Approximate Closed-form Expressions Research Articles

Gel Diffusiophoresis of a Spherical Colloidal Particle

A theoretical framework is established for the gel diffusiophoresis of a spherical colloidal particle moving through an uncharged dilute porous polymer gel medium when an electrolyte concentration gradient field is applied. The network of cross-linked polymer segments is treated as a porous skeleton containing an electrolyte solution using the Brinkman–Debye–Bueche model. We derive a general expression for the gel-diffusiophoretic mobility of a charged spherical colloidal particle. Based on this general mobility expression, we farther derive a closed-form approximate expression for the gel-diffusiophoretic mobility of a weakly charged spherical particle correct to the second order of the particle’s zeta potential. The obtained mobility expression depends on the Debye–Hückel parameter and the Brinkmann parameter.

Torsional buckling of a tape-spring: Review and renew

Tape-springs – straight strips with transversely curved cross-sections – undergo a flattening of the cross-section upon bending, and accompanying torsional buckling in equal-sense bending which then “recedes” at greater deformation. The exact analytical solution for tape-spring bending was solved by Mansfield; however, the resulting expression for the twisting curvature at the onset and recession of torsional buckling cannot be solved in closed-form. Approximate closed-form expressions are therefore sought: firstly, by utilising Taylor series approximations of Mansfield’s extensive functions; and secondly, by assuming a uniform transverse curvature, thereby rendering the problem algebraic in nature, and deriving new governing expressions. Further approximations assume the cross-section does not deform upon bending, and that the tape-spring remains a developable surface in the inextensible limit of zero thickness. All approximate solutions are shown to agree closely with Mansfield’s solution at small curvatures, with the uniform curvature approximation also capturing the recession of torsional buckling to a reasonable degree of accuracy. The accuracy of the uniform curvature approximation is related to a pertinent dimensionless parameter, which governs the size of a boundary layer in Mansfield’s exact solution: as this parameter tends to infinity, the uniform curvature approximation and Mansfield’s exact solution are shown to converge; as it tends to zero, all solutions are shown to converge on the inextensible limit.

Low-Complexity User Scheduling Framework for mmWave Hybrid Beam-forming OFDM System under Non-Ideal CSI

Introduction: User scheduling in millimeter-wave (mmWave) multi-user hybrid beam-forming Orthogonal Frequency Division Multiplexing (OFDM) systems involves the joint optimization of resource block group (RBG) allocation, beam pairing, and user selection. However, choosing the optimal scheduled User Equipment (UE), allocated RBGs, and communicating beams in practical mmWave hybrid beam-forming systems with non-ideal Channel State Information (CSI) remains challenging. Method: In this paper, we propose a low-complexity user scheduling framework. On one hand, two user classification methods are proposed under non-ideal CSI, assisted by beam search and RBG allocation respectively. On the other hand, in order to ensure both the throughput and fairness performance, a novel user selection scheme, called weighted user selection based on feedback threshold (WUSFT) scheme is proposed, and approximate closed-form expression of fairness index for both full feedback and feedback based on threshold are derived. Our proposed methods consist of three steps. First, RBGs are allocated based on the maximum received signal power (MRSP) of each user on each RBG, utilizing quasi-omnidirectional beams. In the second step, the communicating beams are further searched to achieve the MRSP with the allocated RBG. Result: Finally, the allocated RBG, or the determined beam information obtained through the beam search, is used to represent the correlation of user channels and classify the UEs into groups. Only UEs whose reported factor is not less than the feedback threshold will report received information. This simplifies the operation of user scheduling. Moreover, simulation results demonstrate that our proposed schemes can achieve more than 92.3% of the sum rate performance of exhaustive search methods while maintaining relatively low complexity. Conclusion: In addition, the user feedback overhead (FO) reduces obviously, especially with large UE number.

Intra-Beam Interference Mitigation for the Downlink Transmission of the RIS-Assisted Hybrid Millimeter Wave System.

Millimeter-wave (mmWave) communication systems leverage the directional beamforming capabilities of antenna arrays equipped at the base stations (BS) to counteract the inherent high propagation path loss characteristic of mmWave channels. In downlink mmWave transmissions, i.e., from the BS to users, distinguishing users within the same beam direction poses a significant challenge. Additionally, digital baseband precoding techniques are limited in their ability to mitigate inter-user interference within identical beam directions, representing a fundamental constraint in mmWave downlink transmissions. This study introduces an innovative analog beamforming-based interference mitigation strategy for downlink transmissions in reconfigurable intelligent surface (RIS)-assisted hybrid analog-digital (HAD) mmWave systems. This is achieved through the joint design of analog beamformers and the corresponding coefficients at both the RIS and the BS. We first present derived closed-form approximation expressions for the achievable rate performance in the proposed scenario and establish a stringent upper bound on this performance in a large number of RIS elements regimes. The exclusive use of analog beamforming in the downlink phase allows our proposed transmission algorithm to function efficiently when equipped with low-resolution analog-to-digital/digital-to-analog converters (A/Ds) at the BS. The energy efficiency of the downlink transmission is evaluated through the deployment of six-bit A/Ds and six-bit pulse-amplitude modulation (PAM) signals across varying numbers of activated RIS elements. Numerical simulation results validate the effectiveness of our proposed algorithms in comparison to various benchmark schemes.

Secrecy Analysis of ABCom-Based Intelligent Transportation Systems With Jamming

Employing ambient backscatter communication (AmBC) technology in Intelligent Transportation Systems (ITS) has emerged as an appealing solution to boost the awareness of crosswalks. However, the AmBC-based ITS is expected to face serious security threats due to the presence of malicious eavesdroppers. In this paper, we investigate the secure multi-antenna transmission in an AmBC-based ITS coexisting with a passive eavesdropper with jamming. Specifically, a cooperative jammer is placed in the system to deliberately disrupt the eavesdropper without affecting the legitimate receiver. In order to characterize the performance of the proposed scheme, new approximate closed-form expressions of secrecy outage probability (SOP) are derived by adopting the Gauss-Chebyshev quadrature. Additionally, the asymptotic behavior of SOP at the high signal-to-noise ratio (SNR) regime is also studied to provide more insights into the system design. We also derive the asymptotic SOP, when the number of transmit antennas tends to infinity. Monte Carlo simulations are provided to demonstrate the validity of our analytical results and to show that 1) the secrecy performance can be significantly improved by allocating part of the transmit power to perform cooperative jamming and 2) the optimal power allocation factor is related to the total transmit power.

Secure short-packet transmission in uplink massive MU-MIMO assisted URLLC under imperfect CSI

Ultra-reliable and low-latency communication (URLLC) is still in the early stage of research due to its two strict and conflicting requirements, i.e., ultra-low latency and ultra-high reliability, and its impact on security performance is still unclear. Specifically, short-packet communication is expected to meet the delay requirement of URLLC, while the degradation of reliability caused by it makes traditional physical-layer security metrics not applicable. In this paper, we investigate the secure short-packet transmission in uplink massive multiuser multiple-input-multiple-output (MU-MIMO) system under imperfect channel state information (CSI). We propose an artificial noise scheme to improve the security performance of the system and use the system average secrecy throughput (AST) as the analysis metric. We derive the approximate closed-form expression of the system AST and further analyze the system asymptotic performance in two regimes. Furthermore, a one-dimensional search method is used to optimize the maximum system AST for a given pilot length. Numerical results verify the correctness of theoretical analysis, and show that there are some parameters that affect the tradeoff between security and latency. Moreover, appropriately increasing the number of antennas at the base station (BS) and transmission power at user devices (UDs) can increase the system AST to achieve the required threshold.

Tunable SRS-Aware Robust Coordinated Beamforming Design With Channel Aging

Coordinated beamforming holds tremendous potential to maximize network capacity, whose effect largely depends on the period sounding reference signal (SRS)-based channel sounding. However, the huge discrepancy between the sounding channel and the real-time channel is inevitable and nonnegligible with channel aging, which results in severe degradation of coordinated beamforming gain. In this paper, the discrepancy is modeled as a random error term with the effect of channel aging included, and a robust coordinated beamforming design with channel aging is further proposed, exploiting the potential capacity gain through the channel statistical information, which provides the potential for tuning the SRS period. Specifically, the joint correlation properties of channels are modeled in the random error matrix with the static statistical information collected from practical channel samples. To cope with multiple random terms, the deterministic equivalent method is adopted to obtain the approximate closed-form expression of the ergodic capacity, aiming at deriving a robust beamformer that can achieve a stationary point of the expected weighted sum-rate maximization problem. In addition, by adopting the proposed beamforming design, the tuning guidance for the SRS period is provided for practical networks satisfying different capacity requirements and mobile scenarios. Simulation results show that the proposed design can extend the SRS period more than 1× at 10% capacity loss when user speed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$v=30\rm {km/h}$</tex-math></inline-formula> , compared with the traditional beamforming design.

STAR-RIS-Enabled Short-Packet NOMA Systems

This paper analyzes the performance of simultaneous transmitting and reflecting (STAR) reconfigurable intelligent surface (RIS)-enabled short-packet non-orthogonal multiple access (NOMA) systems. A mode-switching protocol combined with partition strategies (MSPS) for the STAR-RIS design is proposed to simultaneously serve multiple actuators along with the discrete phase-shift alignment in order to reduce the STAR-RIS implementation costs. Approximate and asymptotic closed-form expressions for the average block error rate (BLER) and achievable rate have been derived to reveal some useful insights, such as the actuators' diversity order, coding gain, the power allocation factor, packet length, and the number of information bits. Numerical results confirm that: i) the considered system achieves better BLER balance among the actuators than classical RIS-based short-packet NOMA systems and higher achievable sum rate than RIS-based short-packet orthogonal multiple access ones; ii) the STAR-RIS with discrete phase shift design using 3-bit quantization achieves near-optimal performance with the continuous phase shift; iii) there is a trade-off between using bit quantizer and the total number of STAR-RIS elements; and iv) the power allocation strategy provides better average sum rate improvement than block-length increasing strategy.

An analytical model for wearable thermoelectric generators harvesting body heat: An opportunistic approach

Wearable thermoelectric generators (WTEGs) have the potential to transform body heat into electrical energy, thereby establishing a sustainable power reservoir for wearable electronics and offering a promising alternative to traditional chemical batteries. This paper presents an opportunistic analytical model that investigates the energy conversion performance of a human skin-WTEG system. The model incorporates a one-dimensional bioheat transport equation to account for metabolic heat generation within skin tissue and heat exchange resulting from the blood perfusion, while a comprehensive three-dimensional analysis of the WTEG captures convective heat dissipation occurring between the WTEG and ambient surroundings. Analytical solutions for the temperature profile and power density of WTEGs are obtained, along with an approximate closed-form expression for the matched load resistance. The results demonstrate the significant impact of blood perfusion and convective heat dissipation on the WTEG performance. The optimal fill factor and height of thermoelectric couples for maximum power output are also determined. This model will assist in predicting and optimizing the performance of WTEG devices in practical applications.

Closed-Form Expressions of Upper Bound for Polarization-MDCSK System.

The performance analysis of polarization M-ary differential chaos shift keying (P-MDCSK) has been expressed using a tight upper bound with the Q-function. However, evaluating the Q-function directly is not a closed expression and there has been less work on closed expression for the upper bound. In order to solve the problem, this paper presents approximate closed-form expressions on the error probability of P-MDCSK. This expression is derived by employing a polynomial approximation of the Q-function. These closed-form expressions are verified through simulations conducted under both additive white Gaussian noise (AWGN) and multipath Rayleigh fading channels. The simulation results reveal that there exists only a negligible gap between the simulations and the derived closed-form expressions. For example, it is observed that the theoretical approximate closed-form expressions exhibit a marginal deviation of approximately 0.4 dB from the simulations when the bit error rate (BER) reaches 10-4. Although the proposed method can only give approximate closed-form expressions of the upper bound, it provides an effective method for other communication schemes where the exact BER closed-form formula cannot be obtained.

Secrecy performance analysis of IRS-NOMA systems

In this paper, we propose to utilize an intelligent reflecting surface (IRS) as a promising technology to enhance the coverage and physical layer security of non-orthogonal multiple access (NOMA) system. In particular, an IRS-assisted NOMA system is considered with the aid of careful channel ordering of the NOMA users, in which the transmitter sends superposed signals to multiple legitimate users by virtue of the IRS in the presence of multiple eavesdroppers. Meanwhile, the secrecy performance of the IRS-assisted NOMA system is investigated under two wiretapping cases: non-colluding and colluding external eavesdroppers. In the non-colluding case, eavesdroppers operate independently, while in the colluding case, every eavesdroppers can combine their observations to decode the messages. To this end, we derive the approximate closed-form expressions for the secrecy outage probability (SOP) and the asymptotic SOP for each wiretapping case. Also, we assume that the phase of the IRS elements is set by using the ON–OFF control method. Based on analytical results, we show that the secrecy diversity order of the IRS-NOMA at legitimate users is in connection with the number of reflecting elements. From the numerical results, it can be seen that the IRS-NOMA can achieve superior secrecy performance with increasing the number of reflecting elements of the IRS. However, we also find out that using the finite ON state reflective elements can improve the secrecy performance. Actually, increasing the number of ON state reflective elements above five has a negative effect on the system’s secrecy performance.

Survival probability determination of nonlinear oscillators subject to combined deterministic periodic and non-stationary stochastic loads

This paper presents a technique based on Wiener path integrals (WPI) for computing the survival probability of nonlinear oscillators under combined nonstationary stochastic and periodic excitation. Specifically, the equation of motion is equivalently decomposed into a correlated combination of a nonlinear stochastic differential equation (SDE) and a nonlinear deterministic differential equation. The nonstationary response probability density function (PDF) of the stochastic component is obtained using the Wiener path integral concept in conjunction with stochastic averaging/linearization. The approach involves utilizing stochastic averaging/linearization techniques to transform the nonlinear SDE into a linear one with varying equivalent stiffness and damping. Afterwards, using the WPI method and the concept of the most probable path, a closed-form approximate analytical expression of the joint transition probability density function (PDF) is obtained for small intervals corresponding to the equivalent linear SDE. Additionally, the survival probability of the original nonlinear oscillator subject to combined excitation with fixed barriers can be approximated equivalently by using the survival probability with time-varying barriers of the equivalent linear system with time-varying stiffness and damping subject only to the nonstationary stochastic excitation. Finally, by using the analytical expression of the transition PDF of the equivalent linear system and a discrete version of the Chapman–Kolmogorov (C–K) equation, it is possible to determine the response and survival probability of the original oscillator under combined excitation step by step. Examples involving a Duffing oscillator and a vibro-impact oscillator are analyzed to showcase the precision and efficacy of the suggested technique in contrast to relevant Monte Carlo simulations.

Real-time Distributed Computing Model of Low-Voltage Flow Data in Digital Power Grid under Autonomous and Controllable Environments

Motivated by the progress in artificial intelligence and edge computing, this paper proposes a real-time distributed computing model for low-voltage flow data in digital power grids under autonomous and controllable environments. The model utilizes edge computing through wireless offloading to efficiently process and analyze data generated by low-voltage devices in the power grid. Firstly, we evaluate the performance of the system under consideration by measuring its outage probability, utilizing both the received signal-to-noise ratio (SNR) and communication and computing latency. Subsequently, we analyze the system’s outage probability by deriving an analytical expression. To this end, we utilize the Gauss-Chebyshev approximation to provide an approximate closed-form expression. The results of our experimental evaluation demonstrate the effectiveness of the proposed model in achieving real-time processing of low-voltage flow data in digital power grids. Our model provides an efficient and practical solution for the processing of low-voltage flow data, making it a valuable contribution to the field of digital power grids.

Performance Analysis of IRS-Aided Short-Packet NOMA Systems Over Nakagami-$m$ Fading Channels

This paper analyzes the performance of an intelligent reflecting surface (IRS)-aided short-packet communication system 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. In this system, the base station adopts non-orthogonal multiple access (NOMA) to transmit to a near user and a far user, and its transmission to the far user is aided by the IRS. Both continuous phase shifts (CPSs) and discrete phase shifts (DPSs) are considered for the IRS. By using order statistic and moment matching, we approximate the end-to-end channel gain to follow the Gamma distribution and then derive the approximate closed-form expressions for the average block error rate (BLER) in the high signal-to-noise ratio (SNR) regime and then conduct a refined analysis of the interplay among the minimum common blocklength, optimal power allocation, and the related parameters of IRS (e.g., the number of elements of the IRS) for achieving target BLERs at two users. Finally, numerical and simulation results demonstrate the accuracy of theoretical analysis. They also show that the IRS-aided short-packet NOMA system outperforms the system without IRS, and that DPSs with 3-bit quantization achieve a similar performance as the optimal CPSs.

Performance Analysis of Adaptive RIS-Assisted Clustering Strategies in Downlink Communication Systems

Reconfigurable intelligent surfaces (RISs)-assisted Internet of Things (IoT) networks have attracted a great deal of interest due to the potential contributions to the next generation wireless networks. In this article, an adaptive RIS clustering system with weak channels or obstacles is investigated. Ideally, there is a suitable RIS that adopts the self-organized RIS-assisted (SORA) scheme between a base station (BS) and a user equipment (UE) for reflection transmission. However, due to the nonideal channel scenarios, a single RIS may not be capable of connecting the BS with the UE. To tackle these nonideal cases, we propose a complementary RIS-cluster-assisted (C-RCA) strategy by reflecting signals among clusters to improve the performance of the overall system. The performances of the two strategies in terms of outage probability (OP) and ergodic capacity (EC), derived using approximate closed-form expressions, are analyzed. By increasing the numbers of complementary RISs and the reflecting elements on each RIS simultaneously, these performance metrics as well as the effective throughput and energy efficiency (EE) are evaluated under different signal-to-noise ratios (SNRs). The obtained results demonstrated that, in contrast to the ideal case, the network EE of the proposed C-RCA strategy is higher than the SORA scheme under the larger target spectral efficiency (SE).

Analysis and Optimization of Spectral and Energy Efficiency in Underlaid D2D Multi-Cell Massive MISO Over Rician Fading

This paper investigates the uplink spectral efficiency (SE) characterization and energy efficiency (EE) optimization of device-to-device (D2D) communications underlying a multi-cell massive multiple-input single-output (m-MISO) network, assuming that the channels are modeled with Rician fading. First, an analytical expression for the lower-bound of the ergodic capacity of a typical cellular user (CU) is derived with imperfect channel state information in the presence of D2D links’ interference. Then, a closed-form approximate achievable SE expression for a typical D2D pair is derived considering Rician fading between D2D pairs. The asymptotic SE behavior is analyzed in strong line-of-sight (LOS) conditions for CU and D2D pairs. Also, simplified expressions are obtained for the special case of Rayleigh fading. Next, to optimize the total EE, a transmit data power allocation based on the derived rate expressions is formulated. Since the optimization problem has a non-linear and non-convex objective function, it is intractable to be solved straightforwardly. Therefore, we resort to utilizing an iterative algorithm relying on fractional programming. The numerical results show that a stronger LOS component (i.e., larger Rician K-factor) between a typical CU and its serving BS leads to a significant improvement in the system performance, while it has less effect on the D2D network.

Hybrid Long- and Short-Packet Based NOMA Systems With Joint Power Allocation and Beamforming Design

This paper investigates the performance of non-orthogonal multiple access (NOMA) systems with hybrid long- and short-packet communications, where a cell-center user (CU) demands high quality-of-service while a cell-edge user (CE) faces poor channel quality. Targeting at this problem, we propose an effective power-beamforming scheme by jointly allocating high power to CU and configuring a beamforming vector along with a maximum ratio transmission to CE's direction. The impact of imperfect successive interference cancellation on CE is thoughtfully taken into consideration. Approximate and asymptotic closed-form expressions for the block error rate and effective channel sum capacity are derived. Following that, we further quantify the diversity orders of users to reveal more useful insights into system designs. Numerical results are provided to demonstrate the effectiveness of the proposed power-beamforming scheme and to corroborate the accuracy of the derived theoretical analysis.

Wireless Communication System Performance in M2M Nakagami-m Fading Channel

In this paper, we investigate a wireless mobile communication system with a cooperative network operating over a mobile-to-mobile (M2M) fading channel in the presence of co-channel interference. Since there is no direct line of sight between the source (S) and the destination (D) due to various obstacles, the signal transmission takes place by using a relay (R). We modeled and represented the desired signal and co-channel interference by using the Nakagami-m fading distribution from the source (S) to the relay (R), as well as from the relay (R) to the destination (D). As the popularity of M2M communication has increased in 5G cellular networks, the performance analysis of the proposed system is of great importance. We derived closed-form expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the signal-to-interference ratio (SIR) at the input of the destination station. Based on the cumulative distribution function (CDF), we also evaluated the outage probability (Pout) of the proposed communication system. In addition, we derived a closed-form approximate expression for the level crossing rate (LCR) by using the Laplace approximation formula for the three-fold integral. Validity of the derived theoretical results we approved by Monte Carlo simulations.

Dynamic response of beams excited by moving oscillators: Approximate analytical solutions for general boundary conditions

In this paper, the dynamic response of an Euler-Bernoulli beam with general boundary conditions (BCs) and subject to a moving oscillator is examined. Notably, novel approximate closed-form expressions are determined for the vertical responses of both the beam and the moving oscillator, specifically considering the effect of damping in these systems, commonly omitted in standard approaches in the literature. In this regard, a modal superposition procedure is adopted and combined with an appropriate expansion-based approach of the dynamic response of the system, which naturally arises considering the oscillator-beam mass ratio to be reasonably small. Further, general boundary conditions are treated exploiting the use of a suitable set of orthogonal polynomial functions as beam mode shapes. In this manner, novel direct expressions for the response of the system are derived, in which the mode shapes coefficients explicitly appear. This leads to a straightforward application of the proposed solution, irrespective of the chosen BCs. Several numerical examples are presented to assess the reliability and accuracy of the proposed approach, considering different cases of beam BCs, and moving oscillator’s parameters. Results are validated by comparison with the data of finite element analyses, and numerical solutions of the complete system of governing equations.

Index Modulation for STAR-RIS Assisted NOMA System

In this letter, we first incorporate the concept of index modulation (IM) into simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) aided non-orthogonal multiple access (NOMA) system to improve the spectral efficiency. Specifically, the proposed IM aided STAR-RIS-NOMA system enables extra information bits to be transmitted by allocating subsurfaces to different users in a pre-defined subsurface allocation pattern. Furthermore, an approximate closed form expression on bit error rate (BER) is derived. Simulation results demonstrate that the proposed IM aided STAR-RIS-NOMA system is able to acquire transmission rate improvement compared to the conventional STAR-RIS NOMA.