can be achieved when a unitary estimation of the channel is available. However, the conventional estimation algorithms do not guarantee a unitary channel estimation. Therefore, by considering the unitary constraint of the channel, a data-aided channel estimator based on the Kabsch algorithm is proposed, and its performance is numerically evaluated in terms of AIR. Monte Carlo simulations show that Kabsch outperforms the least-square error algorithm. In particular, with complex, Gaussian inputs and eight pilot symbols per block, Kabsch improves the AIR by 0.20 to 0.30 bits/symbol throughout the range of studied signal-to-noise ratios.

Voronoi constellations (VCs) are finite sets of vectors of a coding lattice enclosed by the translated Voronoi region of a shaping lattice, which is a sublattice of the coding lattice. In conventional VCs, the shaping lattice is a scaled-up version of the coding lattice. In this paper, we design low-complexity VCs with a cubic coding lattice of up to 32 dimensions, in which pseudo-Gray labeling is applied to minimize the bit error rate. The designed VCs have considerable shaping gains of up to 1.03 dB and finer choices of spectral efficiencies in practice compared with conventional VCs. A mutual information estimation method and a log-likelihood approximation method based on importance sampling for very large constellations are proposed and applied to the designed VCs. With error-control coding, the proposed VCs can have higher information rates than the conventional scaled VCs because of their inherently good pseudo-Gray labeling feature, with a lower decoding complexity.

In a classical 1983 paper, Conway and Sloane presented fast encoding and decoding algorithms for a special case of Voronoi constellations (VCs), for which the shaping lattice is a scaled copy of the coding lattice. Feng generalized their encoding and decoding methods to arbitrary VCs. Less general algorithms were also proposed by Kurkoski and Ferdinand, respectively, for VCs with some constraints on their coding and shaping lattices. In this work, we design VCs with a cubic coding lattice based on Kurkoski's encoding and decoding algorithms. The designed VCs achieve up to 1.03 dB shaping gains with a lower complexity than Conway and Sloane's scaled VCs. To minimize the bit error rate (BER), pseudo-Gray labeling of constellation points is applied. In uncoded systems, the designed VCs reduce the required SNR by up to 1.1 dB at the same BER, compared with the same VCs using Feng's and Ferdinand's algorithms. In coded systems, the designed VCs are able to achieve lower BER than the scaled VCs at the same SNR. In addition, a Gray penalty estimation method for such VCs of very large size is introduced.

The standard solution for short-haul fiber-optic communications is to deploy noncoherent systems, i.e., to modulate and detect only the light intensity. In such systems, the signal is corrupted with optical noise from amplifiers and with thermal (electrical) noise. The capacity of noncoherent optical links has been studied extensively in the presence of either optical noise or thermal noise. In this paper, for the first time, we characterize the capacity under an average power constraint with both noise sources by establishing upper and lower bounds. In the two extreme cases of zero optical noise or zero thermal noise, we assess our bounds against some well-known results in the literature; improvements in both cases are observed. Next, for amplified fiber-optic systems, we study the trade-off between boosting signal energy (mitigating the effects of thermal noise) and adding optical noise. For a wide spectrum of system parameters and received power levels, we determine the optimal amplification gain. While mostly either no amplification or high-gain amplification is optimal, the best performance is for some parameter intervals achieved at finite gains.

An analytical discrete-time model is introduced for single-wavelength polarization multiplexed nonlinear fiber-optical channels based on the symmetrized split-step Fourier method (SSFM). According to this model, for high enough symbol rates, a fiber-optic link can be described as a linear dispersive channel with additive white Gaussian noise (AWGN) and a complex scaling. The variance of this AWGN noise and the attenuation are computed analytically as a function of input power and channel parameters. The results illustrate a cubic growth of the noise variance with input power. Moreover, the cross effect between the two polarizations and the interaction of amplifier noise and the transmitted signal due to the nonlinear Kerr effect are described. In particular, it is found that the channel noise variance in one polarization is affected twice as much by the transmitted power in that polarization than by the transmitted power in the orthogonal polarization. The effect of pulse shaping is also investigated through numerical simulations. Finally, it is shown that the analytical performance results based on the new model are in close agreement with numerical results obtained using the SSFM for a symbol rate of 28 Gbaud and above.

We optimize modulation formats for the additive white Gaussian noise channel with nonnegative input, also known as the intensity-modulated direct-detection channel, with and without confining them to a lattice structure. Our optimization criteria are the average electrical, average optical, and peak power. The nonnegative constraint on the input to the channel is translated into a conical constraint in signal space, and modulation formats are designed by sphere packing inside this cone. Some dense packings are found, which yield more power-efficient modulation formats than previously known. For example, at a spectral efficiency of 1.5 bit/s/Hz, the modulation format optimized for average electrical power has a 2.55 dB average electrical power gain over the best known format to achieve a symbol error rate of 10-6. The corresponding gains for formats optimized for average and peak optical power are 1.35 and 1.72 dB, respectively. Using modulation formats optimized for peak power in average-power limited systems results in a smaller power penalty than when using formats optimized for average power in peak-power limited systems. We also evaluate the modulation formats in terms of their mutual information to predict their performance in the presence of capacity-achieving error-correcting codes, and finally show numerically and analytically that the optimal modulation formats for reliable transmission in the wideband regime have only one nonzero point.

We optimize modulation formats for the additive white Gaussian noise channel with a nonnegative input constraint, also known as the intensity-modulated direct detection channel, with and without confining them to a lattice structure. Our optimization criteria are the average electrical and optical power. The nonnegativity input signal constraint is translated into a conical constraint in signal space, and modulation formats are designed by sphere packing inside this cone. Some remarkably dense packings are found, which yield more power-efficient modulation formats than previously known. For example, at a spectral efficiency of 1 bit/s/Hz, the obtained modulation format offers a 0.86 dB average electrical power gain and 0.43 dB average optical power gain over the previously best known modulation formats to achieve a symbol error rate of 10-6. This modulation turns out to have a lattice-based structure. At a spectral efficiency of 3/2 bits/s/Hz and to achieve a symbol error rate of 10-6, the modulation format obtained for optimizing the average electrical power offers a 0.58 dB average electrical power gain over the best lattice-based modulation and 2.55 dB gain over the best previously known format. However, the modulation format optimized for average optical power offers a 0.46 dB average optical power gain over the best lattice-based modulation and 1.35 dB gain over the best previously known format.

We present a closed-form maximum likelihood-based data detection algorithm for long-haul optical channels with dominant nonlinear phase noise induced by self-phase modulation. The closed-form detector is evaluated in terms of symbol error rate as a function of input power, and compared with other sub-optimal detectors as well as a non-parametric detector. We show that the performance of the detector deteriorates for high input power levels yielding an optimal operation region. We also provide insights into the behavior of the detector in the highly nonlinear regime.

An exact expression is derived for the symbol error probability when the vertices of the four-dimensional 24-cell are used for digital transmission in Gaussian noise. Corresponding expressions for other regular convex polytopes are summarized. Numerically stable versions of these error probabilities are also obtained.

A multilevel coded modulation (MLCM) system in the presence of nonlinear phase noise for fiber optical communication is introduced. The proposed scheme exploits a 16-point ring constellation with nonlinear post compensation of the self phase modulation produced via the Kerr effect. A new set partitioning based on the Ungerboeck approach is introduced to maintain unequal error protection in amplitude and phase direction. The rate allocation for the MLCM component codes for different fiber lengths and transmit powers are done numerically. Simulation results show that the proposed MLCM system provides up to 2 dB gain over a forward error correcting scheme for a block error rates around 10^{-9}, with the same overhead (7%) and complexity.

We present a new rate allocation scheme for multilevel coded modulation based on the minimization of the total block error rate (BLER). The proposed method uses affine code components and hard decision multistage decoding. Exhaustive search for the rate allocation which minimizes the total BLER justifies the near-optimum performance of the introduced method in moderate to high SNRs. Compared to previous approaches this new rate allocation scheme can improve the performance of the system by 1 dB at BLER = 10-6for 16-QAM with Ungerboeck set partitioning. Interestingly, our results indicate that the optimum rate allocation is a function of the SNR. Finally, the performance of some specific codes are evaluated by simulation and union bounds to verify the theoretical results.