Exact Closed-form Expressions Research Articles

Throughput enhancement for multi-hop decode-and-forward protocol using interference cancellation with hardware imperfection

In this paper, we propose three interference cancellation based multi-hop decode-and-forward (DF) relaying protocols to enhance end-to-end system throughput under constraint of half-duplex and hardware imperfection. In the first proposed protocol, the successive interference cancellation technique is used at each hop to send N signals from a source to a destination. Hence, the first proposed protocol can increase N times of the data rate, as compared with the corresponding conventional multi-hop DF relaying protocol in which only one source data is hop-by-hop relayed to the destination via M intermediate hops. In the second one, the source and relays can at the same time transmit their signals, and the interference cancellation can be performed to remove co-channel interference components. Therefore, the data rate of the second proposed protocol can increase M/2 times, as compared with the corresponding conventional protocol. The third proposed protocol is an efficient combination of the first and second proposed ones, and hence its data rate is higher N times than the second proposed one. Moreover, the imperfect co-channel interference cancellation is also considered in the second and third protocols. For performance evaluation, we derive exact closed-form expressions of the end-to-end throughput for three proposed protocols over Nakagami-m fading channel. We also perform simulations to verify the analytical formulas, and propose an simple transmit power allocation strategy to mitigate impact of the co-channel interference and imperfect co-channel interference cancellation.

Active multi-physical modulation of Poisson’s ratios in composite piezoelectric lattices: On-demand sign reversal

A large number of studies have been reported in literature to analyse various parameters that influence the Poisson’s ratios of multi-functional lattice materials. However, the major limitation in such lattices is that once the lattice is manufactured, the Poisson’s ratios and other elastic properties become fixed corresponding to the particular lattice configuration. This limits the application of such lattices in many advanced multi-functional structures and systems, where on-demand active property modulations are warranted. This paper proposes composite lattices comprising a substrate and piezoelectric materials, wherein it is possible to actively modulate the Poisson’s ratios and other elastic properties as a function of voltage. Considering both axial and transverse deformations of the cell walls along with a unit cell based approach, the exact closed-form expressions of Poisson’s ratios and equivalent Young’s modulus are derived in an expanded design space for the unimorph and bimorph configurations. The study reveals that a sign reversal of the Poisson’s ratios and Young’s modulus can be achieved for specific combinations of cell angle and applied voltage. Contrary to the conventional wisdom, such active sign reversal implies that re-entrant honeycomb lattices can exhibit positive Poisson’s ratios and vice versa.

Analysis of RIS-Based Terrestrial-FSO Link Over G-G Turbulence With Distance and Jitter Ratios

One of the main problems faced by communication systems is the presence of skip-zones in the targeted areas. With the deployment of the fifth-generation mobile network, solutions are proposed to solve the signal loss due to obstruction by buildings, mountains, and atmospheric or weather conditions. Among these solutions, reconfigurable intelligent surfaces (RIS), which are newly proposed modules, may be exploited to reflect the incident signal in the direction of dead zones, increase communication coverage, and make the channel smarter and controllable. This paper tackles the skip-zone problem in terrestrial free-space optical (T-FSO) systems using a single-element RIS. Considering link distances and jitter ratios at the RIS position, we carry out a performance analysis of RIS-aided T-FSO links affected by turbulence and pointing errors, for both heterodyne detection and intensity modulation-direct detection techniques. Turbulence is modeled using the Gamma-Gamma distribution. We analyze the model and provide exact closed-form expressions of the probability density function, cumulative distribution function, and moment generating function of the end-to-end signal-to-noise ratio. Capitalizing on these statistics, we evaluate the system performance through the outage probability, ergodic channel capacity, and average bit error rate for selected binary modulation schemes. Numerical results, validated through simulations, obtained for different RIS positions and link distances ratio values, reveal that RIS-based T-FSO performs better when the RIS module is located near the transmitter.

Exact Analysis of RIS-Aided THz Wireless Systems Over α-μ Fading With Pointing Errors

Reconfigurable intelligent surfaces (RIS) can have an excellent use case for terahertz (THz) wireless transmissions. In this letter, we analyze the performance of a RIS-empowered THz system over the combined effect of <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 and pointing errors. We derive exact closed-form expressions of density and distribution functions of the resultant signal-to-noise ratio (SNR) considering an independent and not identically distributed (i.ni.d.) channel model. Using the derived statistical results, we present an exact analysis on outage probability, ergodic capacity, and average bit-error-rate (BER) of the considered system. We also develop asymptotic analysis on the outage probability and average BER to derive diversity order in terms of system parameters. The proposed analysis provides insights to mitigate the effect of pointing errors and shows that the RIS can significantly improve the performance of THz communications. We validate the analytical results using numerical and Monte Carlo simulations and demonstrate the scaling of system performance with an increase in the number of RIS elements.

Beamforming Using Exact Evaluation of Leakage and Ergodic Capacity of MU-MIMO System.

Closed-form evaluation of key performance indicators (KPIs) of telecommunication networks help perform mathematical analysis under several network configurations. This paper deals with a recent mathematical approach of indefinite quadratic forms to propose simple albeit exact closed-form expressions of the expectation of two significant logarithmic functions. These functions formulate KPIs which include the ergodic capacity and leakage rate of multi-user multiple-input multiple-output (MU-MIMO) systems in Rayleigh fading channels. Our closed-form expressions are generic in nature and they characterize several network configurations under statistical channel state information availability. As a demonstrative example of the proposed characterization, the derived expressions are used in the statistical transmit beamformer design in a broadcast MU-MIMO system to portray promising diversity gains using standalone or joint maximization techniques of the ergodic capacity and leakage rate. The results presented are validated by Monte Carlo simulations.

Impacts of Imperfect CSI and Transceiver Hardware Noise on the Performance of Full-Duplex DF Relay System With Multi-Antenna Terminals Over Nakagami-m Fading Channels

In this paper, we investigate the performance of a full-duplex (FD) relay system where multi-antennas are exploited at source and destination. Unlike previous works, the impacts of imperfect channel state information (I-CSI), transceiver hardware noise (THN), and residual self-interference (RSI) are taken into account. We mathematically derive the exact closed-form expressions of the outage probability (OP), symbol error rate (SER), and ergodic capacity (EC) of the FD relay system with I-CSI, THN, and RSI over Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> fading channels. From the derived expressions, the performance of the considered system under the effects of three negative factors (I-CSI, THN, and RSI) is compared with that system in the case of all ideal factors (perfect channel state information (P-CSI), perfect transceiver hardware (P-TH) and perfect self-interference cancellation (P-SIC)), two ideal factors (P-CSI and P-TH, P-CSI and P-SIC, P-TH and P-SIC), or one ideal factor (P-CSI or P-TH or P-SIC). Numerical results show a strong impact of three negative factors on the OP, SER, and EC of the considered FD relay system, especially when the data transmission rate of the system and signal-to-noise ratio (SNR) are high. In particular, OP, SER, and EC go to the floors in the high SNR regime due to the three negative factors. Therefore, when I-CSI, THNs, and RSI exist in the FD relay system, we should use suitable source and relay transmission power to obtain excellent performance while saving energy consumption. Moreover, when two of the three negative factors are large enough, the remaining factor’s impact becomes weaker and may be neglected in certain circumstances.

Secrecy Performance of FSO Using HD and IM/DD Detection Technique Over F-Distribution Turbulence Channel With Pointing Error

This work is concerned with the secrecy performance of a free-space optical (FSO) communication system over Fisher-Snedecor <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {F}$ </tex-math></inline-formula> -distribution turbulence fading channel. Considering the influence of pointing error, atmospheric turbulence and different detection technique (i.e., the intensity modulation with direct detection and heterodyne detection), we derive the exact closed-form expressions for the average secrecy capacity (ASC), the secrecy outage probability (SOP) and the strictly positive secrecy capacity (SPSC) in terms of Meijer’s G-function. Furthermore, the asymptotic expression for SOP is analyzed as the electrical signal-to-noise ratio approaches infinity. Monte Carlo simulations are provided to validate analytical results.

NOMA-based UAV system under finite blocklength regime with analysis in Rician fading channel

The finite blocklength (FBL) communication has been studied for quite some time for the fifth generation of cellular networks (5G). Though, the research in FBL is still in early stage due to the fact that 5G requires high reliability of almost 99.9999% with latency of less than 1 ms, simultaneously. The FBL has been worked upon to reduce latency such that Shannon's capacity expression is no longer applicable. The 5G claims to provide connectivity everywhere, hence to increase the coverage to remote places Unmanned Ariel Vehicle (UAV) has gained attention due to its low cost, mobility, and air time. In this paper, a scenario where a UAV relays data from a base station (BS) to remote users (RU), which are otherwise not in direct contact, is analyzed. All the communications among BS, UAV, and RU's are considered to be non-orthogonal multiple access (NOMA) based with Finite blocklength (FBL). An exact closed form expression for end-to-end average block error rate (BLER) is derived for above mentioned scenario and verified by extensive simulations. In addition, an asymptotic BLER for remote users at high transmission Signal to noise ratio is also derived. Further, the BLER is minimized to jointly optimize the power allocation coefficients, position of UAV and Blocklength for the communication system. The effects of different parameters on average BLER, position, and power are studied and the coverage provided by UAV for given probability of error is presented.

Charged dielectric spheres interacting in electrolytic solution: A linearized Poisson-Boltzmann equation model.

We present an analytical theory of electrostatic interactions of two spherical dielectric particles of arbitrary radii and dielectric constants, immersed into a polarizable ionic solvent (assuming that the linearized Poisson-Boltzmann framework holds) and bearing arbitrary charge distributions expanded in multipolar terms. The presented development entails a novel two-center re-expansion analytical theory that expands upon and improves the existing ones, bypassing the conventional expansions in modified Bessel functions. On this basis, we develop a specific matrix formalism that facilitates the construction of asymptotic expansions in ascending order of Debye screening terms of potential coefficients, which are then employed to find exact closed-form expressions for the total electrostatic energy. In particular, this work allows us to explicitly and precisely quantify the k-screened terms of the potential coefficients and mutual interaction energy. Specific cases of monopolar and dipolar distributions are described in particular detail. Comprehensive numerical examples and tests of series convergence and the relative balance of leading and higher-order terms of the mutual interaction energy are presented depending on the inter-particle distance and particles' radii. The results of this work find application in soft matter modeling and, in particular, in computational biophysics and colloid science, where the availability of increasingly larger experimental structures at the atomic-level resolution makes numerical treatment challenging and calls for more efficient expressions and an increased range of validity.

Error rate performance of NOMA system with full-duplex cooperative relaying

This paper studies a FD cooperative non-orthogonal multiple access (NOMA) system in terms of pairwise error probability (PEP) while considering imperfect successive interference cancellation (ipSIC). Exact closed-form expressions of the PEP performance for the system users are derived. The PEP expressions are derived under both ideal condition and in the presence of self-interference due to the FD relaying. Moreover, we derive asymptotic PEP expressions to evaluate an effective diversity order for the considered system. Through the derived (asymptotic) expressions and simulation results, we demonstrate that self-interference due to FD relaying has a great influence on the performance of the full-duplex relay.

An improving performance cellular DTV broadcasting with hybrid non-orthogonal LDM and orthogonal eMBMS configuration

In this article, we focus on the hybrid of orthogonal and non-orthogonal broadcasting. The hybrid configuration is raised and proposed for improving cellular digital television broadcasting (CDTB) technology. The hybrid scheme is configured as layered division multiplexing (LDM) multiplexed with an evolved multimedia broadcast multicast service (eMBMS) standalone layer is superimposed. The LDM and eMBMS are sub-sections of power division-non orthogonal multiple access (PD-NOMA) and long-term evolution (LTE) respectively. We assume perfect knowledge of the channel state information (CSI) over the Rayleigh fading multipath channel. The bit error rate (BER) and outage probability performance criteria are investigated. The impact of injection level using the fractional transmit power allocation (FTPA) algorithm is discussed. The proposed hybrid can operate in conditions without a subscriber identity module (SIM) card for uplink beamforming and internet protocol (IP). The broadcast network provider selection is feasible by the user equipment (UE) device. Performance analysis is derived based on the exact closed-form expressions, which allows the theory of hybrid proposed LDM/eMBMS consistent with results. Evaluations and performance will be done based on the Monte Carlo iterative methodology. The results show that the performance and efficiency will be increased when a proposed hybrid LDM/eMBMS configuration is compared with LDM and eMBMS services individually.

Information Freshness Analysis of Slotted ALOHA in Gilbert-Elliot Channels

This letter analyzes a class of information freshness metrics for large IoT systems in which terminals employ slotted ALOHA to access a common channel. Considering a Gilbert-Elliot channel model, information freshness is evaluated through a penalty function that follows a power law of the time elapsed since the last received update, in contrast with the linear growth of age of information. By means of a signal flow graph analysis of Markov processes, we provide exact closed form expressions for the average penalty and for the peak penalty violation probability.

Impacts of Residual Self-Interference, Hardware Impairment and Cascade Rayleigh Fading on the Performance of Full-Duplex Vehicle-to-Vehicle Relay Systems.

In practice, self-interference (SI) in full-duplex (FD) wireless communication systems cannot be completely eliminated due to imperfections in different factors, such as the SI channel estimation and hardware circuits. Therefore, residual SI (RSI) always exists in FD systems. In addition, hardware impairments (HIs) cannot be avoided in FD systems due to the non-ideal characteristics of electronic components. These issues motivate us to consider an FD-HI system with a decode-and-forward (DF) relay that is applied for vehicle-to-vehicle (V2V) communication. Unlike previous works, the performance of the proposed FD-HI-V2V system is evaluated over cascaded Rayleigh fading channels (CRFCs). We mathematically obtain the exact closed-form expressions of the outage probability (OP), system throughput (ST), and ergodic capacity (EC) of the proposed FD-HI-V2V system under the joint and crossed effects of the RSI, HIs, and CRFCs. We validate all derived expressions via Monte-Carlo simulations. Based on these expressions, the OP, ST, and EC of the proposed FD-HI-V2V system are investigated and compared with other related systems, such as ideal hardware (ID) and half-duplex (HD) systems, as well as a system over traditional Rayleigh fading channels (RFCs), to clearly show the impacts of negative factors.

Multihop Weibull-fading communications: Performance analysis framework and applications

The paper presents a comprehensive closed-form performance analysis framework for multihop communications over Weibull fading channels. This framework may be of interest in different applications in the contexts of beyond-5G (B5G) and Internet of Things (IoT) use cases. The analyzed scheme consists generally of multiple regenerative relays, and we also consider generalized high-order quadrature amplitude modulation (M-QAM) transmissions. To take into consideration the channel fading, we adopt the Weibull model for its largely flexible ability to cover different channel conditions in different application contexts. The end-to-end performance is evaluated in terms of outage probability, bit error probability (BER), symbol error probability (SER), block error rate (BLER), ergodic capacity, and energy efficiency (EE). For all the metrics, we present exact closed-form expressions—along with their asymptotic behavior—capitalizing on the powerful generalized hypergeometric functions. To illustrate the utility of the obtained analytical results, we derive two BER- and EE-optimal transmit power allocation strategies, and we discuss the resulting performance gains. The exactness of our analysis is illustrated by numerical examples, and assessed via Monte-Carlo simulations for different system and channel parameters. Finally, as a secondary contribution, and noting the increasing popularity of single and bivariate Fox’s H-function, we provide generalized Matlab codes for computing these functions which are of practical utility in many fields.

A screw dislocation and an edge dislocation interacting with a locally rough bimaterial interface

We apply analytic continuation and conformal mapping techniques to investigate, in detail, the elastic interaction of a screw dislocation and subsequently an edge dislocation with a locally rough bimaterial interface. In the case of a screw dislocation, the locations and Burgers vectors of the image screw dislocations for the half-plane containing the real screw dislocation are determined in an analytical manner. The image force acting on the screw dislocation is then expediently obtained using the Peach-Koehler formula. A sufficiently accurate closed-form expression for the image force is proposed. For the case of an edge dislocation, it is assumed that the two half-planes have identical shear moduli but distinct Poisson’s ratios. An exact closed-form expression for the image force acting on the edge dislocation is obtained. The image force is found to vary with the direction of the Burgers vector of the edge dislocation due to the influence of interface roughness.

Secrecy Performance Analysis of Hybrid RF/VLC Dual-Hop Relaying Systems

In this article, the secrecy performance of a hybrid radio frequency (RF)/visible light communication (VLC) system is studied. In this hybrid system, the source node (i.e., Alice) transmits information to the relay node via the outdoor RF link. Nakagami-m fading and path loss are considered for the RF link. The relay node includes an outdoor component and an indoor component, which are connected by using a wired medium. The outdoor component receives and recovers information by using the decode-and-forward (DF) relaying scheme and then transmits it to the indoor component. The indoor component then converts the received electrical signal into an optical signal by using a light-emitting diode. A legitimate receiver (i.e., Bob) deployed on the floor receives the optical signal. An eavesdropper (i.e., Eve) deployed in the RF or VLC link wiretaps the confidential information. In this study, we use the secrecy outage probability (SOP) and the probability of strictly positive secrecy capacity (SPSC) to evaluate the system performance. We then obtain the closed-form expression for a lower bound on the SOP and an exact closed-form expression for the probability of SPSC when the RF and VLC links are wiretapped, respectively. Numerical results are presented to validate the accuracy of our derivations. We further discuss the effects of the noise standard deviation, the equivalent threshold of the signal-to-noise ratio, and the floor radius on the system secrecy performance when the VLC link is eavesdropped upon. For the case when the RF link is eavesdropped upon, the impacts of the distance between Alice and the relay, the path loss exponent, the fading factor, and the distance between Alice and Eve on secrecy performance are also provided.

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.

Combining RF energy harvesting and cooperative communications for low-power wide-area systems

This paper presents the combination of energy harvesting (EH) at the wireless sensor and cooperative communications for low-power wide-area (LPWA) systems. Firstly, the Internet of Things sensor harvests the energy from the power beacon with multiple transmit antennas via radio frequency signals and then uses the harvested energy to transmit signals to multiple gateways with multiple receive antennas. Then, the cooperative communications is applied at the server based on the gateway outputs. By mathematical analysis, we derive the exact closed-form expressions of outage probabilities (OPs), throughput, and symbol error probabilities (SEPs) of the EH-LPWA system over Nakagami-m fading channel in the cases without and with cooperative communications. Our expressions can be considered as the first results applying EH for LPWA systems with mathematical analysis. Then, we obtain the optimal value of the time switching ratio that minimizes the OPs and SEPs and maximizes the throughput of the considered EH-LPWA system. Numerical results have clarified that, the distances, path loss exponent, and data transmission rate have a strong impact on the OPs, throughput, and SEPs. Particularly, using a half of transmission blocks for EH can maximize the system performance. Moreover, when the number of transmit antennas at power beacon is equal to the number of receive antennas at gateways, the system performance can be improved significantly. Finally, the accuracy of the obtained expressions is demonstrated via Monte-Carlo simulations.

Exact transcendental stiffness matrices of general beam-columns embedded in elastic mediums

Stiffness matrices of beams embedded in an elastic medium and subjected to axial forces are considered. Both the bending and the axial deformations have been incorporated. Two approaches for deriving the element stiffness matrix analytically have been proposed. The first approach is based on the direct force–displacement relationship, whereas the second approach exploits shape functions within the finite element framework. The displacement function within the beam is obtained from the solution of the governing differential equation with suitable boundary conditions. Both approaches result in identical expressions when the exact transcendental displacement functions are used. Exact closed-form expressions of the elements of the stiffness matrix have been derived for the bending and axial deformation. Depending on the nature of the axial force and stiffness of the elastic medium, seven different cases are proposed for the bending stiffness matrix. A unified approach to the non-dimensional representation of the stiffness matrix elements and system parameters that are consistent across all the cases has been developed. Through Taylor-series expansions of the stiffness matrix coefficients, it is shown that the classical stiffness matrices appear as an approximation when only the first few terms of the series are retained. Numerical results shown in the paper explicitly quantify the error in using the classical stiffness compared to the exact stiffness matrix derived in the paper. The expressions derived here gives the most comprehensive and consistent description of the stiffness coefficients, which can be directly used in the context of finite element analysis over a wide range of parameter values.

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.