Accurate Approximate Expressions Research Articles

A General Framework for UAV-aided THz Communications Subject to Generalized Geometric Loss

In this article, we develop a general framework for the performance analysis of unmanned aerial vehicle (UAV) aided terahertz (THz) communications under generalized geometric loss, which considers the fluctuations of the UAV's position and orientation alongside the non-orthogonality of the THz beam with respect to the detector plane. In particular, we derive general and novel analytical expressions for the probability density function and the cumulative distribution function of the instantaneous signal-to-noise ratio (SNR) under generalized fading channels subject to generalized geometric loss. Consequently, novel, general, and accurate analytical expressions for the outage probability, average symbol error rate, and average channel capacity are derived. Furthermore, approximate asymptotic expressions in the high SNR regime are also derived. Moreover, A very accurate approximate expression for the channel capacity is also provided in the low SNR regime. As an example, the system performance under the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> fading channel model is analyzed in terms of the above-mentioned performance metrics.

Performance investigation of FSO communication systems with chromatic dispersion, propagation losses and truncated normal modeled time jitter

FSO systems are identified as very reliable, effective and low-cost, wireless technologies for optical communications. However, the information carriers which are the optical pulses are subject to several effects during their propagation path. The atmospheric channels are heavily affected by the current weather conditions, causing signal fluctuations and power losses. Additionally, the very significant physical effect of the group velocity dispersion is responsible for the pulse shape changes along the propagation, especially for very short pulses, i.e. very high data rate transmission and long propagation distances. Furthermore, the time jitter affects strongly the reliability of the links, causing in practice, large irradiance fluctuations that may lead to signal outages or misdetections. All the above mentioned phenomena are studied jointly in this work and, for the first time to the best of our knowledge, the overall link performance is estimated in terms of links’ probability of fade, outage probability and average bit error rate by means of closed form or very accurate approximate mathematical expressions which can be used in any FSO link is needed. The weak-to-moderate time jitter effect, which is assumed in this work, here, is modeled by the truncated normal distribution. The corresponding numerical results for typical and realistic FSO parameters are presented while the obtained outcomes can be easily used for the estimation and designing of any other FSO under consideration.

Performance analysis of atmospheric optical communication systems with spatial diversity affected by correlated turbulence

This paper presents a complete analytical framework for obtaining the performance associated with a free-space optical (FSO) communication system with a spatial diversity and equal gain-combining technique. The system is affected by gamma–gamma scintillations with different realistic degrees of channel correlation depending only on the physical parameters of the link. We derive new analytical closed-form expressions for the average bit error rate (ABER) considering different scenarios to provide very realistic behavior of the system including different numbers of FSO receivers in several geometric configurations, with different receiving areas, different path lengths, and a variety of turbulence conditions. Furthermore, a very accurate approximate closed-form expression is also derived for the ABER of any generic coding scheme with either a very complex or, directly, no closed-form expression for its associated conditional BER that is first obtained in the ideal case of absence of turbulence. Numerical results via Monte Carlo simulation are provided to corroborate the validity of all the derived analytical expressions.

On the Performance of Spectrum-Sharing Backscatter Communication Systems

Spectrum-sharing backscatter communication (SSBC) systems are among the most prominent technologies for ultralow power and spectrum-efficient communications. In this article, we propose an underlay SSBC system, in which the secondary network is a backscatter communication system. We analyze the performance of the secondary network under a transmit power adaption strategy at the secondary transmitter, which guarantees that the interference caused by the secondary network to the primary receiver is below a predetermined threshold. We first derive a novel analytical expression for the cumulative distribution function (CDF) of the instantaneous signal-to-noise ratio of the secondary network. Capitalizing on the obtained CDF, we derive novel accurate approximate expressions for the ergodic capacity, effective capacity, and average bit-error rate. We further validate our theoretical analysis using the extensive Monte Carlo simulations.

The depletion thickness in solutions of semi-flexible polymers near colloidal surfaces: analytical approximations.

We derive a simple, yet accurate approximate mean-field expression for the depletion thickness δsf of a solution of dilute semi-flexible polymers next to a hard surface. In the case of a hard wall this equation has the simple form δsf = δ0[1 − tanh(psf/δ0)], where psf accounts for the degree of flexibility and δ0 is the depletion thickness in the case of fully flexible polymers. For fixed polymer coil size, increasing the chain stiffness leads to a decrease in the depletion thickness. The approach is also extended to include higher polymer concentrations in the semidilute regime. The analytical expressions are in quantitative agreement with numerical self-consistent field computations. A remarkable finding is that there is a maximum in the depletion thickness as a function of the chain stiffness in the semidilute concentration regime. This also means that depletion attractions between colloidal particles reach a maximum for a certain chain stiffness, which may have important implications for the phase stability of colloid–polymer mixtures. The derived equations could be useful for the description of interactions in- and phase stability of mixtures of colloids and semi-flexible polymers.

Capacity Analysis of UAV Communication System Based on FD-NOMA

In order to improve the communication quality between unmanned aerial vehicle (UAV) and ground users. An UAV communication system model based on full-duplex and non-orthogonal multiple access (NOMA) technology is proposed, and the capacity of the system model in urban scenarios is analyzed. First, the accurate capacity expression of the system model is given; then, the calculation problem of the exponential integral function in the formula is solved by introducing the Q function, and the approximate closed-form expression of the exponential integral function is obtained, and then the approximate closed-form expression of the capacity is obtained; second, the coefficient factor is used to obtain a more accurate approximate closed-form expression; finally, simulation and numerical results show that increasing the number of UAV or NOMA power vector can achieve better capacity performance.

Support region of μ-law logarithmic quantizers for Laplacian source applied in neural networks

The main aim of the paper is to provide effective and accurate solutions for the calculation of the support region of the μ-law logarithmic companding quantizers. A new solution for the starting point of iterative methods will be proposed, that provides very accurate value of the support region (being the main parameter needed for the design of the quantizer) only after one iteration of the iterative method. Based on this new starting point, an accurate closed-form approximate expression for the calculation of the support region will be derived, as one of the main contributions of the paper. To significantly simplify implementation of the μ-law companding quantizer, piecewise linearization is performed. A new linearization method is presented, based on the optimization of the last segments. Derivation of an accurate closed-form formula for the support region of the linearized quantizer is done, as an important contribution. The obtained linearized μ-law companding quantizer is very simple to design (due to closed-form formulas) and to implement (due to linearization), providing at the same time very high performance (due to optimization of the last segments). Due to these and other advantages (robustness, adjustability to the statistical distribution of the input signal), the proposed quantizer can be used in many topical applications, such as in receivers of 5G wireless systems or in neural networks for quantization of weights and activations. The paper provides an application of the designed quantizers for quantization of weights of a neural network, showing significant decreasing of the bit-rate compared to the standard full-precision representation (from 32 bits to just 5 bits), with the same prediction accuracy of the network.

A two-sided SPRT control chart for process dispersion

This paper proposes a two-sided sequential probability ratio test chart for monitoring dispersion of a normally distributed process. Accurate approximate expressions that relate the statistical performance measures of this chart to that of its corresponding one-sided charts are obtained through simulations. An approach to design the chart having the specified in-control performance and the optimal out-of-control performances for detecting the particular upper and lower shifts that are desired to detect as efficiently as possible is provided. It is found through numerical comparisons that the overall statistical performance of the proposed chart is superior to that of the competitor charts in the existing literature. The application of the chart is illustrated through real-data on the thickness of dielectric polyimide coating layer that is applied on a wafer used in an integrated circuit.

Dynamic Self-Consistent Field Approach for Studying Kinetic Processes in Multiblock Copolymer Melts.

The self-consistent field theory is a popular and highly successful theoretical framework for studying equilibrium (co)polymer systems at the mesoscopic level. Dynamic density functionals allow one to use this framework for studying dynamical processes in the diffusive, non-inertial regime. The central quantity in these approaches is the mobility function, which describes the effect of chain connectivity on the nonlocal response of monomers to thermodynamic driving fields. In a recent study, one of us and coworkers have developed a method to systematically construct mobility functions from reference fine-grained simulations. Here we focus on melts of linear chains in the Rouse regime and show how the mobility functions can be calculated semi-analytically for multiblock copolymers with arbitrary sequences without resorting to simulations. In this context, an accurate approximate expression for the single-chain dynamic structure factor is derived. Several limiting regimes are discussed. Then we apply the resulting density functional theory to study ordering processes in a two-length scale block copolymer system after instantaneous quenches into the ordered phase. Different dynamical regimes in the ordering process are identified: at early times, the ordering on short scales dominates; at late times, the ordering on larger scales takes over. For large quench depths, the system does not necessarily relax into the true equilibrium state. Our density functional approach could be used for the computer-assisted design of quenching protocols in order to create novel nonequilibrium materials.

Multi-Hop Wireless Optical Backhauling for LiFi Attocell Networks: Bandwidth Scheduling and Power Control

The backhaul of hundreds of light fidelity (LiFi) base stations (BSs) constitutes a major challenge. Building on an indoor wireless optical backhauling approach, this paper presents the top-down design of a multi-hop wireless backhaul configuration for multi-tier optical attocell networks by proposing super cells. Such cells incorporate multiple clusters of attocells that are connected to the core network via a single gateway. Consequently, new challenges arise for managing the bandwidth and power of the bottleneck backhaul. By putting forward user-based bandwidth scheduling (UBS) and cell-based bandwidth scheduling (CBS) policies, the system-level modeling and analysis of the end-to-end multi-user sum rate is elaborated. In addition, optimal bandwidth scheduling under both UBS and CBS policies are formulated as constrained convex optimization problems, and solved by using the projected subgradient method. Furthermore, the transmission power of the backhaul system is opportunistically reduced using a fixed power control (FPC) strategy. The notion of backhaul bottleneck occurrence (BBO) is introduced. An accurate approximate expression of the probability of BBO is derived, and verified using Monte Carlo simulations. Several insights are provided by studying different aspects of the performance of super cells including the average sum rate, the BBO probability and the backhaul power efficiency (PE).

DIVERSITY SIGNAL PROCESSING OVER WEIBULL FADING CHANNELS

We present a moments-based approach to the performance analysis of L-branch equal-gain combining and maximal-ratio combining receivers, operating in independent or correlated, not necessarily identically distributed, Weibull fading. For both equal-gain combining and maximal-ratio combining receivers the moments of the output signal-to-noise ratio are obtained in closed-form. An accurate approximate expression is derived for the moment-generating function of the output signal-to-noise ratio of the equal-gain combining receiver utilizing the Padé approximants theory, while a closed-form expression for the corresponding MGF of the maximal-ratio combining receiver, is obtained. Significant performance criteria, such as average output signal-to-noise ratio, amount of fading and spectral efficiency at the low power regime, are extracted in closed-forms, using the moments of the output signal-to-noise ratio for both independent and correlative fading. Moreover, using the well-known moment-generating function approach, the outage and the average symbol error probability for several coherent, non-coherent, binary, and multilevel modulation schemes, are studied. The average symbol error probability of dual-branch equal-gain combining and maximal-ratio combining receivers is also obtained when correlative fading is considered in the diversity input branches. The proposed mathematical analysis is illustrated by various numerical results and validated by computer simulations.

Dynamics and the periodic solutions of the delayed non-smooth Internet TCP-RED congestion control system via HB–AFT

This paper has attended to obtain the approximate analytical expressions of periodic solutions in the delayed non-smooth Internet TCP-RED congestion control system employing the semi-analytical method named as the harmonic balance method with alternating frequency/time (HB-AFT) domain technique. For the first time, the methodology presented herein provides the accurate approximate analytical expressions of periodic solutions, including the non-impacting periodic solutions and impacting periodic solutions, in the non-smooth dynamical system with time-delays. They agree very well with the results of numerical simulations by MATLAB. It implies that the proposed method in this paper is accurate and effective. Furthermore, rich dynamics of this delayed non-smooth system is discovered in the present paper. Three kinds of bi-stability, i.e. coexistence of asymptotically stable equilibrium and equilibrium, equilibrium and periodic solution, periodic solution and periodic solution, have been found in the system with the variation of the delay. And the grazing solution, the impacting solution traversing both the lower threshold Tmin and the larger threshold Tmax, and a route to chaos, i.e., intermittent to chaos, have been also obtained by varying the delay employing numerical method.

Approximate expression of beam wander of Gaussian array beams through oceanic turbulence

The general analytical formula of beam wander through oceanic turbulence is derived. An accurate approximate expression of this general formula is given based on the oceanic turbulence parameter characteristics. This approximation clearly indicates the relationship between beam wander and propagation distance, oceanic turbulence parameters (the rate of dissipation of kinetic energy per unit mass of fluid, the rate of dissipation of mean-squared temperature, the ratio of temperature and salinity contributions to the refractive index spectrum) as well as the beam spreading. Furthermore, the Gaussian array beams are considered as an example and another approximation is expressed considering that the laser beam underwater propagation distance is usually short. This approximation has quite simple and clear form which is convenient for engineering estimation. Numerical results verified the accuracy of the two approximation expressions and illustrate the beam wander characteristics under different oceanic turbulence conditions. Our results will be beneficial to evaluate the beam wander characteristics and select beam parameters in optical underwater communication systems.

Noncoherent Backscatter Communications Over Ambient OFDM Signals

In recent years, ambient backscatter communications have gained a lot of interests as a promising enabling technology for the Internet-of-Things and green communications. In ambient backscatter communication systems, ultra-low power devices are able to transmit information by backscattering ambient radio-frequency signals generated by legacy communication systems such as Wi-Fi and cellular networks. This paper is concerned with ambient backscatter communications over legacy orthogonal frequency division multiplexing (OFDM) signals. We propose a backscatter modulation scheme that allows backscattering devices to take advantage of the spectrum structure of ambient OFDM symbols to transmit information. The proposed modulation scheme allows both binary and higher-order modulation using noncoherent energy detection. We investigate the detector design and analyze the error performance of the proposed scheme. We provide an exact expression for the error probability for the binary case, whereas accurate approximate expressions for the error probability are derived for the $M$ -ary case. We corroborate our analysis using Monte–Carlo simulation and investigate the effects of varying the OFDM symbol size, maximum channel delay spread, and the number of receive antennas on the error performance. Our numerical results show that the proposed technique outperforms other techniques available in this paper for backscatter communication over ambient OFDM signals in different scenarios.

Parametric Study of RF Energy Harvesting in SWIPT enabled wireless networks under downlink scenario

In this paper, we derive and analyze the average Energy Harvested and average achievable downlink rate expressions for the formulated SWIPT enabled system. Rician channel modeling is used for simulation because the distance between Base Station and Users are assumed to be close and thus Line of Sight component is included. After initializing appropriate parameters, definite and accurate expressions and approximate expressions are derived. Results are simulated and analyzed to validate the theory and the derived expressions, by varying efficiency, Rician factor and Path loss exponent.

Wireless energy harvesting in cooperative decode‐and‐forward relaying networks over mixed generalized η‐μ and κ‐μ fading channels

AbstractIn this paper, we investigate the performance of dual‐hop relaying networks with wireless energy harvesting at the relay node over generalized η‐μ and κ‐μ fading channels. Our analysis considers decode‐and‐forward relays with a single antenna as well as source and destination nodes equipped with a single antenna. Specifically, we derive an accurate approximate analytical expression for the system outage probability (OP). The derived OP considers 3 different energy harvesting mechanisms: (1) time switching–based relaying, (2) power splitting–based relaying, and (3) ideal relaying receiver. Furthermore, we obtain accurate approximate analytical expressions for the system ergodic capacity and achievable throughput for the time switching–based relaying, power splitting–based relaying, and ideal relaying receiver. In addition, an accurate approximate analytical expression for the average bit error rate for several coherent modulation schemes is obtained. Through some numerical examples, we study the OP, throughput, and average bit‐error‐rate performance of the overall system under different system parameters such as fading parameters, energy harvesting time, power splitting factor, and signal‐to‐noise ratio. The results show that the effect of increasing the fading parameter μ on the system performance is more pronounced than that of increasing the fading parameters η and κ. Besides, increasing the energy harvesting time or the power splitting factor may be either beneficial or harmful for the throughput performance. Our numerical results are accompanied with Monte Carlo simulations to verify the accuracy of our analysis.

Source Transmit Antenna Selection for Space Shift Keying With Cooperative Relays

In this letter, we propose a cooperative multiple-input multiple-output (MIMO) scheme combining transmit antenna selection (TAS) and space shift keying (SSK). In this scheme, source transmit antennas are selected and SSK is applied by using the selected antennas. Besides the direct link transmission, the relays, that decode the source signal correctly, take part in the transmission. Exact expressions and a considerably accurate approximate expression for the symbol error rate of the proposed SSK system are derived. It is shown that the proposed scheme outperforms the SSK system without TAS and also the conventional cooperative MIMO system, which employs source TAS, at practical signal-to-noise ratio values for especially high data rates and sufficient number of receive antennas at the destination.

Managing Counterparty Credit Risk Via BSDEs

We discuss a general dynamic replication approach to counterparty credit risk modeling. This leads to a fundamental jump-process backward stochastic differential equation (BSDE) for the credit risk adjusted portfolio value. We then reduce the fundamental BSDE to a continuous BSDE. Depending on the close out value convention, the reduced fundamental BSDE's solution can be represented explicitly or through an accurate approximate expression. Furthermore, we discuss practical aspects of the approach, important for industry applications: (i) efficient numerical methodology for solving a BSDE driven by a moderate number of Brownian motions, and (ii) factor reduction methodology that allows one to approximately replace a portfolio driven by a large number of risk factors with a portfolio driven by a moderate number of risk factors.

Elastic behavior of weakly cemented contact

SummaryWe investigate the elastic behavior of weakly cemented contact. We show that the radial distribution of stresses and the stiffnesses of a cemented contact are governed by the ratio a/RΛ, where R, a, and Λ are, respectively, the grain radius, the contact size, and the ratio of the elastic moduli of cement and grains. Moreover, we show that a cemented contact is always less stiff than a Hertzian contact having a similar size. Finally, we propose accurate approximate expressions of the contact stiffnesses. Copyright © 2014 John Wiley &amp; Sons, Ltd.

Simple Formulae for Distortional Buckling Loads of Cold-Formed Thin-Walled Rack Members Upright with Rear Flange and Additional Lip Stiffeners

The determination of the elastic distortional buckling loads of the cold-formed lipped channel sections upright with rear flange and additional lip stiffeners subjected to the pure bending was performed based on the analytic solutions combined with CUFSM. The parameter analyses were carried out under the extreme load case of pure bending, leading to approximate formulae for predicting the buckling half-wavelength and the modified factor to take into account the effect of the shear and distortional deformation of the flange section. The rotational restraint, provided by the web to the flange during buckling deformation, was then examined, and a more accurate linear approximate expression for the rotational restraint and the web stress was proposed. Based on these approximate formulae, simplified formulae were finally developed to predict the elastic distortional buckling loads of the cold-formed lipped channel sections upright with rear flange and additional lip stiffeners subjected to the pure bending. The proposed formulae are shown to be accurate through comparison with the results of CUFSM, easy to be applied, may be used in practical applications and taken account into design codes and guidelines.