Finite SNR Research Articles

Finite SNR Diversity-Multiplexing Trade-Off in Hybrid ABCom/RCom-Assisted NOMA Networks

Finite SNR Diversity-Multiplexing Trade-Off in Hybrid ABCom/RCom-Assisted NOMA Networks

Ultra-high-cardinality geometric shaping in the finite SNR regime.

Four-dimensional (4D) constellations with up to 131 072 points (17 bit/4D-sym) are designed for the first time using geometric shaping. The constellations are optimized in terms of mutual information (MI) and generalized MI (GMI) for the additive white Gaussian noise (AWGN) channel, targeting a forward error correction (FEC) rate of 0.8 at finite signal-to-noise ratios. The presented 15-17 bit constellations are currently the highest-performing constellations in the literature, having a gap to the AWGN capacity as low as 0.17 dB (MI) and 0.45 dB (GMI) at 17 bit/4D-sym. For lower cardinalities, our constellations match or closely approach the performance of previously published optimized constellations. We also show that (GMI-)optimized constellations with a symmetry constraint, optimized for a FEC rate of 0.8, perform nearly identical to their unconstrained counterparts for cardinalities above 8 bit/4D-sym. A symmetry constraint for MI-optimized constellations is shown to have a negative impact in general. The proposed procedure relies on a Monte-Carlo-based approach for evaluating performance and is extendable to other (nonlinear) channels. Stochastic gradient descent is used for the optimization algorithm for which the gradients are computed using automatic differentiation. This article is part of the theme issue 'Celebrating the 15th anniversary of the Royal Society Newton International Fellowship'.

Closed-Form Hyper-Rayleigh Mode Analysis of the Fluctuating Double-Rayleigh with Line-of-Sight Fading Channel

The research studies hyper-Rayleigh behavior of a wireless communication system functioning in the presence of the generalized multipath fading. Although the initial metric (hyper-Rayleigh mode (HRM)) is quite informative, it is defined only asymptotically (i.e., for the infinitely increasing SNR). In spite of mathematical simplifications brought by such a definition (i.e., in this case most of the performance characteristics defining the HRM can be easily evaluated), it sufficiently limits its applications since, evidently, the real-life systems function in the presence of a finite (and usually not very high) SNR. The study presents a novel approach to the fading channel analysis (i.e., finite signal-to-noise ratio hyper-Rayleigh mode (fHRM)). The proposed metric (fHRM) is studied on the newly presented channel model-fluctuating double-Rayleigh with Line-of-Sight (fdRLoS) fading model. To accomplish this, the novel expressions for two channel-dependent system characteristics (i.e., the Amount of Fading (AoF) and the Outage Probability (OP)) were derived in exact form valid for arbitrary fading parameters. Based on the derived expressions, the finite SNR hyper-Rayleigh map is obtained, which helps to identify the parameters’ values corresponding to the specific propagation scenarios, which were further deployed to analyze the problem of the communication link physical layer security quantified in terms of the probability of strictly positive secrecy capacity (SPSC). Numerical verification of the derived closed-form expressions was performed. Several peculiarities of the system performance are observed and discussed.

Low-Complexity High-Performance Cyclic Caching for Large MISO Systems

Multi-antenna coded caching is known to combine a global caching gain that is proportional to the cumulative cache size found across the network, with an additional spatial multiplexing gain that stems from using multiple transmitting antennas. However, a closer look reveals two severe bottlenecks; the well-known exponential subpacketization bottleneck that dramatically reduces performance when the communicated file sizes are finite, and the considerable optimization complexity of beamforming multicast messages when the SNR is finite. We here present an entirely novel caching scheme, termed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">cyclic multi-antenna</i> coded caching, whose unique structure allows for the resolution of the above bottlenecks in the crucial regime of many transmit antennas. For this regime, where the multiplexing gain can exceed the coding gain, our new algorithm is the first to achieve the exact one-shot linear optimal DoF with a subpacketization complexity that scales only linearly with the number of users, and the first to benefit from a multicasting structure that allows for exploiting uplink-downlink duality in order to yield optimized beamformers ultra-fast. In the end, our novel solution provides excellent performance for networks with finite SNR, finite file sizes, and many users.

Signal-Aligned Network Coding for Multicell Processing With Limited Cooperation

In this paper, we propose a signal-aligned network coding (SNC) scheme for K-user time-varying multiple-input multiple-output (MIMO) interference channels with limited receiver cooperation. We assume that the receivers are connected to a central processor via wired cooperation links with individual limited capacities. Our SNC scheme determines the precoding matrices of the transmitters so that the transmitted signals are aligned at each receiver. The aligned signals are then decoded into noiseless integer combinations of messages, also known as network-coded messages, by physical-layer network coding. The key idea of our scheme is to ensure that independent integer combinations of messages can be decoded at the receivers. Hence the central processor can recover the original messages of the transmitters by solving the linearly independent equations. We prove that our SNC scheme achieves full degrees of freedom (DoF) by utilizing signal alignment and physical-layer network coding. Simulation results show that our SNC scheme outperforms the compute-and-forward scheme in the finite SNR regime of the two-user and the three-user cases. The performance improvement of our SNC scheme mainly comes from efficient utilization of the signal subspaces for conveying independent linear equations of messages to the central processor.

Robust Regularized ZF in Cooperative Broadcast Channel Under Distributed CSIT

In this work, we consider the sum rate performance of joint processing coordinated multi-point transmission network (JP-CoMP, a.k.a Network MIMO) in a so-called distributed channel state information (D-CSI) setting. In the D-CSI setting, the various transmitters (TXs) acquire a local, TX-dependent, estimate of the global multi-user channel state matrix obtained via terminal feedback and limited backhauling. The CSI noise across TXs can be independent or correlated, so as to reflect the degree to which TXs can exchange information over the backhaul, hence allowing to model a range of situations bridging fully distributed and fully centralized CSI settings. In this context we aim to study the price of CSI distributiveness in terms of sum rate at finite SNR when compared with conventional centralized scenarios. We consider the family of JP-CoMP precoders known as regularized zero-forcing (RZF). We conduct our study in the large scale antenna regime, as it is currently envisioned to be used in real 5G deployments. It is then possible to obtain accurate approximations on so-called deterministic equivalents of the signal to interference and noise ratios. Guided by the obtained deterministic equivalents, we propose an approach to derive a RZF scheme that is robust to the distributed aspect of the CSI, whereby the key idea lies in the optimization of a TX-dependent power level and regularization factor. Our analysis confirms the improved robustness of the proposed scheme with respect to CSI inconsistency at different TXs, even with moderate number of antennas and receivers (RXs).

Discrete Modulation for Interference Mitigation

This paper analyzes the performance of discrete input distributions (coded modulation) in interference channels. This approach is motivated in part by the necessity of using coded modulation in practical systems, and in part by the potential of discrete distributions for interference alignment as well as the importance demonstrated by Dytso et al. of discrete input distributions for transmission over the $2\times 2$ interference channel when treating interference as noise. The contribution of this work includes the establishment of achievable rates subject to discrete modulations. In the process, new bounds involving the minimum distance of the sum of discrete modulations have been developed that are useful for facilitating further work in this area. These bounds are then used for finding achievable rates for $3\times 3$ interference channels. It is also shown that interference alignment can be efficiently employed with discrete modulations at finite SNR even with imperfect channel state information at the transmitter.

Wireless Energy Harvesting-Based Relaying: A Finite-SNR Diversity-Multiplexing Tradeoff Perspective

This paper presents an analytical framework to derive the closed-form expressions for diversity-multiplexing tradeoff (DMT) for wireless energy harvesting (WEH) based amplify-and-forward (AF) and decode-and-forward (DF) protocols in finite signal-to-noise (SNR) regime. The results of this investigation suggest that both AF and DF offer similar performance except for few nuances. At low multiplexing gains, DF offers marginally better performance in the low SNR regime, whereas AF performs better in the high SNR scenarios. However, in the higher multiplexing gain regime, though subtle, DF uniformly dominates AF across all SNRs. Furthermore, an analytical study is presented to evaluate the effect of the fraction of time devoted to WEH (time-sharing parameter, $\epsilon $ ) on the finite SNR DMT (f-DMT). In addition, the impact of relay position on the outage performance is also presented. A distinguishing feature of the proposed work is the characterization of WEH-based f-DMT which reveals the complete interplay between the operating SNR and the time-sharing parameter ( $\epsilon $ ), which is of fundamental importance to system designers. Finally, Monte-Carlo simulations are provided to confirm the veracity of analytical solutions.

Matrix Integral Approach to MIMO Mutual Information Statistics in High-SNR Regime

In this work, an analytical framework for deriving the exact moments of multiple-input- multiple-output (MIMO) mutual information in the high-signal-to-noise ratio (SNR) regime is proposed. The idea is to make efficient use of the matrix-variate densities of channel matrices instead of the eigenvalue densities as in the literature. The framework is applied to the study of the emerging models of MIMO Rayleigh product channels and Jacobi MIMO channels, which include several well-known channels models as special cases. The corresponding exact moments of any order for the high-SNR mutual information are derived. The explicit moment expressions are utilized to construct simple yet accurate approximations to the outage probability. Despite the high-SNR nature, simulation shows usefulness of the approximations with finite SNR values in the scenario of low outage probability relevant to MIMO communications.

On the Diversity of Uncoded OTFS Modulation in Doubly-Dispersive Channels

Orthogonal time frequency space (OTFS) is a 2-dimensional (2D) modulation technique designed in the delay-Doppler domain. A key premise behind OTFS is the transformation of a time varying multipath channel into an almost non-fading 2D channel in delay-Doppler domain such that all symbols in a transmission frame experience the same channel gain. It has been suggested in the recent literature that OTFS can extract full diversity in the delay-Doppler domain, where full diversity refers to the number of multipath components separable in either the delay or Doppler dimension, but without a formal analysis. In this paper, we present a formal analysis of the diversity achieved by OTFS modulation along with supporting simulations. Specifically, we prove that the asymptotic diversity order of OTFS (as SNR $\rightarrow \infty$) is one. However, in the finite SNR regime, potential for a higher order diversity is witnessed before the diversity one regime takes over. Also, the diversity one regime starts at lower BER values for increased frame sizes. We also propose a phase rotation scheme for OTFS using transcendental numbers. We show that OTFS with this proposed scheme extracts the full diversity in the delay-Doppler domain.

Physical-Layer Schemes for Wireless Coded Caching

We investigate the potentials of applying the coded caching paradigm in wireless networks. In order to do this, we investigate physical layer schemes for downlink transmission from a multiantenna transmitter to several cache-enabled users. As the baseline scheme, we consider employing coded caching on the top of max–min fair multicasting, which is shown to be far from optimal at high-SNR values. Our first proposed scheme, which is near-optimal in terms of DoF, is the natural extension of multiserver coded caching to Gaussian channels. As we demonstrate, its finite SNR performance is not satisfactory, and thus we propose a new scheme in which the linear combination of messages is implemented in the finite field domain, and the one-shot precoding for the MISO downlink is implemented in the complex field. While this modification results in the same near-optimal DoF performance, we show that this leads to significant performance improvement at finite SNR. Finally, we extend our scheme to the previously considered cache-enabled interference channels, and moreover we provide an ergodic rate analysis of our scheme. Our results convey the important message that although directly translating schemes from the network coding ideas to wireless networks may work well at high-SNR values, careful modifications need to be considered for acceptable finite SNR performance.

Cache-Aided Communications With Multiple Antennas at Finite SNR

We study the problem of cache-aided communication for cellular networks with multi-user and multiple antennas at finite signal-to-noise ratio. Users are assumed to have non-symmetric links, modeled by wideband fading channels. We show that the problem can be formulated as a linear program, whose solution provides a joint cache allocation along with pre-fetching and fetching schemes that minimize the duration of the communication in the delivery phase. The suggested scheme uses zero-forcing and cached interference subtraction, and hence, allows each user to be served at the rate of its own channel. Thus, this scheme is better than the previously published schemes that are compromised by the poorest user in the communication group. We also consider a special case of the parameters for which we can derive a closed form solution and formulate the optimal power, rate, and cache optimization. This special case shows that the gain of MIMO coded caching goes beyond the throughput. In particular, it is shown that in this case, the cache is used to balance the users such that fairness and throughput are no longer contradicting. More specifically, in this case, strict fairness is achieved jointly with maximizing the network throughput.

Distributed Cooperative Two-Cell Zero-Forcing Precoding With Local Channel Correlation

This study focuses on local channel correlation based zero-forcing (ZF) precoding design for boosting weighted ergodic sum-rate (WESR) in the cooperative two-cell multiple-input single-output system. We first present that the WESRs of the statistical channel state information (SCSI) aided general eigenvector coordinated beamforming (SCSI-GE) and the conventional ZF beamforming with erroneous instantaneous channel state information (ICSI; eICSI-ZF) saturate to constant values at high SNR due to inter-cell interference. Motivated by this, we develop a ZF coordination framework with local SCSI requirement at the transmitter and low data-sharing backhaul load. Additionally, a low-cost but effective two-dimensional ZF precoding design called 2d-SZF is proposed, which involves only distributed computations based on the local channel correlation. Theoretical analysis and simulation results confirm that the proposed local SCSI-based 2d-SZF scheme yields the most desirable WESR while the WESRs of SCSI-GE and eICSI-ZF both saturate in the high regime of SNR, and are respectively sensitive to the level of spatial channel correlation and ICSI accuracy at finite SNR.

Asymptotic Analysis of MIMO Multi-Cell Full-Duplex Networks

We study a multi-cell multi-user MIMO full-duplex network, where each base station (BS) has multiple antennas with full-duplex capability supporting single-antenna users with either full-duplex or half-duplex radios. We characterize the up- and downlink ergodic achievable rates for the case of linear precoders and receivers. The rate analysis includes practical constraints, such as imperfect self-interference cancellation, channel estimation error, training overhead, and pilot contamination. We show that the $2\times $ gain of full duplex over half-duplex system remains in the asymptotic regime, where the number of BS antennas grows infinitely large. We numerically evaluate the finite SNR and antenna performance, which reveals that full-duplex networks can use significantly fewer antennas to achieve spectral efficiency gain over the half-duplex counterparts. In addition, the overall full-duplex gains can be achieved under realistic 3GPP multi-cell network settings despite the increased interference introduced in the full-duplex networks.

Non-Uniform Linear Antenna Array Design and Optimization for Millimeter-Wave Communications

In this paper, we investigate the optimization of non-uniform linear antenna arrays (NULAs) for millimeter wave (mmWave) line-of-sight (LoS) multiple-input multiple-output (MIMO) channels. Our focus is on the maximization of the system effective multiplexing gain (EMG), by optimizing the individual antenna positions in the transmit/receive NULAs. Here the EMG is defined as the number of signal streams that are practically supported by the channel at a finite SNR. We first derive analytical expressions for the asymptotic channel eigenvalues with arbitrarily deployed NULAs when, asymptotically, the end-to-end distance is sufficiently large compared to the aperture sizes of the transmit/receive NULAs. Based on the derived expressions, we prove that, the asymptotically optimal NULA deployment that maximizes the achievable EMG should follow the groupwise Fekete-point distribution. Specifically, the antennas should be physically grouped into K separate uniform linear antenna arrays (ULAs) with the minimum feasible antenna spacing within each ULA, where K is the target EMG to be achieved; in addition, the centers of these K ULAs follow the Fekete-point distribution. We numerically verify the asymptotic optimality of such an NULA deployment and extend it to a groupwise projected arch type (PAT) NULA deployment, which provides a more practical option for mmWave LoS MIMO systems with realistic non-asymptotic configurations. Numerical examples are provided to demonstrate a significant capacity gain of the optimized NULAs over traditional ULAs.

Uplink Downlink Rate Balancing in Cooperating Cellular Networks

Broadcast MIMO techniques can significantly increase the throughput in the downlink of cellular networks, at the price of channel state information (CSI) feedback from the mobiles, sent over the uplink. Thus, these techniques create a mechanism that can tradeoff some uplink capacity for increased downlink capacity. In this paper, we quantify this tradeoff and study the exchange ratio between the feedback rate (over the uplink) and the downlink rate. We study both finite and infinite networks and show that for high enough (but finite) SNR, the uplink rate can be exchanged for increased downlink rate with a favorable exchange ratio. This exchange ratio is an increasing function of the channel coherence time, and a decreasing function of the number of measured base stations. We also show that in finite networks, devoting a constant fraction of the uplink to CSI feedback can increase the downlink multiplexing gain continuously from 0 to 1, as a function of the fraction size. On the other hand, in infinite networks (with infinite connectivity) our lower bound is only able to show doubly logarithmic scaling of the rate with SNR. The presented results prove that the adaptation of the feedback rate can control the balance between the uplink and downlink rates. This capability is very important in modern cellular networks, where the operators need to respond to continuously changing user demands.

Deterministic-Coding Capacity of Networks in the Low-Power Regime

The deterministic-coding capacity of a network is the capacity of a network when nodes are restricted to transmitting reliable information, that is, (asymptotically) deterministic functions of the source messages. This is a generalization of network coding to wireless networks. The main contribution of this paper is to develop new outer bounds on the deterministic-coding capacity in the low-power regime, $ \mathrm {SNR}\to 0$ . This is accomplished by developing a technique for finding the limits of single letter bounds for $ \mathrm {SNR}\to 0$ even when these cannot be evaluated for finite SNR. It is shown that these outer bounds are tight in a number of cases, including the butterfly network of Ahlswede et al. in a wireless setting.

On the Corner Points of the Capacity Region of a Two-User Gaussian Interference Channel

This paper considers the corner points of the capacity region of a two-user Gaussian interference channel (GIC). In a two-user GIC, the rate pairs where one user transmits its data at the single-user capacity (without interference), and the other at the largest rate for which reliable communication is still possible are called corner points. This paper relies on existing outer bounds on the capacity region of a two-user GIC that are used to derive informative bounds on the corner points of the capacity region. The new bounds refer to a weak two-user GIC (i.e., when both crosslink gains in standard form are positive and below 1), and a refinement of these bounds is obtained for the case where the transmission rate of one user is within $\varepsilon > 0$ of the single-user capacity. The bounds on the corner points are asymptotically tight as the transmitted powers tend to infinity, and they are also useful for the case of moderate SNR and INR. Upper and lower bounds on the gap (denoted by $\Delta $ ) between the sum-rate and the maximal achievable total rate at the two corner points are derived. This is followed by an asymptotic analysis analogous to the study of the generalized degrees of freedom [where the SNR and INR scalings are coupled such that $({\log (\text {INR})}/{\log (\text {SNR})}) = \alpha \geq 0$ ], leading to an asymptotic characterization of this gap which is exact for the whole range of $\alpha $ . The upper and lower bounds on $\Delta $ are asymptotically tight in the sense that they achieve the exact asymptotic characterization. Improved bounds on $\Delta $ are derived for finite SNR and INR, and their improved tightness is exemplified numerically.

Space-Time Encoded MISO Broadcast Channel With Outdated CSIT: An Error Rate and Diversity Performance Analysis

Studies of the MISO Broadcast Channel (BC) with delayed Channel State Information at the Transmitter (CSIT) have so far focused on the sum-rate and Degrees-of-Freedom (DoF) region analysis. In this paper, we investigate for the first time, the error rate performance at finite SNR and the diversity-multiplexing tradeoff (DMT) at infinite SNR of a space-time encoded transmission over a two-user MISO BC with delayed CSIT. We consider the so-called MAT protocol obtained by Maddah-Ali and Tse, which was shown to provide 33% DoF enhancement over TDMA. While the asymptotic DMT analysis shows that MAT is always preferable to TDMA, the Pairwise Error Probability analysis at finite SNR shows that MAT is in fact not always a better alternative to TDMA. Benefits can be obtained over TDMA only at a very high rate or once concatenated with a full-rate full-diversity space-time code. The analysis is also extended to spatially correlated channels, and the influence of transmit correlation matrices and user pairing strategies on the performance are discussed. Relying on statistical CSIT, signal constellations are further optimized to improve the error rate performance of MAT and make it insensitive to user orthogonality. Finally, other transmission strategies relying on delayed CSIT are discussed.

Interactive Interference Alignment

We study interference channels (IFCs) where the interaction among sources and destinations is enabled, e.g., both sources and destinations can talk to each other using full-duplex radios. The interaction can come in two ways. First is through in-band interaction where sources and destinations can transmit and listen in the same channel simultaneously, enabling interaction. Second is through out-of-band interaction where destinations talk back to the sources on an out-of-band channel, which is possible from white-space channels. The flexibility afforded by the interaction among sources and destinations allows for the derivation of interference alignment (IA) strategies that have desirable “engineering properties,” i.e., insensitivity to the rationality or irrationality of channel parameters, small block lengths, and finite SNR operations. We show that, for several classes of IFCs, the interactive IA scheme can achieve the optimal degrees of freedom. In particular, we show a simple scheme (having a finite block length for channels having no diversity) for three-user and four-user IFCs with full-duplex radios to achieve the optimal degrees of freedom even after accounting for the cost of interaction. On the technical side, we show using a Gröbner basis argument that, in a general network potentially utilizing cooperation and feedback, the optimal degrees of freedom under linear schemes of a fixed block length is the same for channel coefficients with a probability of 1. Furthermore, a numerical method to estimate this value is also presented. These tools have potentially wider utility in studying other wireless networks as well.