Exact Analytical Closed-form Expressions Research Articles

Performance analysis of UAV assisted triple-hop mixed FSO/RF/UWOC system with altitude optimization

Abstract In this paper, we propose a unmanned aerial vehicle (UAV) assisted triple hop mixed FSO/RF/UWOC system to establish a communication between distant users. In the proposed system, the source sends information signal towards a hovering UAV such as drone acting as a first intermediate node via FSO link modelled by Gamma-Gamma distribution. The drone further transmits the received signal to the second intermediate node located on a floating vehicle (FV) on the ocean surface via a RF link modelled by the Nakagami-m fading distribution and from where it is finally forwarded towards the deep ocean destination via visible light channel (VLC) link which is modelled by Exponential Generalized Gamma (EGG) fading distribution. Both the relay nodes act as decode and forward relays. For this asymmetric triple hop mixed communication system, we obtain a new exact closed form analytical expression for the outage probability and also study its behaviour at high signal to noise ratio (SNR) regime. Additionally, the UAV altitude optimization is also carried out for the proposed system to find the optimal altitude of the UAV and the optimal elevation angle to maximize the system’s performance. The numerical simulation results illustrate the accuracy of the proposed system model and present the influence of the various link parameters such as atmospheric path loss, turbulence, pointing errors, angle-of-arrival fluctuations, multipath fading parameter, water turbulence, and scintillation on the performance of the proposed system.

The exact element stiffness matrices of stochastically parametered beams

Stiffness matrices of beams with stochastic distributed parameters modelled by random fields are considered. In stochastic finite element analysis, deterministic shape functions are traditionally employed to derive stiffness matrices using the variational principle. Such matrices are not exact because the deterministic shape functions are not derived from the exact solution of the governing stochastic partial differential equation with the relevant boundary conditions. This paper proposes an analytical method based on Castigliano’s approach for a beam element with general spatially varying parameters. This gives the exact and a simple closed-form expression of the stiffness matrix in terms of certain integrals of the spatially varying function. The expressions are valid for any integrable random fields. It is shown that the exact element stiffness matrix of a stochastically parametered beam can be expressed by three basic random variables. Analytical expressions of the random variables and their associated coefficient matrices are derived for two cases: when the bending rigidity is a random field and when the bending flexibility is a random field. It is theoretically proved that the conventional stochastic element stiffness matrix is a first-order perturbation approximation to the exact expression. A sampling method to obtain the basic random variables using the Karhunen–Loève expansion is proposed. Results from the exact stiffness matrices are compared with the approximate conventional stiffness matrix. Gaussian and uniform random fields with different correlation lengths are used to illustrate the numerical results. The exact closed-form analytical expression of the element stiffness matrix derived here can be used for benchmarking future numerical methods.

The exact closed-form expressions for optimal design parameters of resonating base isolators

In this paper, the exact closed-form analytical expressions of the optimal design parameters for two different base isolation (BI) configurations, i.e., grounded (GR-BI) and ungrounded (UR-BI), are derived employing H2 and H∞ optimization methods. To compare the performance of resonating isolators fairly with an equivalent base isolator, an additional mass is added to the base isolator’s mass. Frequency domain and time history analysis have been conducted to obtain each isolator’s exact vibration reduction capacity. Results from frequency domain analysis show that H2 optimized ungrounded resonating base isolator has a negligible effect on the vibration mitigation compared to the equivalent base isolator having equal total mass. The response reduction capacity of the GR-BI is significantly 23.81% superior to the BI and UR-BI both while implementing H2 optimization method. On the other hand, for H∞ optimization, the response reduction capacity of the GR-BI is significantly 26.1% and 34.98% superior to the BI and UR-BI, respectively. Subsequently, numerical studies employing the Newmark-beta method have been conducted to obtain time-domain responses using near-field earthquake base excitations. Results from time history analysis depict that the response reduction capacity of H2 optimized GR-BI is significantly 47.80% and 47.51% superior to the response reduction capacity of H2 optimized BI and UR-BI. In contrast, the response reduction capacity of H∞ optimized BI is significantly 56.45% and 43.75%, superior to the response reduction capacity of H∞ optimized UR-BI and GR-BI. However, the response reduction capacity of H∞ optimized GR-BI is significantly 22.58% superior to the response reduction capacity of H∞ optimized UR-BI. The requirement of the damping ratios of all the proposed isolators are within unity; hence these are affordable. The results are mathematically accurate and feasible for practical design purposes.

Evaluation of Full-Duplex SWIPT Cooperative NOMA-Based IoT Relay Networks over Nakagami-m Fading Channels.

In this paper, we investigate the performance of non-orthogonal multiple access (NOMA)-based full-duplex Internet-of-Things (IoT) relay systems with simultaneous wireless information and power transfer (SWIPT) over Nakagami-m fading channels to improve the performance of a cell-edge user under perfect and imperfect successive interference cancellation (SIC). Two scenarios, i.e., direct and non-direct links, between the source node and cell-edge user are examined. The exact closed-form analytical and approximate expressions for the outage probability, system throughput, energy efficiency, and ergodic capacities are derived and validated via Monte Carlo simulations to characterize the proposed system performance. To further improve the system performance, we also provide a low-complexity algorithm to maximize the system throughput over-optimizing the time-switching factor. The results show that our proposed NOMA system can achieve superior performance compared to its orthogonal multiple access (OMA) counterpart under perfect SIC and with a low-to-medium signal-to-noise ratio under imperfect SIC, according to the level of residual self-interference and the quality of links.

Wireless Powered Cognitive NOMA-Based IoT Relay Networks: Performance Analysis and Deep Learning Evaluation

In this article, we study novel wireless powered cognitive nonorthogonal multiple access (NOMA)-based Internet-of-Things (IoT) relay networks to improve the performance of a cell-edge user under perfect and imperfect successive interference cancelation (SIC). In the secondary networks, a source node communicates with a cell-center user via direct link and with a cell-edge user through the assistance of a master IoT node under cognitive radio constraint. Exact closed-form analytical expressions for the outage probability (OP) of NOMA users and the overall system throughput are derived. To provide further insights, a performance floor analysis is also carried out considering two power-setting scenarios: 1) the transmit powers at the power beacon goes to infinity and 2) the maximum allowable power constraint goes to infinity. Moreover, we develop two iterative algorithms for minimizing OP users and maximizing system throughput subject to time-switching and power-allocation factors in two-hop transmission. Direct derivation of the closed-form expression for the ergodic capacity (EC) becomes unfeasible due to the high complexity of the proposed system model. To overcome this issue, we design a deep neural network (DNN) framework for the EC prediction toward real-time configurations. Our results show that the predicted results based on this DNN framework perfectly align with the simulations, validating our design framework. In addition, the DNN approach exhibits the lowest root-mean-square error and low run-time predictions among other regression models.

A general analytical framework for the mechanics of heterogeneous hexagonal lattices

The in-plane mechanics of two-dimensional heterogeneous hexagonal lattices are investigated. The heterogeneity originates from two physically realistic considerations: different constituent materials and different wall thicknesses. Through the combination of multi-material and multi-thickness elements, the most general form of 2D heterogeneous hexagonal lattices is proposed in this paper. By exploiting the mechanics of a unit cell with multi-material and multi-thickness characteristics, exact closed-form analytical expressions of equivalent elastic properties of the general heterogeneous lattice have been derived. The equivalent elastic properties of the 2D heterogeneous lattice are Young’s modulli and Poisson’s ratios in both directions and the shear modulus. Two distinct cases, namely lattices with thin and thick constituent members, are considered. Euler–Bernoulli beam theory is employed for the thin-wall case, and Timoshenko beam theory is employed for the thick-wall case. The closed-form expressions are validated by independent finite element simulation results. The generalized expressions can be considered as benchmark solutions for validating future numerical and experimental investigations. The conventional single-material and single-thickness homogeneous lattice appears as a special case of the heterogeneous considered here. By introducing the Material Disparity Ratio (MDR) and Geometric Disparity Ratio (GDR), variability in the equivalent elastic properties has been graphically demonstrated. As opposed to classical homogeneous lattices, heterogeneous lattices significantly expand the design space for 2D lattices. Orders-of-magnitude of variability in the equivalent elastic properties is possible by suitably selecting material and geometric disparities within the lattices. The general closed-form expressions proposed in this paper open up the opportunity to design next-generation heterogeneous lattices with highly tailored effective elastic properties.

The eigenbuckling analysis of hexagonal lattices: closed-form solutions

Elastic instability such as the buckling of cellular materials plays a pivotal role in their analysis and design. Despite extensive research, the quantifi- cation of critical stresses leading to elastic instabi- lities remains challenging due to the inherent nonlinearities. We develop an analytical approach considering the spectral decomposition of the elasticity matrix of two-dimensional hexagonal lattice materials. The necessary and sufficient condition for the buckling is established through the zeros of the eigenvalues of the elasticity matrix. Through the analytical solution of the eigenvalues, the conditions involving equivalent elastic properties of the lattice were directly connected to the mathematical requirement of buckling. The equivalent elastic properties are expressed in closed form using geometric properties of the lattice and trigonometric functions of a non-dimensional axial force parameter. The axial force parameter was identified for four different stress cases, namely, compressive stress in the longitudinal and transverse directions separately and together and torsional stress. By solving the resulting nonlinear equations, we derive exact analytical expressions of critical eigenbuckling stresses for these four cases. Crucial parameter combinations leading to minimum buckling stresses are derived analytically. The exact closed-form analytical expressions derived in the paper can be used for quick engineering design calculations and benchmarking related experimental and numerical studies.

A general frequency adaptive framework for damped response analysis of wind turbines

Dynamic response analysis of wind turbine towers plays a pivotal role in their analysis, design, stability, performance and safety. Despite extensive research, the quantification of general dynamic response remains challenging due to an inherent lack of the ability to model and incorporate damping from a physical standpoint. This paper develops a frequency adaptive framework for the analysis of the dynamic response of wind turbines under general harmonic forcing with a damped and flexible foundation. The proposed method is founded on an augmented dynamic stiffness formulation based on a Euler-Bernoulli beam-column with elastic end supports along with tip mass and rotary inertia arising from the nacelle of the wind turbine. The dynamic stiffness coefficients are derived from the complex-valued transcendental displacement function which is the exact solution of the governing partial differential equation with appropriate boundary conditions. The closed-form analytical expressions of the dynamic response derived in the paper are exact and valid for higher frequency ranges. The proposed approach avoids the classical modal analysis and consequently the ad-hoc use of the modal damping factors are not necessary. It is shown that the damping in the wind turbine dynamic analysis is completely captured by seven different physically-realistic damping factors. Numerical results shown in the paper quantify the distinctive nature of the impact of the different damping factors. The exact closed-form analytical expressions derived in the paper can be used for benchmarking related experimental and finite element studies and at the initial design/analysis stage.

Reliability and Security Performance Analysis of Hybrid Satellite-Terrestrial Multi-Relay Systems With Artificial Noise

Land mobile satellite communication and physical layer security are considered as the promising paradigms in beyond 5G networks. In this paper, we establish the multi-relay hybrid satellite-terrestrial systems (HSTSs) with artificial noise (AN) to investigate the reliability and security, in which the satellite sends signals to the terrestrial destination through multiple decode-and-forward relays and there exists an eavesdropper trying to overhear the useful information. In order to improve the overall system performance, two relay selection (RS) schemes: i) optimal relay-receiver pair selection (ORRPS) and ii) suboptimal relay selection (SRS) are considered. ORRPS selects an optimal relay-receiver pair and an AN signal to maximize the corresponding signal-to-interference-plus-noise ratios of ${R_{k}} \to D\left ({{R_{k} \to E} }\right)$ and the SRS refers to selecting a relay that optimize the instantaneous channel gain ${\left |{ {{h_{R_{k}D}}} }\right |^{2}}\left ({{{{\left |{ {{h_{R_{k}E}}} }\right |}^{2}}} }\right)$ among $K$ relays, with the traditional round-robin selection scheme as a benchmark for comparison. Furthermore, the exact closed-form analytical expressions of outage probability (OP) and intercept probability (IP) under different RS schemes are derived for AN and non-AN conditions. To obtain more insights, the asymptotic analysis for high signal-to-noise-ratio regime is carried out. Based on the asymptotic analysis of OP, the diversity orders of the HSTSs are obtained. The numerical results and theoretical analysis shown that: i) The reliability of TRRS scheme is the worst among the three RS schemes, and the reliability of ORRPS scheme is the best; ii) The AN technology can significantly improve the security of the system, although it has a little adverse affect on the reliability; iii) Infrequent light shadowing has the best reliability, frequent heavy shadowing has the worst outage performance, and the change of security is opposite; iv) The OP of SRS and ORRP schemes decreased with the increase of $K$ , while the IP changed on the contrary, and the reliability and security of TRRS scheme did not change obviously.

On HARQ Schemes in Satellite-Terrestrial Transmissions

As an efficient supplement for terrestrial communications, e.g., fifth-generation communication networks, satellite communication has been proved as a promising choice to realize seamless coverage, due to its inherent superiority over common wireless communication systems, like large-scale coverage area and flexibility of deployment. In this article, a satellite-terrestrial transmission (STT) system including a satellite transmitter (S) and a group of terrestrial receivers (D) is focused on. Considering the randomness of D and employing stochastic geometry, the outage performance and diversity gain of three hybrid automatic repeat request (HARQ) schemes, generalized slotted ALOHA (GSA), repetition time diversity (RTD) and general incremental redundancy (IR), are respectively studied. In detail, the exact closed-form analytical expression for the outage probability (OP) is derived for the GSA scheme, while the approximated expression for the OP of RTD scheme is presented and a useful expression for the OP of IR scheme is derived. Furthermore, the diversity gains of the three considered HARQ schemes are respectively investigated in STT scenarios. Finally, some numerical results are provided to validate the proposed analysis models.

Average bit error rate and channel capacity of terahertz wireless line-of-sight links with pointing errors under combined effects of turbulence and snow.

The estimation of the performance of terahertz wireless communication under the effect of various weather conditions is vital. In this work, the combined effects of cold weather conditions such as snow, as well as random effects of turbulence and pointing errors (PEs) between the transmitter and receiver, on the performance of terahertz wireless line-of-sight links have been evaluated. The exponentiated Weibull distribution has been employed to derive exact analytical closed-form expressions in the presence of combined channels. Our predictions indicate that snow can have various influences on the average bit error rate (BER) and the average channel capacity of terahertz wireless links. Dry and wet snow, based on its liquid-water content, shows different effects in terms of link parameters such as distance and frequency compared to that of rain. Random turbulence and PEs further deteriorate the link performance. It is concluded that the channel capacity is less affected than the BER under the effects of turbulence and PEs, while snow can have a remarkable effect. All of the predictions through the derived expressions are validated using Monte Carlo simulations.

Secrecy Outage Analysis for Satellite-Terrestrial Downlink Transmissions

This letter investigates the secrecy outage performance of satellite-to-terrestrial downlink transmissions consisting of one legitimate receiver ( ${D}$ ) and one eavesdropper ( ${E}$ ). To reflect the practical application scenarios, it is assumed that ${D}$ and ${E}$ are randomly distributed in the footprint of the satellite. Furthermore, in the considered downlink scenario, ${D}$ and ${E}$ are equipped with multiple antennas, and maximal ratio combining (MRC) technique is adopted at both of them to seek the maximum receiving diversity gain. To reveal the impacts of the randomness of the positions of ${D}$ and ${E}$ , and MRC scheme on the secrecy outage performance of the considered satellite-terrestrial system, the exact and asymptotic closed-form analytical expressions for secrecy outage probability are derived by using stochastic geometry method. Finally, simulation results are given to confirm the accuracy of our proposed analytical models.

Outage of Cognitive Electric Vehicle Networks Over Mixed RF/VLC Channels With Signal-Dependent Noise and Imperfect CSI

In this paper, we propose a novel jamming-robust communication technique for the outdoor cognitive EV-enabled network over mixed radio-frequency (RF)/visible light communication (VLC) channels with signal-dependent noise. One EV acts as a relaying node to allow an aggregator to reach the jammed EV and, at the same time, operates in both RF and VLC spectrum bands while satisfying interference constraints imposed by the primary network entities. We derive an exact closed-form analytical expression for the outage probability and also provide its asymptotic analysis while considering various channel state information quality scenarios. Finally, simulation results validate the accuracy of our analysis.

Outage Analysis of Cognitive Electric Vehicular Networks Over Mixed RF/VLC Channels

Modern transportation infrastructures are considered as one of the main sources of the greenhouse gases emitted into the atmosphere. This situation requires the decision-making players to enact the mass use of electric vehicles (EVs) which, in turn, highly demand novel secure communication technologies robust to various cyber-attacks. Therefore, in this paper, we propose a novel jamming-robust communication technique for different outdoor cognitive EV-enabled network cases over mixed radio-frequency (RF)/visible light communication (VLC) channels. One EV acts as a relaying node to allow an aggregator to reach the jammed EV and, at the same time, operates in both RF and VLC spectrum bands while satisfying interference constraints imposed by the primary network entities. We derive exact closed-form analytical expressions for the outage probability and also provide their asymptotic analysis while considering various channel state information quality scenarios. Moreover, we quantify the outage reduction achievable by deploying such mixed VLC/RF channels. Finally, analytical and simulation results validate the accuracy of our analysis.

Performance assessment of correlated Rayleigh‐inverse Gaussian fading channel over distributed MIMO systems with ZF detectors

Composite fading channel modelling of terrestrial wireless propagation is crucial for the design of communication system and its performance analysis. In this study, a systematic characterisation of Rayleigh-inverse Gaussian (RIG) distribution is provided for modelling the channel of distributed multiple-input multiple-output (MIMO) systems with zero-forcing (ZF) receiver in the presence of transmit antenna correlation. At the outset, the closed-form expression for probability density function of instantaneous signal-to-noise ratio (SNR) is derived followed by its associated moments and cumulative density function. In the subsequent analysis, exact closed-form analytical expression of ergodic capacity with its bounds are derived and their asymptotic expressions are obtained in high and low SNR regime. Then the closed-form expressions are deduced for average symbol error rate (ASER) and outage probability (OP). Additionally, the authors propose to formulate simplified expressions for asymptotic ASER as well as OP and assess their approximation accuracy. Numerical results are presented to compare the performance of RIG model with the existing generic models and illustrate that RIG is superior in terms of ergodic capacity, ASER and OP. Finally, the derived analytical expressions are validated through Monte Carlo simulations which demonstrate the effectiveness of RIG model in the diverse environmental scenario for various system and channel parameters.

On the Performance Analysis of WPT-Based Dual-Hop AF Relaying Networks in <inline-formula> <tex-math notation="LaTeX">$\alpha$ </tex-math> </inline-formula>-<inline-formula> <tex-math notation="LaTeX">$\mu$ </tex-math> </inline-formula> Fading

In this paper, a two-hop amplify-and-forward relaying system, where an energy-constrained relay node entirely depends on the energy scavenged from the source signal, is investigated. This paper analyzes the performance of the energy-harvesting (EH) protocols, namely, ideal relaying receiver, power-splitting relaying (PSR), and time-switching relaying (TSR), over independent but not identically distributed (i.n.i.d.) $\alpha $ - $\mu $ fading channels in terms of the ergodic capacity and ergodic outage probability (OP). We derive exact unified and closed-form analytical expressions for the performance metrics with the aforementioned protocols over i.n.i.d. $\alpha $ - $\mu $ channels. Three fading scenarios, such as Weibull, Nakagami- $m$ , and Rayleigh channels, are investigated. Provided simulation and numerical results validate our analysis. It is demonstrated that the optimal EH time-switching and power-splitting factors of the corresponding TSR and PSR protocols are critical in achieving the best system performance. Finally, we analyzed the impact of the fading parameters $\alpha $ and $\mu $ on the achievable ergodic OP.

Cooperative Communications With Wireless Energy Harvesting Over Nakagami- $m$ Fading Channels

In this paper, a dual-hop decode-to-forward cooperative system is considered where multiple relays are with finite energy storage and can harvest energy from the destination. In our analysis, the relays are spatially randomly located with invoking stochastic geometry. In an effort to improve spectral efficiency, an optimal source-relay link scheme is employed. Assuming Nakagami- $m$ fading, two different scenarios are considered: 1) the single-antenna source with perfect channel state information (CSI) and 2) the multiple-antenna source with transmit antenna selection and imperfect CSI. In both scenarios, the destination node is equipped with a single transmit antenna to forward power via frequency radio signal to the relay candidates. For improving the system performance, multiple antennas at the destination are considered to process the multiple copies of the received signal from the best relay. For characterizing the performance of the proposed scenarios, exact closed-form analytical expressions for the outage probability are derived. To obtain further insights, we carry out diversity gain analysis by adopting asymptotic relative diversity. We also derive the exact closed-form analytical expression for the system throughput. Finally, simulation results are presented to corroborate the proposed analysis and to show that: 1) the system performance is improved by enlarging the area of the circle and the density of the relays and 2) the energy storage size has impacts on the performance of considered networks, which determines the maximal transmit power at relays.

Radiation force and torque of light-sheets

The aim of this work is to provide exact analytical closed-form expressions for the longitudinal and transverse optical radiation force and axial spin torque components, for a 2D surface cross-section with arbitrary shape in the field of light-sheet beams of arbitrary wavefront. Generalized partial-wave series expressions for the longitudinal and transverse optical radiation forces and torque are derived based on the multipole expansion in cylindrical wave functions, stemming from series expansions for the incident and scattered electromagnetic fields. The incident light-sheet wavefields are expressed using generalized series involving the beam-shape coefficients (BSCs), and the scattered fields are given in terms of series involving the scattering coefficients of the object. Numerical illustrative examples on a dielectric absorptive circular cylindrical cross-section are provided for different wavefronts, ranging from plane waves, as well as non-paraxial scalar Airy and Gaussian light-sheet beams. The BSCs are derived based on the angular spectrum decomposition method, which provides adequate means to evaluate the radiation force and torque components when the cylinder cross-section is centered on the beam, or shifted off-axially with respect to the incident axis of wave propagation. In essence, the present theoretical analysis provides a complete formalism in the framework of the generalized Lorenz–Mie theory in 2D based upon exact closed-form series expressions to compute the optical force and torque components induced by 2D light-sheets of arbitrary wavefronts, illuminating a scatterer with an arbitrary geometrical cross-section (in 2D). Possible applications are in particle transport and rotation.

Performance Analysis of Digital Communication Systems Over $\alpha{-}\eta{-}\mu$ Fading Channels

This paper analyzes and evaluates the performance of digital communication systems that operate over the $\alpha{-}\eta{-}\mu$ fading channels. More specifically, we derived novel, unified, and exact closed-form analytical expressions for the cumulative density function (CDF), the moment-generating function (MGF), the average channel capacity, and the average symbol error probability (SEP) for several coherent and non-coherent modulation schemes. Note that the derived expressions are valid for arbitrary values of the fading parameters. The derived expressions are then used to study the implication of the fading parameters on system performance. In addition, the performance over other well-known fading channels, such as $\eta{-}\mu$ and $\alpha{-}\mu$ , as well as their inclusive special cases, can be analyzed using our results. To validate the correctness of our derivations, the numerical results are compared with Monte Carlo simulation results. Both results are in perfect agreement over a wide range of average signal-to-noise ratio (SNR) and different values of the fading parameters.

Average BER performance of FSO SIM-QAM systems in the presence of atmospheric turbulence and pointing errors

This paper presents the exact average bit error rate (BER) analysis of the free-space optical system employing subcarrier intensity modulation (SIM) with Gray-coded quadrature amplitude modulation (QAM). The intensity fluctuations of the received optical signal are caused by the path loss, atmospheric turbulence and pointing errors. The exact closed-form analytical expressions for the average BER are derived assuming the SIM-QAM with arbitrary constellation size in the presence of the Gamma–Gamma scintillation. The simple approximate average BER expressions are also provided, considering only the dominant term in the finite summations of obtained expressions. Derived expressions are reduced to the special case when optical signal transmission is affected only by the atmospheric turbulence. Numerical results are presented in order to illustrate usefulness of the derived expressions and also to give insights into the effects of different modulation, channel and receiver parameters on the average BER performance. The results show that the misalignment between the transmitter laser and receiver detector has the strong effect on the average BER value, especially in the range of the high values of the average electrical signal-to-noise ratio.