Asymptotic Scaling Laws Research Articles

Secure Cooperative Communications With an Untrusted Relay: A NOMA-Inspired Jamming and Relaying Approach

We propose a novel non-orthogonal multiple access (NOMA)-inspired jamming and relaying scheme to enhance the physical layer security of untrusted relay networks. Particularly, during the first phase, with the application of the downlink NOMA principle, the source sends a superimposed version of a desired signal and a jamming signal, where the jamming signal is designed to deliberately confuse the untrusted relay by exploiting the beamforming design and adapting the transmission rate at the source. During the second phase, relying on the uplink NOMA principle, the untrusted relay forwards its received signals and, simultaneously, the source transmits a new desired signal that cannot be wiretapped by the untrusted relay to maximize the secrecy sum rate. Scenarios of single-antenna and multiple-antenna relaying are considered, and the impact of different antenna configurations at the source and the untrusted relay on the secrecy performance is investigated. Analytical expressions of an ergodic secrecy sum rate (ESSR) lower bound and an asymptotic ESSR scaling law are derived to evaluate the secrecy performance of the proposed NOMA-inspired jamming and relaying scheme. Computer simulations are presented to validate the accuracy of the derived analytical results, and demonstrate the significant ESSR improvement of the proposed scheme over the conventional orthogonal multiple access-based relaying schemes.

Achievable Diversity Order of HARQ-Aided Downlink NOMA Systems

The combination between non-orthogonal multiple access (NOMA) and hybrid automatic repeat request (HARQ) is capable of realizing ultra-reliability, high throughput and many concurrent connections particularly for emerging communication systems. This paper focuses on characterizing the asymptotic scaling law of the outage probability of HARQ-aided NOMA systems with respect to the transmit power, i.e., diversity order. The analysis of diversity order is carried out for three basic types of HARQ-aided downlink NOMA systems, including Type I HARQ, HARQ with chase combining (HARQ-CC) and HARQ with incremental redundancy (HARQ-IR). The diversity orders of three HARQ-aided downlink NOMA systems are derived in closed-form, where an integration domain partition trick is developed to obtain the bounds of the outage probability specially for HARQ-CC and HARQ-IR-aided NOMA systems. The analytical results show that the diversity order is a decreasing step function of transmission rate, and full time diversity can only be achieved under a sufficiently low transmission rate. It is also revealed that HARQ-IR-aided NOMA systems have the largest diversity order, followed by HARQ-CC-aided and then Type I HARQ-aided NOMA systems. Additionally, the users’ diversity orders follow a descending order according to their respective average channel gains. Furthermore, we expand discussions on the cases of power-efficient transmissions and imperfect channel state information (CSI). Monte Carlo simulations finally confirm our analysis.

High-Mobility Wideband Massive MIMO Communications: Doppler Compensation, Analysis and Scaling Laws

In this paper, we apply angle-domain Doppler compensation for high-mobility wideband massive multi-input multi-output (MIMO) uplink communications. The time-varying multipath channel is considered between high-speed terminal and static base station (BS), where multiple Doppler frequency offsets (DFOs) are associated with distinct angle of departures (AoDs). With the aid of large-scale uniform linear array (ULA) at the transmitter, we design a beamforming network to generate multiple parallel beamforming branches, each transmitting signal pointing to one particular angle. Then, the transmitted signal in each branch will experience only one dominant DFO when passing over the time-varying channel, which can be easily compensated before transmission starts. We theoretically analyze the Doppler spread of the equivalent uplink channel after angle-domain Doppler compensation, which takes into account both the mainlobe and sidelobes of the transmit beam in each branch. It is seen that the channel time-variation can be effectively suppressed if the number of transmit antennas is sufficiently large. Interestingly, the asymptotic scaling law of channel variation is obtained, which shows that the Doppler spread is proportional to the maximum DFO and decreases approximately as $1/\sqrt {M}$ ( $M$ is the number of transmit antennas) when $M$ is sufficiently large. The numerical results are provided to corroborate the proposed scheme.

Security and Reliability Performance Analysis of Cooperative Multi-Relay Systems With Nonlinear Energy Harvesters and Hardware Impairments

In this paper, we investigate the reliability and security performance of cooperative multi-relay systems, where both source and relay nodes are energy-constrained nonlinear energy harvesters, scavenging energy from a power beacon nearby. Our analysis is based on practical model since residual hardware impairments (RHIs) and channel estimation errors (CEEs) are considered. Aiming at improving the system efficiency, three representative relay selection strategies are considered: 1) random relay selection (RRS); 2) suboptimal relay selection (SRS); and 3) optimal relay selection (ORS). To characterize the security performance of the considered strategies, we derive closed-form analytical expressions of the reliability and security in terms of outage probability (OP) and intercept probability (IP). We further discuss the asymptotic expressions and scaling laws of OP with the number of relays. The IP is analyzed for non-colluding and colluding scenarios. The numerical results illustrate that: 1) There is a tradeoff between reliability and security, that is when the outage constraint is relaxed, the IP can be enhanced, and vice versa; ii) The outage performance of the ORS and SRS schemes outperform RRS, indicating that relay selection can enhance reliability performance; iii) There are error floors for the OP due to the CEEs; iv) Colluding eavesdroppers can enhance eavesdropping attacks by sharing their intercepted information; and v) Although RHIs and CEEs have deleterious effects on the OP, they can protect the information transmission against eavesdropping attacks.

Optical response induced by bound states in the continuum in arrays of dielectric spheres

We consider optical response induced by bound states in the continuum (BICs) in arrays of dielectric spheres. By combining quasi-mode expansion technique with coupled mode theory (CMT) we put forward a theory of the optical response by high-Q resonance surrounding BICs in momentum space. The central result are analytical expressions for the CMT parameters, which can be easily calculated from the eigenfrequencies and eigenvectors of the interaction matrix of the scattering systems. The results obtained are verified in comparison against exact numerical solutions to demonstrate that the CMT approximation is capable of reproducing Fano features in the spectral vicinity of the BIC. Based on the quasi-mode expansion technique we derived the asymptotic scaling law for the CMT parameters in the vicinity of the $\Gamma$-point. It is rigorously demonstrated that the line width in the CMT approximation exhibits different asymptotic behaviour depending on the symmetry of the BIC.

Asymptotic scaling laws and semi-similarity solutions for a finite-source spherical blast wave

A spherical blast wave generated by a sudden release of a sphere of compressed gas is an important model problem to understand blast phenomena such as volcanic eruptions and explosive detonations. The resulting explosion flow physics, such as the instability at the gas contact discontinuity and the interaction between the shock wave and the gas contact, are dictated by the initial pressure and sound-speed ratios between the compressed gas and the ambience. Since the initial pressure and sound-speed ratios vary over a wide range in practical applications, it is of interest to investigate the scaling laws and similarity solutions for the spherical symmetric explosion flow. In the present study, numerical simulation of a spherical blast wave is performed. A long-term length scale that incorporates the initial charge radius and the initial pressure ratio is introduced. The trajectories of the main shock normalized by the long-term length scale for a wide range of parameters collapse after a short transition time, indicating an asymptotic similarity solution exists for the far field in the long term. With the assistance of this similarity solution, the full evolution of the main shock can be obtained semi-analytically. For near-field features, i.e. the gas contact and the secondary shock wave, only semi-similarity solutions are observed, which depend on the initial sound-speed ratio but not the initial pressure ratio. The gas contact and the secondary shock share the same scaling relations. Asymptotic analysis is performed to obtain the short-term dynamics of the gas contact, including the gas contact acceleration and the Atwood number, which are the key parameters determining the Rayleigh–Taylor instability development at the gas contact. The asymptotic contact radius as$t\rightarrow \infty$is also obtained, which is found to be well represented by the long-term length scale and thus only depends on the initial pressure ratio. A simple model of an oscillating bubble is employed to explain the scaling relation of the asymptotic gas contact radius.

Lability of Nanoparticulate Metal Complexes at a Macroscopic Metal Responsive (Bio)interface: Expression and Asymptotic Scaling Laws

The lability of metal complexes expresses the extent of the dissociative contribution of the complex species to the flux of metal ions toward a macroscopic metal-responsive (bio)interface, for example, an electrodic sensor or an organism. While the case of molecular ligands is well-established, it is only recently that a definition was elaborated for the lability of metal complexes with nanoparticles (NPs) in aqueous dispersions. The definition includes the thickness of the nonequilibrium reaction layer operational at the (bio)interface and the extent of geometrical exclusion of NPs therefrom. In this work, an explicit expression is derived for the lability of nanoparticulate metal complexes (M–NP) toward a macroscopic reactive (bio)interface. Interpretation accounts for the M–NP chemodynamic properties that depend on the NP size, electrostatics, metal diffusion and dehydration rates, and density of metal binding sites for various NP types, for example, soft/core–shell and hard NPs having volume and surfa...

Performance Scaling Law for Multicell Multiuser Massive MIMO

This paper provides a comprehensive scaling law-based performance analysis for multicell multiuser massive multiple-input-multiple-output (MIMO) downlink systems. Imperfect channel state information (CSI), pilot contamination, and channel spatial correlation are all considered. First, a sum-rate lower bound is derived by exploiting the asymptotically deterministic property of the received signal power, while keeping the random nature of other components in the signal-to-interference-plus-noise-ratio (SINR) intact. Via a general scaling model on important network parameters, including the number of users, the channel training energy and the data transmission power, with respect to the number of base station antennas, the asymptotic scaling law of the effective SINR is obtained, which reveals quantitatively the tradeoff of the network parameters. More importantly, pilot contamination and pilot contamination elimination (PCE) are considered in the analytical framework. In addition, the applicability of the derived asymptotic scaling law in practical systems with large but finite antenna numbers are discussed. Finally, sufficient conditions on the parameter scalings for the SINR to be asymptotically deterministic in the sense of mean square convergence are provided, which covers existing results on such analysis as special cases and shows the effect of PCE explicitly.

On the large-Q2 behavior of the pion transition form factor

We study the transition of non-perturbative to perturbative QCD in situations with possible violations of scaling limits. To this end we consider the singly- and doubly-virtual pion transition form factor π0→γγ at all momentum scales of symmetric and asymmetric photon momenta within the Dyson–Schwinger/Bethe–Salpeter approach. For the doubly virtual form factor we find good agreement with perturbative asymptotic scaling laws. For the singly-virtual form factor our results agree with the Belle data. At very large off-shell photon momenta we identify a mechanism that introduces quantitative modifications to Efremov–Radyushkin–Brodsky–Lepage scaling.

Performance Analysis and Scaling Law of MRC/MRT Relaying With CSI Error in Multi-Pair Massive MIMO Systems

This paper provides a comprehensive scaling law and performance analysis for multi-user massive multiple-input-multiple-output (MIMO) relay networks, where the relay is equipped with a massive antenna array and uses maximal-ratio combining/maximal-ratio transmission (MRC/MRT) for low-complexity processing. Imperfect channel state information (CSI) is considered for both source-relay and relay-destination channels. First, a sum-rate lower bound is derived, which manifests the effect of system parameters, including the numbers of relay antennas and users, the CSI quality, and the transmit powers of the sources and the relay. Via a general scaling model on the parameters with respect to the relay antenna number, the asymptotic scaling law of the signal-to-interference-plus-noise-ratio (SINR) is obtained, which shows quantitatively the tradeoff of the network parameters. In addition, a sufficient condition on the parameter scalings for the SINR to be asymptotically deterministic is given, which covers existing results on such analysis as special cases. Then, the scenario where the SINR increases linearly with the relay antenna number is studied. The sufficient and necessary condition on the parameter scaling for this scenario is proved. It is shown that in this case, the interference power is not asymptotically deterministic, and then, the average bit error rate is analyzed.

Beta-plane turbulence above monoscale topography

Using a one-layer quasi-geostrophic model, we study the effect of random monoscale topography on forced beta-plane turbulence. The forcing is a uniform steady wind stress that produces both a uniform large-scale zonal flow $U(t)$ and smaller-scale macroturbulence characterized by standing and transient eddies. The large-scale flow $U$ is retarded by a combination of Ekman drag and the domain-averaged topographic form stress produced by the eddies. The topographic form stress typically balances most of the applied wind stress, while the Ekman drag provides all of the energy dissipation required to balance the wind work. A collection of statistically equilibrated numerical solutions delineate the main flow regimes and the dependence of the time average of $U$ on parameters such as the planetary potential vorticity (PV) gradient $\unicode[STIX]{x1D6FD}$ and the statistical properties of the topography. We obtain asymptotic scaling laws for the strength of the large-scale flow $U$ in the limiting cases of weak and strong forcing. If $\unicode[STIX]{x1D6FD}$ is significantly smaller than the topographic PV gradient, the flow consists of stagnant pools attached to pockets of closed geostrophic contours. The stagnant dead zones are bordered by jets and the flow through the domain is concentrated into a narrow channel of open geostrophic contours. In most of the domain, the flow is weak and thus the large-scale flow $U$ is an unoccupied mean. If $\unicode[STIX]{x1D6FD}$ is comparable to, or larger than, the topographic PV gradient, then all geostrophic contours are open and the flow is uniformly distributed throughout the domain. In this open-contour case, there is an ‘eddy saturation’ regime in which $U$ is insensitive to large changes in the wind stress. We show that eddy saturation requires strong transient eddies that act effectively as PV diffusion. This PV diffusion does not alter the kinetic energy of the standing eddies, but it does increase the topographic form stress by enhancing the correlation between the topographic slope and the standing-eddy pressure field. Using bounds based on the energy and enstrophy power integrals, we show that as the strength of the wind stress increases, the flow transitions from a regime in which the form stress balances most of the wind stress to a regime in which the form stress is very small and large transport ensues.

The rolling suitcase instability: a coupling between translation and rotation

A two-wheel suitcase or trolley can exhibit undamped rocking oscillations from one wheel to the other when pulled fast enough. We study this instability both experimentally—with a toy model of a suitcase rolling on a treadmill—and theoretically. The suitcase oscillates only if a finite perturbation is applied. This is because intrinsic dissipation occurs when the supporting wheel switches. When unstable, the suitcase either increasingly rocks until overturning or reaches a stable limit cycle. The friction force at the rolling wheels constrains wheels to roll without slipping. This constraint imposes a coupling between the translational motion and the three-dimensional rotational motion of the suitcase that drives the rocking instability. The same behaviours are observed in the experiments and in the simulations. The asymptotic scaling laws we observe in the simulations are explained by means of a simplified model where the coupling force is explicit.

Particle acceleration in MHD turbulence

MHD Turbulence describes dynamics of astrophysical plasmas on large scale. It is characterized by energy transfer between different scales and the exchange of energy with nonthermal population – typically cosmic rays. Recent progress in theory regarding almost all basic regimes of turbulence – from the dynamo to the decaying case and the asymptotic scaling laws, allowed us to proceed with more observationally motivated questions. One of them is why almost all strongly magnetized environments are indeed observable, e.g. why such environments are infused with high-energy particles, their distributions stretching to energies orders of magnitude higher than thermal. It turns out that there are generic mechanisms of acceleration in turbulence, both second and first order in v/c, the latter was somehow unnoticed. These generic mechanisms may explain the ubiquity of non-thermal tails in various magnetized astrophysical environments, e.g. solar chromosphere, pulsar magnetospheres, jets from supermassive black holes, gamma-ray bursts, etc.

Stability of vertical magnetic chains.

A linear stability analysis is performed for a pair of coaxial vertical chains made from permanently magnetized balls under the influence of gravity. While one chain rises from the ground, the other hangs from above, with the remaining ends separated by a gap of prescribed length. Various boundary conditions are considered, as are situations in which the magnetic dipole moments in the two chains are parallel or antiparallel. The case of a single chain attached to the ground is also discussed. The stability of the system is examined with respect to three quantities: the number of balls in each chain, the length of the gap between the chains, and a single dimensionless parameter which embodies the competition between magnetic and gravitational forces. Asymptotic scaling laws involving these parameters are provided. The Hessian matrix is computed in exact form, allowing the critical parameter values at which the system loses stability and the respective eigenmodes to be determined up to machine precision. A comparison with simple experiments for a single chain attached to the ground shows good agreement.

IQ Imbalance in Multiuser Systems: Channel Estimation and Compensation

In this paper, we consider the uplink of a single-cell multi-user single-input multiple-output (MU-SIMO) system with in-phase and quadrature-phase imbalance (IQI). In particular, we investigate the effect of receive (RX) IQI on the performance of MU-SIMO systems with large antenna arrays employing maximum-ratio combining receivers. In order to study how IQI affects channel estimation, we derive a new channel estimator for the IQI-impaired model and show that the higher the value of signal-to-noise ratio, the higher the impact of IQI on the spectral efficiency (SE). Moreover, a novel pilot-based joint estimator of the augmented multiple-input multiple-output (MIMO) channel matrix and IQI coefficients is described, and then, a low-complexity IQI compensation scheme is proposed, which is based on the IQI coefficients' estimation and it is independent of the channel gain. The performance of the proposed compensation scheme is analytically evaluated by deriving a tractable approximation of the ergodic SE assuming transmission over Rayleigh fading channels with large-scale fading. Furthermore, we investigate how many mobile stations should be scheduled in massive MIMO systems with IQI and show that the highest SE loss occurs at the optimal operating point. Finally, by deriving asymptotic power scaling laws and proving that the SE loss due to IQI is asymptotically independent of the number of BS antennas, we show that massive MIMO is resilient to the effect of RX IQI.

Nonasymptotic Multicast Throughput and Delay in Multihop Wireless Networks

Previous works on multicast capacity mainly focus on deriving asymptotic order results in large-scale wireless networks. While they can explore the general scaling laws of throughput capacity, it is also of great interest for practical concern to predict the exact achievable throughput in networks with an arbitrary finite number of nodes. In this paper, we investigate the nonasymptotic throughput and delay of multihop wireless networks for multicast applications wherein for each source node, $k$ nodes are randomly selected as receivers. It is challenging for the exact performance analysis since multicast transmission has a dynamic nature due to the following factors: 1) random distribution of nodes; 2) bursty traffic arrivals; and 3) different timescales for transient analysis. To tackle the problem, we propose an explicit analytical model and develop a multicast routing scheme, which accounts for these aspects. With our proposed model, we derive lower bound (LB) and upper bound (UB) on nonasymptotic multicast throughput and delay using stochastic network calculus. We show that the performance results hold for all timescales and network sizes and are strongly correlated to data burstiness and the number of receivers. While we investigate from a nonasymptotic point of view, our results can also cover the asymptotic scaling laws. Simulations are conducted to further verify the accuracy of the analytical bounds.

Analytical Descriptions of Magnetic Properties and Magnetoresistance in n-Type HgCr2Se4**Supported by the National Natural Science Foundation of China under Grant Nos 61425015, 11474330 and 11374337, the National Basic Research Program of China under Grant Nos 2012CB921703 and 2015CB921102, and the Chinese Academy of Sciences.

We present a detailed investigation of magnetic properties of colossal magnetoresistance material HgCr2Se4. While spontaneous magnetization and zero-field magnetic susceptibility are found to follow asymptotic scaling laws for a narrow range of temperatures near the critical point, two methods with connections to the renormalization group theory provide analytical descriptions of the magnetic properties for much wider temperature ranges. Based on this, an analytical formula is obtained for the temperature dependence of the low field magnetoresistance in the paramagnetic phase.

Reynolds-number dependence of the longitudinal dispersion in turbulent pipe flow.

In Taylor's theory, the longitudinal dispersion in turbulent pipe flows approaches, on long time scales, a diffusive behavior with a constant diffusivity K_{L}, which depends empirically on the Reynolds number Re. We show that the dependence on Re can be determined from the turbulent energy spectrum. By using the intimate connection between the friction factor and the longitudinal dispersion in wall-bounded turbulence, we predict different asymptotic scaling laws of K_{L}(Re) depending on the different turbulent cascades in two-dimensional turbulence. We also explore numerically the K_{L}(Re) dependence in turbulent channel flows with smooth and rough walls using a lattice Boltzmann method.

Asymptotic Scaling Laws of Wireless Ad Hoc Network With Physical Layer Caching

We propose a physical layer (PHY) caching scheme for wireless ad hoc networks. The PHY caching exploits cache-assisted multihop gain and cache-induced dual-layer CoMP gain, which substantially improves the throughput of wireless ad hoc networks. In particular, the PHY caching scheme contains a novel PHY transmission mode called the cache-induced dual-layer CoMP, which can support homogeneous opportunistic CoMP in the wireless ad hoc network. Compared with traditional per-node throughput scaling results of $\Theta\left(1/\sqrt{N}\right)$ , we can achieve $O(1)$ per node throughput for a cached wireless ad hoc network with $N$ nodes. Moreover, we analyze the throughput of the PHY caching scheme for regular wireless ad hoc networks and study the impact of various system parameters on the PHY caching gain.

Asymptotic cross section and scaling law: positronium formation in Rydberg states in positron–hydrogen collisions

Positronium formation in Rydberg states from the ground state of the hydrogen atom by positron impact has been studied within the framework of a distorted wave theory which includes static dipole polarization potential. The distorted wave scattering amplitude has been obtained in a closed form. A detailed investigation has been made on the differential and total cross sections in the energy range 25–300 eV of incident positron. It has been found that asymptotic cross sections for the positronium formation into different angular momentum states obey a simple law.