Analytical Mathematical Expressions Research Articles

Capacity Performance Analysis for Terrestrial THz Channels

The outdoor terrestrial terahertz (THz) communication links have recently attracted great research and commercial interest in response to the emerging bandwidth-hungry demands for extremely high-speed wireless data transmissions. However, their development is hindered by the random behavior of the atmospheric channel due to the molecular attenuation, adverse weather effects, and atmospheric turbulence (along with free space path loss (FSPL) and pointing errors) due to the stochastic misalignments between the transmitter and the receiver. Thus, in this work, we investigate the joint influence of these detrimental effects on both capacities, i.e., average (ergodic) and outage, of such a typical line of sight (LOS) THz communication link. Specifically, atmospheric turbulence-induced intensity fluctuations can be modeled by using either the suitable gamma or the well-known gamma–gamma distribution for weak and moderate to strong turbulence conditions, respectively. Additionally, weak to strong stochastic misalignment-induced intensity fluctuations, due to generalized pointing errors with non-zero boresight (NZB), are emulated by the appropriate Beckman distribution. Taking into additional consideration the unavoidable presence of FSPL and the different but realistic water vapor concentration values along with the influence of weather conditions, an outage performance analysis has been conducted. Considering the abovementioned significant effects, novel analytical mathematical expressions have been extracted for both average (ergodic) and outage capacity, which are critical metrics that first incorporate the total influence of all of the above significant effects on the THz links’ performance. Through the derived expressions, proper analytical results verified by simulations are presented and demonstrate the validity of our analysis. It is notable that the derived expressions can accommodate realistic parameter values involved in all the above-mentioned major effects and link characteristics. In this context, they provide encouraging quantitative results and outcomes for both capacity metrics under investigation. The latter enables the design and the establishment of modern and future high-speed THz links, which are expected to cover longer propagation distances and thus become even more vulnerable to atmospheric turbulence effect. This is modeled and incorporated in our analysis and expressions contrary to most of the previous works in the open technical literature.

PDE Surface-Represented Facial Blendshapes

Partial differential equation (PDE)-based geometric modelling and computer animation has been extensively investigated in the last three decades. However, the PDE surface-represented facial blendshapes have not been investigated. In this paper, we propose a new method of facial blendshapes by using curve-defined and Fourier series-represented PDE surfaces. In order to develop this new method, first, we design a curve template and use it to extract curves from polygon facial models. Then, we propose a second-order partial differential equation and combine it with the constraints of the extracted curves as boundary curves to develop a mathematical model of curve-defined PDE surfaces. After that, we introduce a generalized Fourier series representation to solve the second-order partial differential equation subjected to the constraints of the extracted boundary curves and obtain an analytical mathematical expression of curve-defined and Fourier series-represented PDE surfaces. The mathematical expression is used to develop a new PDE surface-based interpolation method of creating new facial models from one source facial model and one target facial model and a new PDE surface-based blending method of creating more new facial models from one source facial model and many target facial models. Some examples are presented to demonstrate the effectiveness and applications of the proposed method in 3D facial blendshapes.

Determination of Logistic Model Parameters From AM2 Model for Methane Production for Batch Bioreactors

Modeling the estimation of cumulated methane production generated from batch anaerobic bioreactors is of paramount importance. In this context, there are two main modeling approaches. The first approach is based on developing mathematical expressions representing the processes involved in the bioreactors. The AM1 model is known as the most complete one. However, it is a complicated model as it requires about 80 parameters to be tuned. A model named AM2 which is a simplified version of AM1 has also been developed. It is based on only two microbial growth processes and requires only 13 parameters. Nevertheless, both AM1 and AM2 models do not provide explicit mathematical expressions that enable to estimate the temporal evolution of methane production with respect to the parameters involved in the considered models. These models are much more useful for simulations and graphical visualizations of the dynamical behavior of the state variables including methane production. On the other hand, the second approach suggests semi-empirical or data-driven models which are based on simple explicit mathematical expressions that provide an estimate of the cumulated methane production (Logistic model, Gompertz model, etc.). But, this type of models require the identification of few parameters which are extracted from experimental data. Usually, they are simplistic and use only one growth process and thus cannot exhibit the influence of the many parameters involved in such complex dynamic biotechnological systems. In this paper, to overcome the complexity of the first type of models and to avoid the dependency on experimental data in the second type; an explicit analytical mathematical expression is proposed for estimating the cumulated methane production for batch anaerobic bioreactors. This analytical expression is derived via the adoption of some appropriate approximations performed on the set of differential equations characterizing the AM2 model. Therefore, the proposed analytical expression can be considered as an approximation of the AM2 model itself and this is the first contribution of this paper. Graphical profiles of the cumulated methane production are presented showing that of the proposed logistic expression and that of the AM2 model considered as a reference. To the best of the author’s knowledge, no such approach and result have been encountered in the literature. On the other hand, this expression resembles formally to the semi-empiric logistic model. However, the equivalent parameters of the proposed expression as self-defined by the parameters of the AM2 model and do not require experimental data to be identified as it is for the semi-empiric logistic model. Moreover, by comparing the proposed logistic model to the semi-empiric logistic model, an identification of the parameters of the semi-empiric model is linked to the parameters of the AM2 model, providing more insight into the methane production. This can be considered as a second contribution of this paper.

Analytical Warburg Impedance Model for EIS Analysis of the Gas Diffusion Layer with Oxygen Depletion in the Air Channel of a PEMFC

In this study, an analytical mathematical expression to simulate the low frequency impedance spectrum that represents the gas diffusion layer (GDL) and air channel of a proton-exchange membrane fuel cell (PEMFC) has been developed. The analytical expression considers the impedance response of the GDL with oxygen depletion in the air channel/GDL interface. Parameters of the finite-length Warburg impedance (ZW) reported in the literature and estimated from EIS measurements carried out in a PEMFC at different oxygen stoichiometry are considered as a base-line to simulate the GDL-channel impedance spectrum. The results demonstrate that non-stationary depletion of oxygen at the air channel/GDL interface during AC conditions yields the formation of a second loop on the GDL-channel impedance spectrum at low frequencies. In addition, it is demonstrated that the contribution of the AC current amplitude increases the second loop of the GDL-channel impedance spectrum and drifts the GDL-channel impedance spectrum away from its steady-state value which is related to the real component of the impedance response as the frequency is approaching zero. The resulting analytical expression of the GDL-channel impedance is analogous to the general mathematical expression of the finite-length Warburg component. The developed GDL-channel impedance model can simulate and separate the contribution of the impedance response of the GDL from the impedance response associated to oxygen depletion in the channel/GDL interface. This developed model could support other studies focusing on the estimation of diffusion parameters of the GDL from EIS measurements carried out in PEMFCs at different operating conditions.

Error Performance Estimation of Modulated Retroreflective Transdermal Optical Wireless Links with Diversity under Generalized Pointing Errors

Recent developments in both optical wireless communication (OWC) systems and implanted medical devices (IMDs) have introduced transdermal optical wireless (TOW) technology as a viable candidate for extremely high-speed in-body to out-of-body wireless data transmissions, which are growing in demand for many vital biomedical applications, including telemetry with medical implants, health monitoring, neural recording and prostheses. Nevertheless, this emerging communication modality is primarily hindered by skin-induced attenuation of the propagating signal bit carrier along with its stochastic misalignment-induced fading. Thus, by considering a typical modulated retroreflective (MRR) TOW system with spatial diversity and optimal combining (OC) for signal reception in this work, we focus, for the first time in the MRR TOW literature, on the stochastic nature of generalized pointing errors with non-zero boresight (NZB). Specifically, under these circumstances, novel analytical mathematical expressions were derived for the total average bit error rate (BER) of various system configurations. Their results revealed significant outage performance enhancements when spatial diversity was utilized. Moreover, taking into consideration the total transdermal pathloss along with the effects of stochastic NZB pointing errors, the critical average signal-to-noise ratio (SNR) metric was evaluated for typical power spectral-density values.

An entropic model for the rock water absorption process

The interaction between rock and surrounding water is an important research topic in geotechnical engineering. In this study, the temporal variation in rock moisture content when immersed in water is investigated using the probability method based on the entropy theory. An analytical mathematical expression is derived by assuming the rock moisture content to be a random variable, maximizing the entropy function subject to a constraint condition and adopting a hypothesis regarding the rock moisture content cumulative distribution function. The proposed entropy-based expression models the temporal evolution of rock moisture content as immersion time progresses from null to infinity. Furthermore, the derived model is tested against eight experimental data sets from the literature, with good agreement, a high correlation coefficient (0.9689) and a low mean absolute relative error (0.0664). Based on previous experimental results, the impacts of three main factors—rock type, rock porosity and outer water pressure on the maximum capacity for moisture content growth, a key parameter of the proposed expression—are discussed. The entropy-based method is a useful tool for predicting the variation of rock moisture content with different immersion times during the water absorption process.

Indexing grazing-incidence X-ray diffraction patterns of thin films: lattices of higher symmetry.

Grazing-incidence X-ray diffraction studies on organic thin films are often performed on systems showing fibre-textured growth. However, indexing their experimental diffraction patterns is generally challenging, especially if low-symmetry lattices are involved. Recently, analytical mathematical expressions for indexing experimental diffraction patterns of triclinic lattices were provided. In the present work, the corresponding formalism for crystal lattices of higher symmetry is given and procedures for applying these equations for indexing experimental data are described. Two examples are presented to demonstrate the feasibility of the indexing method. For layered crystals of the prototypical organic semiconductors di-indeno-perylene and (ortho-di-fluoro)-sexi-phenyl, as grown on highly oriented pyrolytic graphite, their yet unknown unit-cell parameters are determined and their crystallographic lattices are identified as monoclinic and orthorhombic, respectively.

Evaluating the impact of transport inertia on the electrochemical response of lithium ion battery single particle models

The description of the transport mechanisms in operating Li ion battery cells is of key importance for a correct evaluation of their performance and for their optimization.In this work, we revise the Fickian approach for the description of the lithium transport in intercalation-type active materials. We adopt the Maxwell-Cattaneo-Vernotte (MCV) theory to capture the impact of lithium transport inertia on the electrochemical response of graphitic materials, taken here as an application example. We formalize this theory by means of an analytical mathematical expression which allows extracting the values of the lithium diffusion coefficient DMCV and the inertia characteristic time τ from potentiostatic intermittent titration technique (PITT) experiments. The implications of adopting the MCV theory in single particle models to calculate transient current response during the graphite lithiation are discussed (i) on the basis of the fitting of the calculations with in house PITT results and, (ii) by comparing the estimated diffusion coefficients with the ones resulting from the fitting using the classical Fickian approach.

A new look at energy harvesting half-duplex DF power splitting protocol relay network over rician channel in case of maximizing capacity

<span>In this paper, the system performance in term of the ergodic capacity of a half-duplex decode-and-forward relaying network over Rician Fading Channel is investigated. The power splitting protocol is proposed for the system model. For this purpose, the analytical mathematical expressions of the ergodic capacity in cases of maximize and no-maximize ergodic capacity are derived and discussed. Furthermore, the effect of various system parameters on the system performance is rigorously studied. Finally, the analytical results are also demonstrated by Monte-Carlo simulation in comparison with the analytical expressions. The research results show that the analytical mathematical and simulated results match for all possible parameter values for both schemes. </span>

On the Performance of Power Splitting Energy Harvested Wireless Full-Duplex Relaying Network with Imperfect CSI over Dissimilar Channels

The energy harvesting amplify-and-forward full-duplex relaying network over the dissimilar fading environments in imperfect CSI condition is investigated. In this system model, the energy, and information are transferred from the source to the relay nodes by the power splitting protocol with helping of the full-duplex relay node. Firstly, the outage probability, achievable throughput, and the optimal power splitting factor in terms of the analytical mathematical expressions were proposed, analyzed, and demonstrated. Furthermore, the system performance of the proposed model on the connection with all system parameters is rigorously studied. Finally, the numerical results demonstrated and convinced one that the analytical and the simulation results are matched well with each other for all system parameter values using Monte-Carlo simulation. The results show that the system performance degrades significantly but is still in a permissible interval while the channel estimation error increases and the system performance of the mixing scenarios is better in comparison with the Rayleigh-Rayleigh scenario.

New fluid approaches for studying the performance of elastic traffic under class based weighed fair queuing system

SummaryAlthough TCP is typically designed to carry data (transfer of documents such as files or web pages), it is also suitable, today, for transporting most commercial video‐streaming traffic such as YouTube or Netflix traffic. The class‐based weighted fair queuing is still an important router discipline that allows different classes of elastic flows (generally, TCP flows) to be transported together in a truly converged network. Such system has been extensively studied in packet level by evaluating several criteria of effectiveness such as the mean queue length and the average queue waiting time, without proposing a general model that captures the flow‐level dynamics and the real coupling aspect between different queues. Moreover, most studies limited their works to a simple two‐queue system where performance evaluation is very much easier. Even the few works focusing on providing extended results did not give accurate results for very loaded systems. Proposed packet‐level models seem, then, to be not convenient to predict the performance of large‐scale operator networks with millions of users, millions of flows, and unexpected user behaviors. This paper aims to overcome these limitations by presenting new analytical explicit mathematical expressions evaluating the flow‐level performance metrics of elastic traffic under a general class‐based weighted fair queuing system. The core of our analysis is based on some approximations proven for balanced fairness allocation, which provides a reasonable framework for estimating bandwidth sharing among elastic traffic for best‐effort allocations. Detailed packet level simulations are used to verify the accuracy and the effectiveness of the proposed approaches.

Indexing of grazing-incidence X-ray diffraction patterns: the case of fibre-textured thin films.

Crystal structure solutions from thin films are often performed by grazing-incidence X-ray diffraction (GIXD) experiments. In particular, on isotropic substrates the thin film crystallites grow in a fibre texture showing a well defined crystallographic plane oriented parallel to the substrate surface with random in-plane order of the microcrystallites forming the film. In the present work, analytical mathematical expressions are derived for indexing experimental diffraction patterns, a highly challenging task which hitherto mainly relied on trial-and-error approaches. The six lattice constants a, b, c, α, β and γ of the crystallographic unit cell are thereby determined, as well as the rotation parameters due to the unknown preferred orientation of the crystals with respect to the substrate surface. The mathematical analysis exploits a combination of GIXD data and information acquired by the specular X-ray diffraction. The presence of a sole specular diffraction peak series reveals fibre-textured growth with a crystallographic plane parallel to the substrate, which allows establishment of the Miller indices u, v and w as the rotation parameters. Mathematical expressions are derived which reduce the system of unknown parameters from the three- to the two-dimensional space. Thus, in the first part of the indexing routine, the integers u and v as well as the Laue indices h and k of the experimentally observed diffraction peaks are assigned by systematically varying the integer variables, and by calculating the three lattice parameters a, b and γ. Because of the symmetry of the derived equations, determining the missing parameters then becomes feasible: (i) w of the surface parallel plane, (ii) the Laue indices l of the diffraction peak and (iii) analogously the lattice constants c, α and ß. In a subsequent step, the reduced unit-cell geometry can be identified. Finally, the methodology is demonstrated by application to an example, indexing the diffraction pattern of a thin film of the organic semiconductor pentacenequinone grown on the (0001) surface of highly oriented pyrolytic graphite. The preferred orientation of the crystallites, the lattice constants of the triclinic unit cell and finally, by molecular modelling, the full crystal structure solution of the as-yet-unknown polymorph of pentacenequinone are determined.

Two-Way Half Duplex Decode and Forward Relaying Network with Hardware Impairment over Rician Fading Channel: System Performance Analysis

In this paper, the system performance analysis of a two-way decode and forward (DF) relaying network over the Rician fading environment under hardware impairment effect is proposed, analyzed and demonstrated. In this analysis, the analytical mathematical expressions of the achievable throughput, the outage probability, and ergodic capacity were proposed, analyzed and demonstrated. After that, the effect of various system parameters on the system performance is deeply studied with closed-form expressions for the system performance. Finally, the analytical results are also demonstrated by Monte-Carlo simulation in comparison with the closed-form expressions. The numerical results demonstrated and convinced the effect of the system parameters on the system performance of the two-way DF relaying network. The results show that the analytical mathematical and simulated results match for all possible parameter values.DOI: http://dx.doi.org/10.5755/j01.eie.24.2.20639

The System Performance of Half-Duplex Relay Network under Effect of Interference Noise

In recent years, harvesting energy from radio frequency (RF) signals has drawn significant research interest as a promising solution to solve the energy problem. In this paper, we analyze the effect of the interference noise on the wireless energy harvesting performance of a decode-and-forward (DF) relaying network. In this analysis, the energy and information are transferred from the source to the relay nodes in the delay-limited transmission and Delay-tolerant transmission modes by two methods: i) time switching protocol and ii) power splitting protocol. Firstly, due to the constraint of the wireless energy harvesting at the relay node, the analytical mathematical expressions of the achievable throughput, outage probability and ergodic capacity of both schemes were proposed and demonstrated. After that, the effect of various system parameters on the system performance is rigorously studied with closed-form expressions for system throughput, outage probability, and ergodic capacity. Finally, the analytical results are also demonstrated by Monte-Carlo simulation. The results show that the analytical mathematical and simulated results agree with each other. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

System Performance Analysis of Half-Duplex Relay Network over Rician Fading Channel

In this paper, the system performance of an amplify-and-forward (AF) relaying network over Rician Fading Channel is proposed, analyzed and demonstrated. For details this analysis, the energy and information are transferred from the source to the relay nodes by two methods: 1) time switching protocol and 2) power splitting protocol. Firstly, due to the constraint of the wireless energy harvesting at the relay node, the analytical mathematical expressions of the achievable throughput and the outage probability of both schemes were proposed and demonstrated. After that, the effect of various system parameters on the system performance is rigorously studied with closed-form expressions for the system performance. Finally, the analytical results are also demonstrated by Monte-Carlo simulation in comparison with the closed-form expressions. The numerical results demonstrated the effect of various system parameters, such as energy harvesting time, power splitting ratio, source transmission to noise power, and the threshold value, on the system performance of AF wireless relay nodes. The results show that the analytical mathematical and simulated results match for all possible parameter values for both schemes.

An Accurate Computational Tool for Performance Estimation of FSO Communication Links over Weak to Strong Atmospheric Turbulent Channels

The terrestrial optical wireless communication links have attracted significant research and commercial worldwide interest over the last few years due to the fact that they offer very high and secure data rate transmission with relatively low installation and operational costs, and without need of licensing. However, since the propagation path of the information signal, i.e., the laser beam, is the atmosphere, their effectivity affects the atmospheric conditions strongly in the specific area. Thus, system performance depends significantly on the rain, the fog, the hail, the atmospheric turbulence, etc. Due to the influence of these effects, it is necessary to study, theoretically and numerically, very carefully before the installation of such a communication system. In this work, we present exactly and accurately approximate mathematical expressions for the estimation of the average capacity and the outage probability performance metrics, as functions of the link’s parameters, the transmitted power, the attenuation due to the fog, the ambient noise and the atmospheric turbulence phenomenon. The latter causes the scintillation effect, which results in random and fast fluctuations of the irradiance at the receiver’s end. These fluctuations can be studied accurately with statistical methods. Thus, in this work, we use either the lognormal or the gamma–gamma distribution for weak or moderate to strong turbulence conditions, respectively. Moreover, using the derived mathematical expressions, we design, accomplish and present a computational tool for the estimation of these systems’ performances, while also taking into account the parameter of the link and the atmospheric conditions. Furthermore, in order to increase the accuracy of the presented tool, for the cases where the obtained analytical mathematical expressions are complex, the performance results are verified with the numerical estimation of the appropriate integrals. Finally, using the derived mathematical expression and the presented computational tool, we present the corresponding numerical results, using common parameter values for realistic terrestrial free space optical communication systems.

Time-resolved thermal lens spectroscopy with a single-pulsed laser excitation beam: an analytical model for dual-beam mode-mismatched experiments.

Pulsed laser beam excitations are more commonly used in thermal lens spectroscopy (TLS) than continuous-wave (CW) ones, because CW excitations limit the measurement to linear absorption processes [J. Opt. A5, 256 (2003)]. In this work, we present a new and full analytical model for a single-pulsed laser excitation dual-beam mode-mismatched TLS for low absorption solid-state and liquid samples. Our model has been based on a new solution of time-dependent heat equation for a finite-radius cylindrical sample exposed to a single-pulsed excitation laser beam. For low absorbent samples, unlike previous models, all aberration terms associated in the thermal lens were taken into account in Fresnel integration. Besides, the model provides a full analytical mathematical expression for the temperature rise, normalized signal intensity, and Z-scan photothermal lens signal. The model was confirmed with experimental data of distilled deionized water with excellent agreement. Therefore, the model allows us to extract thermo-optical properties of samples in an analytical and more accurate way.

Thermal modelling of N partially covered photovoltaic thermal (PVT) – Compound parabolic concentrator (CPC) collectors connected in series

In the present study, the thermal energy balance equations have been taken for N partially covered photovoltaic thermal–compound parabolic concentrator collectors (N-PVT–CPC) connected in series. An analytical mathematical expression for Nth outlet water temperature (TfoN) and efficiency of solar cell at Nth collector (ηcN) have been derived in terms of climate and design parameters. Numerical computations have been carried out for two Indian climatic condition namely (a) January (winter) and (b) June (summer), respectively. Effect of number of PVT–CPC collectors connected in series, mass flow rate and the percentage of PV coverage on various temperatures, an overall thermal energy and overall exergy have been carried. It has been observed that an overall exergy of proposed system is maximum for winter condition due to less thermal losses.

Effect of Power Line Conductor Resistance-to-Reactance Ratio on Voltage Magnitude during Two-Phase Faults at Electric Energy Grids

This paper deals with the effect of power line impedance characteristics on the retained voltage magnitude after a two-phase fault at electric power distribution or transmission networks. Specifically, the effect of resistance-to-reactance (R/X) ratio of power line conductors or cables on voltage magnitude during two-phase faults is studied with analytical mathematical expressions. The basic aim of this investigation is to create a tool for the automatic recognition of the type and the location of faults on electric power networks.

An effort to formulate frequency modulation constraints in ship-electrical systems with pulsed loads

One of the most significant power supply quality problems in ship electric networks is ‘frequency modulation’. This is a phenomenon most frequently met in extensively electrified ships and it is caused by the operation of certain electric loads, known as ‘pulsed loads’, which require high power repeated on a regular or random basis. To the best of the authors' knowledge, no well-based theoretical analysis of the phenomenon exists in the literature, although any existing constraints limiting the operation of pulsed loads are not properly explained in the respective standards. Hence, in this study, an effort is made, first to establish a theoretical explanation of the modulation phenomenon via an analytical mathematical expression comprising all the critical parameters that affect it, and then highlight the appropriate procedure of setting limits via standards. In this way, the frequency modulation is expressed as a function of several parameters of the ship electric system. Furthermore, using the obtained expression, a sensitivity analysis of the particular effect of each of these system parameters is made. The proposed method is a useful design tool for ship electric power systems enabling the assessment of acceptable pulsed loading level without the need of complicated simulation models comprising many parameters usually not readily available.