Discrete Input Distributions Research Articles

Downlink Transmission With Heterogeneous URLLC Services: Discrete Signaling With Single-User Decoding

The problem of designing downlink transmission schemes for supporting heterogeneous ultra-reliable low-latency communications (URLLC) and/or with other types of services is investigated. We consider the broadcast channel, where the base station sends superimposed signals to multiple users. Under heterogeneous blocklength constraints, strong users who are URLLC users cannot wait to receive the entire transmission frame and perform successive interference cancellation (SIC) due to stringent latency requirements, in contrast to the conventional infinite blocklength cases. Even if SIC is feasible, SIC may be imperfect under finite blocklength constraints. To cope with the heterogeneity in latency and reliability requirements, we propose a practical downlink transmission scheme with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">discrete signaling</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">single-user decoding (SUD), i.e., without SIC</i> . We carefully design the discrete input distributions to enable efficient SUD by exploiting the structural interference. Furthermore, we derive the second-order achievable rate under heterogenous blocklength and error probability constraints and use it to guide the design of channel coding and modulations. It is shown that in terms of achievable rate under short blocklength, the proposed scheme with regular quadrature amplitude modulations and SUD can operate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">extremely close</i> to the benchmark schemes that assume perfect SIC with Gaussian signaling.

Channel capacity and power allocation of MIMO visible light communication system

In this paper, the channel capacity of the multiple-input multiple-output (MIMO) visible light communication (VLC) system is investigated under the peak, average optical and electrical power constraints. Finding the channel capacity of MIMO VLC is shown to be a mixed integer programming problem. To address this open problem, we propose an inexact gradient projection method to find the channel capacity-achieving discrete input distribution and the channel capacity of MIMO VLC. Also we derive both upper and lower bounds of the capacity of MIMO VLC with the closed-form expressions. Furthermore, by considering practical discrete constellation inputs, we develop the optimal power allocation scheme to maximize transmission rate of MIMO VLC system. Simulation results show that more discrete points are needed to achieve the channel capacity as SNR increases. Both the upper and lower bounds of channel capacity are tight at low SNR region. In addition, comparing the equal power allocation, the proposed power allocation scheme can significantly increase the rate for the low-order modulation inputs.

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.

Dynamic Interpolation Search revisited

A new dynamic Interpolation Search (IS) data structure is presented that achieves O(log⁡log⁡n) search time with high probability on unknown continuous or even discrete input distributions with measurable probability of element collisions, including power law and Binomial distributions. No such previous result holds for IS when the probability of element collisions is measurable. Moreover, our data structure exhibits O(1) search time with high probability (w.h.p.) for a wide class of input distributions that contains all those for which o(log⁡log⁡n) expected search time was previously known.

Artificial Noise-Based Beamforming for the MISO VLC Wiretap Channel

This paper investigates the secrecy performance of the multiple-input single-output visible light communication (VLC) wiretap channel. The considered system model comprises three nodes: a transmitter (Alice) equipped with multiple fixtures of LEDs, a legitimate receiver (Bob), and an eavesdropper (Eve), each equipped with one photo-diode. The VLC channel is modeled as a real-valued amplitude-constrained Gaussian channel. Eve is assumed to be randomly located in the same area as Bob. Due to this, artificial noise-based beamforming is adopted as a transmission strategy in order to degrade Eve’s signal-to-noise ratio. Assuming discrete input signaling, we derive an achievable secrecy rate in a closed-form expression as a function of the beamforming vectors and the input distribution. We investigate the average secrecy performance of the system using stochastic geometry to account for the location randomness of Eve. We also adopt the truncated discrete generalized normal (TDGN) as a discrete input distribution. We present several examples through which we confirm the accuracy of the analytical results via Monte Carlo simulations. The results also demonstrate that the TDGN distribution, albeit being not optimal, yields performance close to the secrecy capacity.

On the IID Capacity-Achieving Input for Binding Channels With Multiple Ligand Receptors

This paper studies the molecular communication system where the transmitter has limited productivity and the receiver employs ligand-binding receptors. By simplifying the release and propagation process of the information particles, the ligand-binding process is regarded as a binding channel with peak and average constraints and modeled by a finite-state Markov chain. It is proved that the capacity of the constrained independent and identically distributed (IID) binding channel, defined as the IID capacity, is achieved by a discrete input distribution. The sufficient and necessary conditions of an IID input distribution being capacity-achieving is presented. Moreover, an algorithm called modified steepest ascent cutting-plane algorithm is proposed to efficiently compute the IID capacity-achieving distributions. The numerical results show that the IID capacity is a tight lower bound of the capacity for the binding channel.

Optical Wiretap Channel With Input-Dependent Gaussian Noise Under Peak- and Average-Intensity Constraints

This paper studies the optical wiretap channel with input-dependent Gaussian noise, in which the main distortion is caused by an additive Gaussian noise whose variance depends on the current signal strength. Subject to nonnegativity and peak-intensity constraints on the channel input, we first present a practical optical wireless communication scenario for which the considered wiretap channel is stochastically degraded. We then study the secrecy-capacity-achieving input distribution of this wiretap channel and prove it to be discrete with a finite number of mass points, one of them located at the origin. Moreover, we show that the entire rate-equivocation region of this wiretap channel is also obtained by discrete input distributions with a finite support. Similar to the case of the Gaussian wiretap channel under a peak-power constraint, here too, we observe that under nonnegativity and peak-intensity constraints, there is a tradeoff between the secrecy capacity and the capacity in the sense that both may not be achieved simultaneously. Furthermore, we prove the optimality of discrete input distributions in the presence of an additional average intensity constraint. Finally, we shed light on the asymptotic behavior of the secrecy capacity in the low- and high-intensity regimes. In the low-intensity regime, the secrecy capacity scales quadratically with the peak-intensity constraint. On the other hand, in the high-intensity regime, the secrecy capacity does not scale with the constraint.

Secrecy Rate Closed-Form Expressions for the SISO VLC Wiretap Channel With Discrete Input Signaling

We study in this letter the secrecy performance of a degraded single-input single-output visible-light communication wiretap channel. It has been shown in the literature that the secrecy capacity-achieving input distribution for this system is discrete with a finite support set. However, neither the secrecy capacity nor its achieving distribution has been derived in closed-form. In this letter, we derive in closed-form the achievable secrecy rate as a function of the discrete input distribution. We propose a class of discrete input distributions in an effort to enhance the achievable secrecy rate and approach the secrecy capacity. We also derive expressions for the achievable secrecy rate for different scenarios including the one in which the locations of the terminals are randomly located. We provide several examples that demonstrate the accuracy of the derived expressions and show the substantial secrecy rate improvements provided by the proposed scheme over existing ones.

A Lattice-Partition Framework of Downlink Non-Orthogonal Multiple Access Without SIC

In this paper, a novel lattice-partition-based downlink non-orthogonal multiple access framework is proposed. This framework is motivated by recognizing the algebraic structure behind the previous scheme recently proposed by Shieh and Huang as a lattice partition in $ \mathbb {Z}$ and is in fact a generalization of the scheme to any base lattice. The schemes in the proposed framework enjoy many desirable properties such as explicit and systematic design and discrete input distributions. Moreover, the proposed method only requires a limited knowledge of channel parameters. The rates achieved by the proposed scheme with any base lattice and with single-user decoding (i.e., without successive interference cancellation) are analyzed, and a universal upper bound on the gap to the multiuser capacity is obtained as a function of the normalized second moment of the base lattice. Since the proposed framework has a substantially larger design space than that of the previous scheme of Shieh and Huang whose base lattice is a 1-D lattice, one can easily find instances in larger dimensions that can provide superior performance. Design examples with the base lattices $A_{2}$ , $D_{4}$ , $E_{8}$ , and Construction A lattices, respectively, are provided, and both theoretical and simulation results exhibit smaller gaps to the multiuser capacity as dimensions increase.

Capacity Bounds for the Gaussian IM-DD Optical Multiple-Access Channel

Optical wireless communications (OWC) is a promising technology for closing the mismatch between the growing number of connected devices and the limited wireless network capabilities. Similar to downlink, uplink can also benefit from OWC for establishing connectivity between such devices and an optical access point. In this context, the incoherent intensity-modulation and direct-detection (IM-DD) scheme is desirable in practice. Hence, it is important to understand the fundamental limits of communication rates over an OWC uplink employing IM-DD, i.e., the channel capacity. This uplink, modeled as a Gaussian multiple-access channel (MAC) for indoors OWC, is studied in this paper, under the IM-DD constraints, which form the main difference with the standard Gaussian MAC commonly studied in the radio-frequency context. Capacity region outer and inner bounds for this channel are derived. The bounds are fairly close at high signal-to-noise ratio (SNR), where a truncated-Gaussian input distribution achieves the capacity region within a constant gap. Furthermore, the bounds coincide at low SNR showing the optimality of ON–OFF keying combined with successive cancellation decoding in this regime. At moderate SNR, an optimized uniformly spaced discrete input distribution achieves fairly good performance.

Characteristics and Performance of Image Sensor Communication

A complementary metal-oxide semiconductor (CMOS) image sensor can be used for communication, in addition to imaging. This paper investigates noise characteristics and system performance for CMOS image sensor communication. Different signal-dependent and independent noise sources are categorized and comprehensively analyzed in conjunction with sensor measurements. Accordingly, a unified channel model in the pixel-matched case is proposed, and the communication signal-to-noise ratio is defined and demonstrated in different parameter settings. Moreover, the channel capacity with mixed noise and bounded input is derived and is proven to be achievable by a discrete input distribution of finite number of probability mass points. Simulation shows that capacity of more than 7 b/s/Hz per pixel or 8–10 b/s/Hz per block of multiple pixels is achievable.

A Simple Scheme for Realizing the Promised Gains of Downlink Nonorthogonal Multiple Access

In this paper, the downlink nonorthogonal multiple access (NOMA) system is studied where purely discrete input distributions are found that achieve the capacity region to within a constant gap without successive interference cancellation (SIC). The approach is a two-step approach where the corresponding linear deterministic model is first studied and the results are then systematically translated into purely discrete input distributions for the original model. A simple yet powerful coding scheme, which adopts off-the-shelf turbo codes with pulse amplitude modulations (PAM) is then used to simulate the proposed input distributions. Simulation results show that the proposed simple scheme under turbo decoding, both with and without SIC, can operate close to information-theoretic bounds of the proposed input distributions, which lies outside the achievable rate region of any orthogonal multiple access (OMA)-type scheme.

Gaussian Wiretap Channel With Amplitude and Variance Constraints

We consider the Gaussian wiretap channel with amplitude and variance constraints on the channel input. We first show that the entire rate-equivocation region of the Gaussian wiretap channel with an amplitude constraint is obtained by discrete input distributions with finite support. We prove this result by considering the existing single-letter description of the rate-equivocation region, and showing that discrete distributions with finite support exhaust this region. Our result highlights an important difference between the peak power (amplitude) constrained and the average power (variance) constrained cases. Although, in the average power constrained case, both the secrecy capacity and the capacity can be achieved simultaneously, our results show that in the peak power constrained case, in general, there is a tradeoff between the secrecy capacity and the capacity, in the sense that, both may not be achieved simultaneously. We also show that under sufficiently small amplitude constraints the possible tradeoff between the secrecy capacity and the capacity does not exist and they are both achieved by the symmetric binary distribution. Finally, we prove the optimality of discrete input distributions in the presence of an additional variance constraint.

Capacity Bounds for Wireless Optical Intensity Channels With Gaussian Noise

Lower and upper bounds on the capacity of wireless optical intensity pulse amplitude modulation channels under nonnegativity and average optical power constraints are derived. A lower bound is derived based on source entropy maximization over a family of discrete nonuniform distributions with equally spaced mass points. A closed form for the maxentropic discrete input distribution is provided. Compared to previously reported bounds, the derived lower bound is tight at both low and high signal-to-noise ratios (SNRs). In addition, a closed-form upper bound is derived based on signal space geometry via a sphere packing argument. The proposed bound is tight at low SNRs and incurs a small gap to the channel capacity at high SNRs. The derived bounds asymptotically describe the optical intensity channel capacity at low SNRs, where a majority of such links operate.

Channel capacity and non-uniform signalling for free-space optical intensity channels

This work considers the design of capacity-approaching, non-uniform optical intensity signalling in the presence of average and peak amplitude constraints. Although it is known that the capacity-achieving input distribution is discrete with a finite number of mass points, finding it requires complex non-linear optimization at every SNR. In this work, a simple expression for a capacity-approaching distribution is derived via source entropy maximization. The resulting mutual information using the derived discrete non-uniform input distribution is negligibly far away from the channel capacity. The computation of this distribution is substantially less complex than previous optimization approaches and can be easily computed at different SNRs. A practical algorithm for non-uniform optical intensity signalling is presented using multi-level coding followed by a mapper and multi-stage decoding at the receiver. The proposed signalling is simulated on free-space optical channels and outage capacity is analyzed. A significant gain in both rate and probability of outage is achieved compared to uniform signalling, especially in the case of channels corrupted by fog.

NANOLAB—A Tool for Evaluating Reliability of Defect-Tolerant Nanoarchitectures

As manufacturing technology reaches nanoscale, architectural designs need to accommodate the uncertainty inherent at such scales. These uncertainties are germane in the minuscule dimension of the devices, quantum physical effects, reduced noise margins, system energy levels reaching computing thermal limits, manufacturing defects, aging, and many other factors. Defect-tolerant architectures and their reliability measures will gain importance for logic and micro-architecture designs based on nanoscale substrates. Recently, the Markov random field has been proposed as a model of computation for nanoscale logic gates. This opens up new possibilities in designing logic and architecture where the conventional Boolean logic is replaced by a notion of an energy distribution function based on Gibbs distribution. In this computational scheme, probabilities of energy levels at various gate inputs and interconnects are considered, and belief propagation is used to propagate these probability distributions from the primary inputs to the primary outputs of a Boolean network. In this paper, we take this approach further by automating this computational scheme and belief propagation algorithm. We have developed MATLAB-based libraries for fundamental logic gates that can compute probability distributions and entropies at the outputs for specified discrete input distributions and in the presence of noise at the inputs and interconnects. Our tool automates the evaluation of reliability measures of combinational logic blocks. The effectiveness of this automation is illustrated in this paper by automatically deriving various reliability results for defect-tolerant architectures, such as triple modular redundancy (TMR), cascaded TMR, and multistage iterations of these. Also, signal noise is modeled as uniform and Gaussian distributions at the inputs and interconnects, so as to evaluate reliability-redundancy tradeoffs of these architectural configurations taking into account such noise models.

How wrong can it be? Understanding uncertainty in petroleum systems modelling

Petroleum systems modelling is used in industry to understand and predict the interplay between the main elements of a petroleum system. This is done using numerical modelling because it cannot possibly be done experimentally given the geological time- and space scales. Such models describe the geological processes forward in time and, therefore, generate a single unique solution for a given input dataset. The evidence from data is, in most cases, insufficient to remove any ambiguities. Therefore, it is more interesting to know how much this uncertainty can be reduced, or how wrong it still might be, rather than knowing the right answer. Given the uncertainty of model outcomes, the appropriate solution should, therefore, be probabilistic rather than deterministic, where the variance expresses the remaining uncertainty in the outcome covering all possible valid geological scenarios. This paper describes how experimental design and response surface analysis can be used to perform sensitivity and uncertainty analysis in petroleum generation and migration modelling, focusing on trapped volumes derived from its drainage area. Map properties are parameterized as end-member scenarios, either as discrete or linearly varying maps. Regression coefficients indicate a good representation of trapped hydrocarbon volume by the response surface. In the example presented, hydrocarbon volume is most sensitive to fault characteristics, total organic carbon content and, to a lesser degree, to thickness of the source rock and heat flow. Well data allow a significant reduction of uncertainties, which can be shown by the probability distribution that becomes narrower but remains similar in form. The volume probability distribution of the prospect is multimodal for a discrete parameter scenario and log-normal for both discrete or uniform input distributions. Volume distributions result directly from the deterministic modelling without any prior statistical assumption on the form of the distribution.

Capacity of fading channelwith no side information

A supremum to the Shannon capacity of a Rayleigh fading channel without side information (SI) is computed using variational methods and is shown to be asymptotically achievable for signal-to-noise ratios (SNRs) approaching either zero or infinity. A lower bound is obtained from the mutual information corresponding to a specific discrete input distribution.

Approaching capacity of a continuous channel by discrete input distributions

In this paper memoryless channels with general alphabets and input constraint are considered. Sufficient conditions are given that channel capacity can be approached by discrete input distributions or by uniform input distributions with finite support. As an example, the additive white Gaussian noise channel is considered.