Finite Input Constellations 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.

A Low-Complexity Linear Precoding for MIMO Channels With Finite Constellation Inputs

This letter investigates linear precoding scheme that maximizes the mutual information (MI) of multiple-input multiple-output (MIMO) channel with arbitrary input constellations. Since traditional precoding schemes which rely on the relationship between MI and minimum mean-square error (MMSE) are too time consuming, this letter first proposes a lower bound of MI with which the computational complexity can be significantly reduced compared with direct computation of MI, and we prove that the bound is asymptotically optimal with a constant shift. Based on this lower bound, a low-complexity precoding scheme is obtained. Numerical results verify the effectiveness and high-performance of the proposed precoding method compared with traditional precoding schemes.

Optimal Linear Detection for MIMO Systems With Finite Constellation Inputs

To realize the high data rates with low complexity in MIMO systems, linear detections are commonly used where the received signal vector is first transformed to a new vector and then detected on per symbol basis. However, the typical linear detectors such as zero forcing detector and minimum mean-square error detector are not optimal in terms of maximizing the achievable data rate of MIMO system under linear detection constraint. In this letter, we investigate the optimal linear transformer that can maximize such rate with finite constellation inputs. By capitalizing on the relationship between mutual information and the MMSE between channel inputs and outputs, the optimal linear detector (OLD) can be expressed as a fixed-point equation and be solved iteratively. Analytical results prove the consistence between the proposed OLD and the joint maximum likelihood detector (JMLD) in the low- and high-SNR region. The further simulation results verify the consistence, and moreover, at moderate SNR, OLD can significantly improve the achievable data rate and is close to that of JMLD.

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.

Uplink Waveform Channel With Imperfect Channel State Information and Finite Constellation Input

This paper investigates the capacity limit of an uplink waveform channel assuming imperfect channel state information at the receiver (CSIR). Various realistic assumptions are incorporated into the problem, which make the study valuable for performance assessment of real cellular networks to identify potentials for performance improvements in practical receiver designs. We assume that the continuous-time received signal is first discretized by mismatched filtering based on the imperfect CSIR. The resulting discrete-time signals are then decoded considering two different decoding strategies, i.e., an optimal decoding strategy based on specific statistics of channel estimation errors and a sub-optimal decoding strategy treating the estimation error signal as additive Gaussian noise. Motivated by the proposed decoding strategies, we study the performance of the decision feedback equalizer for finite constellation inputs, in which inter-stream interferences are treated either using their true statistics or as Gaussian noise. Numerical results are provided to exemplify the benefit of exploiting the knowledge on the statistics of the channel estimation errors and inter-stream interferences. Simulations also assess the effect of the CSI imperfectness on the achievable rate, which reveal that finite constellation inputs are less sensitive to the estimation accuracy than Gaussian input, especially in the high SNR regime.

On Precoding for Constant <inline-formula> <tex-math notation="TeX">$K$</tex-math></inline-formula>-User MIMO Gaussian Interference Channel With Finite Constellation Inputs

This paper considers linear precoding for the constant channel-coefficient K-user MIMO Gaussian interference channel (MIMO GIC) where each transmitter-i (Tx-i) requires the sending of d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> independent complex symbols per channel use that take values from fixed finite constellations with uniform distribution to receiver-i (Rx-i) for i=1,2,..., K. We define the maximum rate achieved by Tx-i using any linear precoder as the signal-to-noise ratio (SNR) tends to infinity when the interference channel coefficients are zero to be the constellation constrained saturation capacity (CCSC) for Tx-i. We derive a high-SNR approximation for the rate achieved by Tx-i when interference is treated as noise and this rate is given by the mutual information between Tx-i and Rx-i, denoted as I[X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ;Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ]. A set of necessary and sufficient conditions on the precoders under which [IX <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ; Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ] tends to CCSC for Tx-i is derived. Interestingly, the precoders designed for interference alignment (IA) satisfy these necessary and sufficient conditions. Furthermore, we propose gradient-ascent-based algorithms to optimize the sum rate achieved by precoding with finite constellation inputs and treating interference as noise. A simulation study using the proposed algorithms for a three-user MIMO GIC with two antennas at each node with d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> =1 for all i and with BPSK and QPSK inputs shows more than 0.1-b/s/Hz gain in the ergodic sum rate over that yielded by precoders obtained from some known IA algorithms at moderate SNRs.

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.

Achievable Secrecy Rates for the Broadcast Channel with Confidential Message and Finite Constellation Inputs

This paper considers the Broadcast Channel with Confidential Message (BCCM) where the sender attempts to send altogether a common message to two receivers and a confidential message to one of them. The achievable rate regions are derived for the power-constrained Gaussian BCCM with finite input alphabet using various transmission strategies. Namely, time sharing, superposition modulation and superposition coding are used as broadcast strategies. For superposition modulation and superposition coding, the maximal achievable rate regions are obtained by maximizing over both constellation symbol positions and the joint probability distribution. The maximization of the secrecy rate for wiretap channels is also studied as a particular case of the BCCM problem. We compare the considered transmission strategies in terms of percentage gains in achievable rates. We concentrate on the impact of the finite alphabet constraint on achievable rates, and show that this constraint may change well known results obtained in the Gaussian case. We show also that the secrecy constraint can change the shape of the achievable rate region in superposition modulation used in some standards when symbols are equiprobable. On a more practical side, it is shown that a performance close to the optimum can be obtained by strategies with reduced complexity.

Improved Perfect Space-Time Block Codes

Perfect space-time block codes (STBCs) are based on four design criteria-full-rateness, nonvanishing determinant, cubic shaping, and uniform average transmitted energy per antenna per time slot. Cubic shaping and transmission at uniform average energy per antenna per time slot are important from the perspective of energy efficiency of STBCs. The shaping criterion demands that the generator matrix of the lattice from which each layer of the perfect STBC is carved be unitary. In this paper, it is shown that unitariness is not a necessary requirement for energy efficiency in the context of space-time coding with finite input constellations, and an alternative criterion is provided that enables one to obtain full-rate (rate of nt complex symbols per channel use for an nt transmit antenna system) STBCs with larger normalized minimum determinants than the perfect STBCs. Further, two such STBCs, one each for 4 and 6 transmit antennas, are presented and they are shown to have larger normalized minimum determinants than the comparable perfect STBCs which hitherto had the best-known normalized minimum determinants.

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.

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.

Analysis and Optimization of Interleave-Division Multiple-Access Communication Systems

The recently proposed interleave-division multiple-access (IDMA) system is a flexible spread-spectrum air-interface technique featuring low receiver complexity and high spectral efficiency. In IDMA, each user's chip sequence is interleaved by a distinct chip-level interleaver. The receiver employs a simple chip-level iterative multiuser detector to decode the user's data. Here we focus on the analysis and optimization of such an IDMA system. We show that IDMA can be viewed as a limit case of the conventional CDMA employing repetition code and random spreading. Two analytical tools, namely, the large-system performance approximation and the EXIT chart technique are tailored in the context of IDMA chip-level iterative detector to facilitate the system performance analysis and optimization. The spectral efficiencies of a low-rate coded IDMA system with equal power setting in both single-cell and multi-cell scenarios are then analyzed, from which it is seen that the coded IDMA system with turbo receiver is a spectral efficient multiple-access scheme. Finally, we consider optimal power allocation among users in IDMA to maximize the spectral efficiency with finite-alphabet constellation. The differential evolution technique for nonlinear optimization is adopted to solve the power profile optimization problem. With optimized power profiles, the low-rate coded IDMA system can approach the optimal spectral efficiency with finite input constellations