Constellation-constrained Capacity Research Articles

Performance Improvement on Nonorthogonal Multiple Access without CSIT

In this paper, a downlink virtual-channel-optimization nonorthogonal multiple access (VNOMA) without channel state information at the transmitter (CSIT) is proposed. The novel idea is to construct multiple complex virtual channels by jointly adjusting the amplitudes and phases to maximize the minimum Euclidean distance (MED) among the superposed constellation points. The optimal solution is derived in the absence of CSIT. Considering practical communications with finite input constellations in which symbols are uniformly distributed, we resort to the sum constellation constrained capacity (CCC) to evaluate the performance. For MED criterion, the maximum likelihood (ML) decoder is expected at the receiver. To decrease the computational cost, we propose a reduced-complexity bitwise ML (RBML) decoder. Experimental results are presented to validate the superior of our proposed scheme.

Design and Analysis of Rotated-QAM Based Probabilistic Shaping Scheme for Rayleigh Fading Channels

Probabilistic shaping (PS) combined with bipolar amplitude shift keying modulation is a promising transmission scheme. However, such 1-dimensional PS scheme inevitably suffers shaping loss in Rayleigh fading channels due to the lack of signal diversity. In this paper, we design a rotated quadrature amplitude modulation based PS strategy. With the discrete inputs, the ergodic constellation constrained (CC) capacity is derived based on the mutual information conditioned on full channel state information available at the receiver, which is found to be mutually influenced by both the probability mass function (PMF) and the rotation angle (RA) of the input signals. Furthermore, the closed-form cutoff rate is also derived to simplify the analysis, which is the lower bound of the mutual information with finite decoding complexity. Correspondingly, the optimizations of RA and PMF are both considered for maximizing the ergodic CC capacity and the cutoff rate, respectively, where the single parameter RA is firstly obtained by the exhaustive search. And then, the optimal PMFs of the subproblem for the fixed RAs and the fixed constellation scaling factor are obtained. Furthermore, with this non-equiprobable transmission scheme, the closed-form pairwise error probability is derived for the first time, which also validates the superiority of the proposed PS scheme in comparison with the uniform schemes with elaborate rotations.

Design and Analysis of Probabilistic Shaping Scheme for Uplink Nonorthogonal Multiple Access Systems

Probabilistic shaping (PS) combined with bipolar amplitude shift keying modulation is a promising transmission scheme to increase spectral efficiency. Different from the existing design of the point to point communications, we propose a PS-based uplink nonorthogonal multiple access (NOMA) scheme for fading channels to support multiple connections simultaneously, where successive interference cancellation (SIC)-based maximum a posterior (MAP) detection is implemented at the receiver. To exploit the shaping gain for each user, we first theoretically analyze the ergodic channel capacity and the constellation constrained (CC) capacity with discrete-constellation inputs, respectively. And then, a multi-step optimization scheme for maximizing the ergodic CC capacity is proposed to obtain the optimal probability mass function (PMF) of these input signals. The order of optimization is inverse to the SIC decoding order so that for each specific user, the remaining multiuser interference is calculated based on the optimal PMF of the later decoded users. Furthermore, based on this non-equiprobable transmission scheme, the closed-form pairwise error probability expressions for each user are obtained for the first time. Finally, compared with the uniform scheme, the simulations show that the proposed PS NOMA strategies achieve more than 1 dB gain in terms of the error performance.

Optimal Minimum Euclidean Distance-Based Precoder for NOMA With Finite-Alphabet Inputs

Non-orthogonal multiple access (NOMA) scheme with Gaussian input signals, which have no constellation constraint on the input alphabets, has been investigated intensively, whereas the research on finite input constellations is relatively few. This paper focuses on the power allocation in practical NOMA application with finite-alphabet inputs. We propose a practical power allocation scheme for downlink NOMA scenario based on the merit of maximizing the minimum Euclidean distance (MMED) by adopting the constellation-constrained (CC) capacity. Applying the nearest neighbor approximation and Jensen's inequality, we prove that optimal CC capacity can be achieved under the MMED criterion at high signal-to-noise ratio (SNR). Instead of updating power allocation coefficient instantaneously, a preset fixed power factor α MMED derived from the MMED criterion is preferred and thus reduces the complexity of overall system implementation. We take two cases into consideration, namely, Gaussian broadcast channel (GBC) and fading broadcast channel (FBC). We model close channel conditions, which are challenging for conventional NOMA as the GBC scenario, and model near-far effect as well Rayleigh fading as the FBC scenario. In these two cases, we show that the optimal performance can be guaranteed at high SNR without the knowledge of channel state information (CSI). We study the power coefficient pairs for the most commonly used Mary quadrature amplitude modulation (M-QAM) constellations based on the MMED criterion. We explore the relationship between the power allocation coefficients and the sizes of constellations. The numerical simulations on CC sum capacity and capacity regions are provided to validate our analysis.

Broadcast Coded Modulation: Multilevel and Bit-Interleaved Construction

The capacity of the AWGN broadcast channel is achieved by superposition coding, but the superposition of individual coded modulations expands the modulation alphabet and distorts its configuration. Hierarchical modulations, which appear in the literature mostly in the context of unequal error protection, can approach only a few isolated points on the boundary of the broadcast capacity region. This paper studies multilevel coding (MLC) for constellation-constrained broadcast-coded modulation. The conditions under which multilevel codes can achieve the constellation-constrained capacity of the AWGN broadcast channel are derived. We propose a pragmatic multilevel design technique with near-constellation-constrained-capacity performance where the coupling of the superposition inner and outer codes is localized to each bit-level. It is shown that this can be further relaxed to a code coupling on only one bit-level, with a little or no penalty under natural labeling. The rate allocation problem between the bit levels of the two users is studied and a pragmatic method is proposed, again with near-capacity performance. In further pursuit of lower complexity, a hybrid MLC-bit-interleaved-coded modulation is proposed, whose performance is shown to be very close to the boundary of the constellation-constrained capacity region. Simulation results show that good point-to-point LDPC codes produce excellent performance in the proposed coded modulation framework.

Design and Analysis of Bit-Interleaved Coded Modulation With Pseudorandom Sequence

In this paper, we design and analyze an improved bit-interleaved coded modulation (CM) scheme with a pseudorandom sequence (BICM-P). For the proposed scheme, the pseudorandom sequence is introduced into a systematic encoder at the transmitter and serves as side information at the receiver. With this design, the receiver can enjoy the benefit of increased Hamming distance when applying this side information at the receiver. In particular, we present a new constellation mapping principle to take advantage of the side information in the modulator, which leads to different signal constellations for the information bits and the parity-check bits. We then analytically derive the constellation-constrained capacity for BICM-P and quantify its gain as a nonlinear function associated with the two constellations. For the practical use of BICM-P, this paper indicates that the side information makes the regular low-density parity-check (LDPC) code irregular. Accordingly, the density evolution analysis tool is developed to design and optimize the regular LDPC code for BICM-P. The performance of LDPC-coded BICM-P on additive white Gaussian noise (AWGN) channels and fast Rayleigh fading channels is analyzed using extensive simulations. We observe that, at high code rates $(\geqslant \mbox{1}/\mbox{2})$ , LDPC-coded BICM-P has a substantial gain (0.1117–0.6558 dB) compared with LDPC-coded BICM on the AWGN channel. The performance gains will become more significant (0.7541–2.3481 dB) on the fast Rayleigh fading channel.

Approaching Capacity Region for Two-user GBC with Bit Division Multiplexing

A two-user Gaussian broadcast channel (GBC) with finite input constellations is addressed in this paper. The constellation-constrained capacity region for the two-user GBC by recently proposed bit-division multiplexing (BDM) is derived. BDM can be seen as a practical implementation of constellation-constrained nonlinear superposition coding (SC). For comparison, the constellation-constrained capacity region for the two-user GBC by traditional constellation-constrained linear SC is also computed. With similar implementation complexity, BDM outperforms traditional constellation-constrained linear SC with lower shaping loss in the capacity defined in this paper. Analytical results show that the upper boundary of the capacity region for the two-user GBC can be well approached by BDM with amplitude phase-shift keying modulation, which could be considered as the preliminary demonstration on the applicability of BDM in practical systems.

A Novel Power Allocation Scheme for Two-User GMAC with Finite Input Constellations

Constellation Constrained (CC) capacity regions of two-user Gaussian Multiple Access Channels (GMAC) have been recently reported, wherein an appropriate angle of rotation between the constellations of the two users is shown to enlarge the CC capacity region. We refer to such a scheme as the Constellation Rotation (CR) scheme. In this paper, we propose a novel scheme called the Constellation Power Allocation (CPA) scheme, wherein the instantaneous transmit power of the two users are varied by maintaining their average power constraints. We show that the CPA scheme offers CC sum capacities equal (at low SNR values) or close (at high SNR values) to those offered by the CR scheme with reduced decoding complexity for QAM constellations. We study the robustness of the CPA scheme for random phase offsets in the channel and unequal average power constraints for the two users. With random phase offsets in the channel, we show that the CC sum capacity offered by the CPA scheme is more than the CR scheme at high SNR values. With unequal average power constraints, we show that the CPA scheme provides maximum gain when the power levels are close, and the advantage diminishes with the increase in the power difference.

Two-User Gaussian Interference Channel with Finite Constellation Input and FDMA

In the two-user Gaussian Strong Interference Channel (GSIC) with finite constellation inputs, it is known that relative rotation between the constellations of the two users enlarges the Constellation Constrained (CC) capacity region. In this paper, a metric for finding the approximate angle of rotation to maximally enlarge the CC capacity is presented. It is shown that for some portion of the Strong Interference (SI) regime, with Gaussian input alphabets, the FDMA rate curve touches the capacity curve of the GSIC. Even as the Gaussian alphabet FDMA rate curve touches the capacity curve of the GSIC, at high powers, with both the users using the same finite constellation, we show that the CC FDMA rate curve lies strictly inside the CC capacity curve for the constellations BPSK, QPSK, 8-PSK, 16-QAM and 64-QAM. It is known that, with Gaussian input alphabets, the FDMA inner-bound at the optimum sum-rate point is always better than the simultaneous-decoding inner-bound throughout the Weak Interference (WI) regime. For a portion of the WI regime, it is shown that, with identical finite constellation inputs for both the users, the simultaneous-decoding inner-bound enlarged by relative rotation between the constellations can be strictly better than the FDMA inner-bound.

Linear Precoding for MIMO Multiple Access Channels with Finite Discrete Inputs

In this paper, we study linear precoding for multiple-input multiple-output (MIMO) multiple access channels (MAC) with finite discrete inputs. We derive the constellation-constrained capacity region for the MIMO MAC with an arbitrary number of users and find that the boundary can be achieved by solving the problem of weighted sum rate maximization with constellation and individual power constraints. Due to the non-concavity of the objective function, we obtain a set of necessary conditions for the optimization problem through Karush-Kuhn-Tucker analysis. To find the optimal precoding matrices for all users, we propose an iterative algorithm utilizing alternating optimization strategy. In particular, each iteration of the algorithm involves the gradient descent update with backtracking line search. Numerical results show that when inputs are digital modulated signals and the signal-to-noise ratio is in the medium range, our proposed algorithm offers considerably higher sum rate than non-precoding and the traditional method which maximizes Gaussian-input sum capacity. Furthermore, a low-density parity-check coded system with iterative detection and decoding for MAC is presented to evaluate the bit error rate (BER) performance of precoders. BER results also indicate that the system with the proposed linear precoder achieves significant gains over the non-precoding system and the precoder designed for Gaussian inputs.

On Two-User Gaussian Multiple Access Channels With Finite Input Constellations

Constellation Constrained (CC) capacity regions of two-user Single-Input Single-Output (SISO) Gaussian Multiple Access Channels (GMAC) are computed for several Non-Orthogonal Multiple Access schemes (NO-MA) and Orthogonal Multiple Access schemes (O-MA). For NO-MA schemes, a metric is proposed to compute the angle(s) of rotation between the input constellations such that the CC capacity regions are maximally enlarged. Further, code pairs based on Trellis Coded Modulation (TCM) are designed with PSK constellation pairs and PAM constellation pairs such that any rate pair within the CC capacity region can be approached. Such a NO-MA scheme which employs CC capacity approaching trellis codes is referred to as Trellis Coded Multiple Access (TCMA). Then, CC capacity regions of O-MA schemes such as Frequency Division Multiple Access (FDMA) and Time Division Multiple Access (TDMA) are also computed and it is shown that, unlike the Gaussian distributed continuous constellations case, the CC capacity regions with FDMA are strictly contained inside the CC capacity regions with TCMA. Hence, for finite constellations, a NO-MA scheme such as TCMA is better than FDMA and TDMA which makes NO-MA schemes worth pursuing in practice for two-user GMAC. Then, the idea of introducing rotations between the input constellations is used to construct Space-Time Block Code (STBC) pairs for two-user Multiple-Input Single-Output (MISO) fading MAC. The proposed STBCs are shown to have reduced Maximum Likelihood (ML) decoding complexity and information-losslessness property. Finally, STBC pairs with reduced sphere decoding complexity are proposed for two-user Multiple-Input Multiple-Output (MIMO) fading MAC.

Coded multipulse pulse-position modulation for free-space optical communications

In this letter, we address two questions concerning the application of multipulse pulse-position modulation (MPPM) for free-space optical (FSO) communications: (1) under what condition would MPPM offer a notable advantage over conventional pulse-position modulation (PPM), and (2) what practical coding scheme should be used to realize that advantage. Focusing on bandwidth-efficient FSO transmission, we find decimated MPPM constellations which are suitable for combining with binary codes and offer significant gains in terms of constellation-constrained capacity over PPM under simultaneous peak-power, average-power, and bandwidth constraints. We then consider labeling design for the popular bit-interleaved coded modulation (BICM) and devise a new multilevel coding (MLC) architecture for MPPM, which we refer to as reduced-layer MLC (RLMLC). Considering the Poisson FSO channel model as a relevant example, we provide simulative evidence that RL-MLC MPPM with off-the-shelf low-density parity-check codes can outperform any PPM scheme under the same transmission constraints.

Serial Concatenated TCM With an Inner Accumulate Code—Part II: Density-Evolution Analysis

In a companion paper, we showed the existence of decoding thresholds for maximum-likelihood (ML) decoding of a serial concatenated trellis-coded modulation (SCTCM) system with one or more inner accumulate codes. In this paper, we compute the decoding thresholds for an iterative, non-ML decoder by density evolution (DE), assuming infinite blocklengths. We also derive a stability condition for the particular case of an outer parity-check code and a single inner accumulate code. We show that, for equiprobable signaling, the bit-wise log-likelihood ratio densities for higher order constellations are symmetric. Furthermore, when used in DE, these densities can be averaged without significantly affecting the resulting threshold values. For an outer single parity-check code, the lowest decoding thresholds are achieved with two inner accumulate codes. For an outer repeat code, a single inner accumulate code gives the lowest thresholds. At code rates r/sub c/>2/3, the decoding thresholds for the SCTCM system are within 1 dB of the constellation-constrained channel capacity for additive white Gaussian noise channels, and within 1.5 dB for independent, identically distributed Rayleigh channels. Simulation results verify the computed thresholds.

Design of multidimensional signal constellations

An algorithmic technique intended for signal constellation design for an N-dimensional Euclidean signal space is presented. Such signals are used for reliable and efficient digital communications on an additive white Gaussian noise channel. The minimum Euclidean distance between signals and the constellation-constrained capacity are used as performance measures. The basic idea is to spread out the signal vectors using an iterative method modelling the behaviour of equally charged particles in free space. Initially, the vectors are randomly distributed. The vectors reach equilibrium after a few iterations. Numerical calculations for N=2, 3, 4 and 5 dimensions and M=N+1, N+2, …,20 signals show that the generated codes are nearly as good as previously known codes, and in some cases even better.