Approximate Closed-form Expressions Research Articles

Short-Packet Two-Way Amplify-and-Forward Relaying

This letter investigates an amplify-and-forward two-way relay network (TWRN) for short-packet communications. We consider a classical three-node TWRN consisting of two sources and one relay. Both two time slots (2TS) scheme and three time slots (3TS) scheme are studied under the finite blocklength regime. We derive approximate closed-form expressions of sum-block error rate (BLER) for both schemes. Simple asymptotic expressions for sum-BLER at high signal-to-noise ratio (SNR) are also derived. Based on the asymptotic expressions, we analytically compare the sum-BLER performance of 2TS and 3TS schemes, and attain an expression of critical blocklength, which can determine the performance superiority of 2TS and 3TS in terms of sum-BLER. Extensive simulations are provided to validate our theoretical analysis. Our results discover that 3TS scheme is more suitable for a system with lower relay transmission power, higher differences between the average SNR of both links and relatively lower requirements on data rate and latency.

Full Duplex AF and DF Relaying Under Channel Estimation Errors for V2V Communications

In this paper, we investigate the outage performance of full-duplex (FD) and half-duplex (HD) vehicle to vehicle (V2V) networks. We consider two vehicular communication scenarios with the assistance of roadside unit (RSU), i.e., RSU-to-vehicle-to-vehicle (RVV) and vehicle-to-RSU-to- vehicle (VRV) scenarios. In particular, we analyze the outage performance of vehicular networks with the amplify-and-forward (AF) and decode-and-forward (DF) protocols under the effect of residual loop-back-interference (LBI) and channel estimation errors. For RVV scenario, the upper bounds for the outage probability of the AF based FD and HD relaying over double Rayleigh fading channels are derived. Moreover, we derive the approximate closed form expressions for the outage probability of the DF based FD and HD relaying protocol over Rayleigh fading channels; For VRV scenario, the exact closed-form expressions for the outage probability of the AF and DF relaying protocols are presented in case of FD and HD modes. Furthermore, we derive the approximate closed-form expressions for the outage probability of hybrid FD/HD mode on AF and DF relaying protocols. The numerical and simulation results corroborate our theoretical ones and quantify the impact of the data rates, channel estimation errors and the residual LBI on the system performance.

Geometry-Based Statistical Modeling of Non-WSSUS Mobile-to-Mobile Rayleigh Fading Channels

In this paper, we present a novel geometry-based statistical model for small-scale non-wide-sense stationary uncorrelated scattering (non-WSSUS) mobile-to-mobile (M2M) Rayleigh fading channels. The proposed model builds on the principles of plane wave propagation to capture the temporal evolution of the propagation delay and Doppler shift of the received multipath signal. This is different from existing non-WSSUS geometry-based statistical channel models, which are based on a spherical wave propagation approach, that in spite of being more realistic is more mathematically intricate. By considering an arbitrary geometrical configuration of the propagation area, we derive general expressions for the most important statistical quantities of nonstationary channels, such as the first-order probability density functions of the envelope and phase, the four-dimensional (4-D) time-frequency correlation function (TF-CF), local scattering function (LSF), and time-frequency-dependent delay and Doppler profiles. We also present an approximate closed-form expression of the channel's 4-D TF-CF for the particular case of the geometrical one-ring scattering model. The obtained results provide new theoretical insights into the correlation and spectral properties of non-WSSUS M2M Rayleigh fading channels.

Performance Analysis of a Dual-Hop Wireless-Power Line Mixed Cooperative System

Wireless communications and power line communications (PLC) are essential components for smart grid communications. This paper analyses the performance of a dual-hop wireless-power line mixed communication setup employing a decode-and-forward relay in terms of analytical average bit error rate (BER), outage probability, and average channel capacity. The Nakagami- $m$ distribution captures the wireless channel fading; whereas the PLC channel gain is characterized by the Log-normal distribution. The additive PLC channel noise is assumed to be Bernoulli-Gaussian distributed. Approximate closed-form expression of the average BER and exact closed-form expression of the outage probability are derived for the considered system. Further, we obtain an approximate closed-form expression of the capacity of the wireless-power line mixed system in terms of the Meijer-G function. It is observed that the system performance deteriorates as the impulsive noise index and the arrival probability of the impulsive component of the PLC additive noise increase.

Power line communications under Rayleigh fading and Nakagami noise: novel insights on the MIMO and multi‐hop techniques

This work targets a comprehensive understanding of the impact of the serial-relaying and multiple-input-multiple-output (MIMO) fading mitigation techniques on the performance of indoor power line communication (PLC) systems. While these techniques were studied extensively under additive Gaussian noise, we aim at highlighting the implications of Nakagami-like noise on the achievable performance levels. The author derive approximate closed-form expressions for the bit error rate (BER) under Rayleigh fading and Nakagami-m background noise where the obtained approximations are very close to the exact BERs. The derived expressions are useful for evaluating the achievable diversity orders and for relating the BER to the main parameters of the PLC system in a direct and intuitive way. In this context, we analytically prove that MIMO doubles the diversity order with different combining schemes while the diversity order of a system under Nakagami-like noise is only half the diversity order that would have been obtained under Gaussian noise. The author also shows that multi-hop relaying does not enhance the diversity order.

Quantized Feedback-Based Diagonal Precoding for $N \times 1$ MISO System With Generalized Orthogonal Space-Time Block Codes

In this paper, we propose an imperfect quantized feedback-based diagonal precoding scheme for generalized orthogonal space-time block codes in an $N \times 1$ multiple-input single-output (MISO) wireless communication system employing $M$ -QAM constellation. It is well known that a feedback-based precoding scheme aids channel adaptive signaling in wireless communication system and capitulates large improvements in almost all performance metrics. However, a practical feedback link is susceptible to error due to wireless channel fading and leads to wrong precoder selection. Therefore, a signal-to-noise ratio (SNR) adaptive error-tolerant weighting scheme is introduced which provides significant gain in comparison with another proposed arbitrarily fixed SNR error-tolerant weighting scheme. At first, a tight approximate closed-form expression of average symbol-error-rate (SER) and an exact analytical expression of outage probability are derived under imperfect feedback information with the help of order statistics. By minimizing the derived average SER, optimized transmit weights for the diagonal precoding matrix (based on feedback bits) are obtained. Later, a closed-form expression of the ergodic capacity with erroneous feedback bits for arbitrary MISO system is also provided by using the moment generating function approach. Further, numerical results show that the considered error-tolerant schemes outperform the uniform power allocation scheme in term of all the investigated performance metrics.

Energy beamforming for full-duplex wireless-powered communication networks

In this paper, we consider a full-duplex (FD) wireless-powered communication network (WPCN), where one FD hybrid access point (HAP) equipped with multiple antennas simultaneously transmits energy to and receives information from multiple users. Firstly, we propose a space division wireless energy allocation scheme and calculate the harvested energy in downlink wireless energy transfer (WET) for each user. Secondly, we derive an approximate closed-form expression of user’s achievable ergodic rate in uplink wireless information transfer (WIT). Thirdly, the energy allocation for different users is optimized under the max–min user fairness constraint, and a closed-form solution is obtained. Numerical results show that the simulation and the approximation of achievable rates are well matched, and energy beamforming can effectively suppress self-interference (SI), and improve rates as well as fairness among users. Moreover, FD-WPCNs are shown to outperform half-duplex (HD) WPCNs in rates with same number of antennas.

On the Optimal Power Allocation for Two-Way Full-Duplex AF Relay Networks

This paper investigates the optimal power allocation that maximizes the system utility for a full-duplex (FD) amplify-and-forward (AF) two-way relay network (TWRN) with a rate outage and a power constraint. Both cases of an individual and a sum power constraint are considered. To formulate the corresponding optimization problem, closed-form expressions of outage probabilities are needed. With FD transceivers, the derivation of such closed-form expressions become too involved. Thus, approximate closed-form expressions were derived instead. The resulting optimization problem is still nonconvex and difficult to solve. Via solving a series of approximate convex problems, a successive convex approximation (SCA) algorithm was proposed. Our simulation results demonstrate the accuracy of the approximate closed-form expressions of outage probabilities and that the proposed SCA algorithm achieves near-optimal performance and significantly outperforms the full power allocation under the individual power constraint and uniform power allocation under the sum power constraint both in system utility and in power consumption. Our results further show that simultaneous power transmission for all relay nodes with judicious power allocation generated by the SCA algorithm achieves $174\%$ of performance gain under the individual power constraint, and $85 \%$ of performance gain under the sum power constraint over the relay selection scheme. A tradeoff between the FD system and the half-duplex (HD) system with respect to the residual interference was also observed. We can also observe that not only the residual self-interference but also the cross-link interference is a factor that degrades the system performance for an FD system.

Achievable Rates of Full-Duplex Massive MIMO Relay Systems Over Rician Fading Channels

We analytically study the achievable rates of full-duplex massive multiple-input multiple-output (MIMO) relay systems over Rician fading channels. The decode-and-forward protocol is adopted at the relay station, where the channel state information (CSI) is assumed to be imperfect. We demonstrate that the system bottleneck, self-interference (SI), can be significantly canceled by zero-forcing (ZF) processing at the relay station, which is equipped with massive receive and transmit arrays. Because no active SI cancellation is deployed at the relay station, there is no need for SI channel estimation. We derive an approximate closed-form achievable rate expression for ZF processing with statistical CSI. When the number of antennas of the transmit and receive arrays at the relay station is sufficiently large, the transmit powers of each source and the relay station can be significantly scaled down proportionally to the number of antennas. Since Rayleigh fading is a special case of Rician fading, the results in this paper hold for Rayleigh fading channels. The theoretical analysis is verified by the numerical results.

STATISTICS OF SIGNAL TO INTERFERENCE RATIO PROCESS AT OUTPUT OF MOBILE-TO-MOBILE RAYLEIGH FADING CHANNEL IN THE PRESENCE OF COCHANNEL INTERFERENCE

Dual-hop cooperative communications in interference-limited Rayleigh fading channel will be investigated in this paper. The paper considers the first- and second-order statistics of the signal to interference ratio process at the input of the destination mobile station. The exact closed-form expressions for the first-order statistical measures, the probability density function and cumulative distribution function, will be derived. We will also derive approximate closed form expressions for the second-order statistics, the level crossing rate and the average fade duration. The obtained theoretical results will be verified by the Monte-Carlo simulations.

Aperture Distributions for Maximum Endfire Directivity From a Continuous Line Source With a Uniform Phase Progression

An optimized aperture distribution is investigated to achieve the maximum endfire directivity for a continuous line source with a uniform phase progression. The optimization method has the advantages of simplicity and efficiency, in which the aperture distribution is expanded into a set of cosine functions and the optimization reduces to solving a constrained linear-least-squares problem. Based on the copious computed results from the optimization, an approximate closed-form expression is derived for the maximum directivity. The expression can serve as a bound for the endfire directivity for a line source with a uniform phase progression. It is found that the optimum directivity increases almost linearly with an increase of the normalized phase constant, but with a decrease in the radiated power. Nevertheless, the enhancement in the directivity is much higher than that in previously studied superdirectivity, under the same constraint in the ratio of reactive to radiated power. The current distribution does not oscillate widely as is does for superdirectivity, and is therefore more practical than for superdirectivity.

Energy Harvesting in Multi-relay Multiuser Networks based on Two-step Selection Scheme

In this paper, we analyze average capacity of an amplify-and-forward (AF) cooperative communication system model in multi-relay multiuser networks. In contrast to conventional cooperative networks, relays in the considered network have no embedded energy supply. They need to rely on the energy harvested from the signals broadcasted by the source for their cooperative information transmission. Based on this structure, a two-step selection scheme is proposed considering both channel state information (CSI) and battery status of relays. Assuming each relay has infinite or finite energy storage for accumulating the energy, we use the infinite or finite Markov chain to capture the evolution of relay batteries and certain simplified assumptions to reduce computational complexity of the Markov chain analysis. The approximate closed-form expressions for the average capacity of the proposed scheme are derived. All theoretical results are validated by numerical simulations. The impacts of the system parameters, such as relay or user number, energy harvesting threshold and battery size, on the capacity performance are extensively investigated. Results show that although the performance of our scheme is inferior to the optimal joint selection scheme, it is still a practical scheme because its complexity is much lower than that of the optimal scheme.

Orthotropic hydraulic permeability of arrays of parallel cylinders.

Approximate analytical methods are presented to calculate the overall orthotropic hydraulic permeability of a flow with low Reynolds number, passing through a bundle of parallel circular cylinders. Two particular distributions are considered: (i) arrays with ordered rectangular lattices and (ii) irregular nonrandom distributions for which the unit cell cross sections are elliptical. The standard unit cell models, originally developed by Happel and Kuwabara for a random distribution of cylinders, are adapted to the case of nonrandom distributions. The drag force on a representative cylinder in a direction perpendicular to its axis is obtained based on the standard unit cell model: the actual unit cell of rectangular or elliptical cross section is replaced with an "equivalent" cylindrical unit cell of diameter equal to the maximum width of the actual unit cell. Using the obtained drag forces and referring back to the original geometry of the unit cell, closed-form approximate expressions for the overall permeabilities in the perpendicular directions are obtained. Numerical comparisons with more sophisticated approaches confirm the good efficiency of the presented approach, especially in the range of low solid volume fraction, i.e., of high porosity. Previous studies have revealed that, for the parallel fluid flow, the variation of permeability with aspect ratio (or in general the lateral arrangement) of parallel cylinders is generally weak. These observations suggest that Happel's model for parallel permeability in a random distribution of cylinders could be a good approximation for parallel permeabilities in nonrandom distributions with the same volume fraction.

Cooperative Non-Orthogonal Multiple Access with Energy Harvesting

In this paper, we analyze the outage probability of an amplify-and-forward (AF) cooperative non-orthogonal multiple access (NOMA) model in multi-relay multiuser networks. In contrast to conventional cooperative networks, relays in the considered network have no embedded energy supply; they need to rely on the energy harvested from the signals broadcasted by the source for their cooperative NOMA transmission. Based on this structure, a new relay selection scheme is proposed, considering both channel state information (CSI) and battery status of relays. Assuming each relay has infinite or finite energy storage for accumulating energy, we use the infinite or finite Markov chain to capture the evolution of relay batteries and certain simplified assumptions to reduce computational complexity of the Markov chain analysis. The approximate closed-form expressions for the outage probability of the proposed scheme are derived therefrom. All theoretical results are validated by numerical simulations. The impacts of the system parameters, such as relay number, energy harvesting threshold and battery size, on the performance are extensively investigated.

An Approximation for BER of Optical Wireless System under Weak Atmospheric Turbulence using Point Estimate

Abstract Obtaining analytically closed-form expression for bit error rate (BER) of an optical wireless communication (OWC) system under weak atmospheric turbulence is extremely difficult and intractable. In this paper, a simple closed-form approximate expression for average BER of OWC system under weak turbulence condition is obtained using point estimate method. The proposed expression is used to compute the BER for on-off keying (OOK) and sub carrier intensity modulated (SIM) binary phase shift keying (BPSK) modulation schemes. Results obtained using proposed expression are compared with those obtained using Gauss-Hermite quadrature approximation and Monte-Carlo simulations.

Influence of technological deviations on the basic operational characteristics of hydrodynamic bearings

The main purpose of this paper is to define a methodology to determine the analytical approximate closed-form expression of probabilistic characteristics of load capacity of liquid-lubricated journal bearings in the case of a steady flow regime. Influence of random change of basic geometrical parameters of journal bearings (gap value and eccentricity) on the load capacity of bearing is considered. It is shown that the actual value of hydrodynamic force in bearing can substantially differ from a calculation one. The got results confirm the necessity of application of such approach at consideration both geometrical and operating parameters of bearing, for example, axes misalignment, deviation of surface form, roughness and others, which have casual nature too.

Effects of diversity schemes and correlated channels on OWC systems performance

The salient features and advantages of free-space optical (FSO) system are very appealing for different applications in a number of communication network sectors. In spite of the advantages and diverse applications of FSO communication, its extensive use is hindered by the atmospheric turbulence-induced fading in real-life scenarios. Spatial diversity technique is one of effective means of mitigating turbulence-induced fading and, consequently, improves the system performance. In this paper, we study the spatial diversity schemes for mitigating turbulence-induced fading in the FSO communication systems using the bit error rate (BER) as a performance metric. The schemes considered are based on orthogonal space–time block codes and repetition codes (RCs). We derive simple approximate closed-form expressions for the error probability of the log-normal FSO links with intensity modulation and direct detection (IM/DD). Furthermore, we also investigate the effects of spatial correlation between the transmit apertures on the system performance. We achieve this using the exponential model for determining the correlations between the apertures. We observe that the proposed BER expressions are able to quantify the effects of spatial diversity schemes and spatial correlations on the system.

A unified response surface framework for the interval and stochastic finite element analysis of structures with uncertain parameters

In this paper, finite element analysis of structures with uncertain properties is addressed within both a probabilistic and non-probabilistic framework. Specifically, uncertainties affecting structural parameters are modelled either as interval or random variables. In both cases, uncertainty propagation analysis is performed by applying a ratio of polynomial response surface which enables to derive approximate closed-form expressions of the main descriptors of interval and random response variability by requiring just a few deterministic analyses at selected sampling points. A unified response surface framework for interval and stochastic finite element analysis is thus developed which allows comparisons of structural response variability under different uncertainty models. Numerical results focusing on the comparison between the interval and stochastic response of structures with uncertain Young’s modulus are presented.

Exploiting Direct Links in Multiuser Multirelay SWIPT Cooperative Networks With Opportunistic Scheduling

In this paper, we analyze the downlink outage performance of opportunistic scheduling in dual-hop cooperative networks consisting of one source, multiple radio-frequency energy harvesting relays, and multiple destinations. To this end, two low-complexity, suboptimal, yet efficient, relay-destination selection schemes are proposed, namely direct links plus opportunistic channel state information (CSI)-based selection (DOS) scheme and direct links plus partial CSI-based selection (DPS) scheme. Considering three relaying strategies, i.e., decode-and-forward (DF), variable-gain amplify-and-forward (VG-AF), and fixed-gain amplify-and-forward (FG-AF), the performance analysis in terms of outage probability (OP) is carried out for each selection scheme. For the DF and VG-AF strategies, exact analytical expressions and tight closed-form approximate expressions for the OP are derived. For the FG-AF strategy, an exact closed-form expression for the OP is provided. Additionally, we propose a gradient-based search method to find the optimal values of the power-splitting ratio that minimizes the attained OPs. The developed analysis is corroborated through Monte Carlo simulation. Comparisons with the optimal joint selection scheme are performed and it is shown that the proposed schemes significantly reduce the amount of channel estimations while achieving comparable outage performance. In addition, regardless of relaying strategy used, numerical results show that the DOS scheme achieves full diversity gain, i.e., $M+K$ , and the DPS scheme achieves the diversity gain of $M+1$ , where $M$ and $K$ are the numbers of destinations and relays, respectively.

Two Novel Adaptive Transmission Schemes in a Decode-and-Forward Relaying Network

Adaptive transmission in a cooperative network with a half-duplex relay operating in decode-and-forward mode is considered. The main purpose of the paper is maximizing the spectral efficiency of the system, which is reduced by using half-duplex relaying, while the bit error performance is kept below an appropriate threshold. The source transmits its data adaptively using quadratic amplitude modulation. Two adaptive transmission schemes are proposed: the first scheme is named simple adaptive transmission scheme (SATS), and the second one is called high-performance spectral efficiency scheme (HPSES). The SATS has a low complexity system at the destination which does not combine received signals from the source and relay. However, the HPSES uses a linear combination at the destination which is a novel detector to take the possibility of error at the relay into account. Then, we derive exact closed-form expression for the average spectral efficiency and outage probability of the system and an approximate closed-form expression for the average bit error probability. The simulation results corroborates theoretical results. Furthermore, it is shown that despite much lower complexity, the performance of the SATS is close to other well-known schemes. Moreover, the HPSES outperforms other methods of adaptive transmissions in sense of the spectral efficiency and outage probability.