This paper examines the coverage characteristics of large inclined low Earth orbit (LEO) satellite constellations, focusing on equatorial regions. The study uses stochastic geometry to model satellites and users as uniformly distributed along their respective orbits and latitude ranges. The central angle and distance distributions between users and nth nearest satellite are derived to estimate line-of-sight (LoS) and coverage probability. Analytical models are validated with Monte Carlo simulations to provide practical insights for optimizing satellite deployments for equatorial regions. Results highlight the impact of satellite altitude and constellation density, revealing how lower inclination angles enhance satellite density over equatorial regions, thus improving coverage performance.

This paper studies a high-altitude platform (HAP) network supported by reconfigurable intelligent surfaces (RISs). The practical irregular placement of HAPs and RISs is modeled using homogeneous Poisson point processes, while buildings that cause blockages in urban areas are modeled as a Boolean scheme of rectangles. We introduce a novel approach to characterize the statistical channel based on generalized Beta prime distribution. Analytical expressions for coverage probability and ergodic capacity in an interference-limited system are derived and validated through Monte Carlo simulations. The findings show notable performance improvements and reveal the impact of various system parameters, including blockages effect which contribute in mitigating interference from the other visible HAPs. This proposed system could enhance connectivity and enable effective data offloading in urban environments.

This paper proposes a three-dimensional (3D) satellite-terrestrial communication network assisted with reconfigurable intelligent surfaces (RISs). Using stochastic geometry models, we present an original framework to derive tractable yet accurate closed-form expressions for coverage probability and ergodic capacity in the presence of fading. A homogeneous Poisson point process models the satellites on a sphere, while RISs are randomly deployed in a 3D cylindrical region. We consider nonidentical channels that correspond to different RISs and follow the $\kappa-\mu$ fading distribution. We verify the high accuracy of the adopted approach through Monte Carlo simulations and demonstrate the significant improvement in system performance due to using RISs. Furthermore, we comprehensively study the effect of the different system parameters on its performance using the derived analytical expressions, which enable system engineers to predict and optimize the expected downlink coverage and capacity performance analytically.

This paper investigates the ergodic capacity of reflecting intelligent surface (RIS)-aided single-input single-output communication systems with spatially correlated Rayleigh-fading channels. The ergodic capacity for such systems does not admit an exact closed-form expression. Therefore, we consider two alternative fading distributions to approximate the systems’ statistical characterization to enable the derivation of closed-form expressions for the ergodic capacity. We further simplify the ergodic capacity by proposing novel and unified approximations in the form of a weighted sum of logarithmic functions with optimized coefficients. We validate the effectiveness and the high accuracy of the adopted schemes and the proposed approximations through numerical results. Performance analysis to study the impact of several system parameters on the ergodic capacity is also conducted. Deploying an RIS to the communication system can significantly increase the ergodic capacity which increases even further with increasing the number of reflecting elements equipped on the RIS, and this effect is best seen when the direct path is weak.

We present tight yet tractable approximations and bounds for the ergodic capacity of any communication system in the form of a weighted sum of logarithmic functions, with the focus on the Nakagami and lognormal distributions that represent key building blocks for more complicated systems. A minimax optimization technique is developed to derive their coefficients resulting in uniform absolute or relative error. These approximations and bounds constitute a powerful tool for the statistical performance analysis as they enable the evaluation of the ergodic capacity of various communication systems that experience small-scale fading together with the lognormal shadowing effect and allow for simplifying the complicated integrals encountered when evaluating the ergodic capacity in different communication scenarios. Simple and tight closed-form solutions for the ergodic capacity of many classic and timely application examples are derived using the logarithmic approximations. The high accuracy of the proposed approximations is verified by numerical comparisons with existing approximations and with those obtained directly from numerical integration methods.

In this paper, we present simple and tight approximations for the integer powers of the Gaussian Q-function, in the form of exponential sums. They are based on optimizing the corresponding coefficients in the minimax sense using the Remez exchange algorithm. In particular, the best exponential approximation is characterized by the alternation of its absolute error function, which results in extrema that alternate in sign and have the same magnitude of error. The extrema are described by a system of nonlinear equations that are solved using Newton- Raphson method in every iteration of the Remez algorithm, which eventually leads to a uniform error function. This approximation can be employed in the evaluation of average symbol error probability (ASEP) under additive white Gaussian noise and various fading models. Especially, we present several application examples on evaluating ASEP in closed forms with Nakagami-m, Fisher-Snedecor F, η - μ, and κ - μ channels. The numerical results show that our approximations outperform the existing ones with the same form in terms of the global error. In addition, they achieve high accuracy for the whole range of the argument with and without fading, and it can even be improved further by increasing the number of exponential terms.

In this paper, we present a comprehensive overview of (perhaps) all possible approximations resulting from applying the most common numerical integration techniques on the Gaussian Q-function. We also present a unified method to optimize the coefficients of the resulting exponential approximation for any number of exponentials and using any numerical quadrature rule to produce tighter approximations. Two new tight approximations are provided as examples by implementing the Legendre numerical rule with Quasi-Newton method for two and three exponential terms. The performance of the different numerical integration techniques is evaluated and compared, and the accuracy of the optimized ones is verified for the whole argument-range of interest and in terms of the chosen optimization criterion.

This paper presents a novel systematic methodology to obtain new simple and tight approximations, lower bounds, and upper bounds for the Gaussian Q-function, and functions thereof, in the form of a weighted sum of exponential functions. They are based on minimizing the maximum absolute or relative error, resulting in globally uniform error functions with equalized extrema. In particular, we construct sets of equations that describe the behaviour of the targeted error functions and solve them numerically in order to find the optimized sets of coefficients for the sum of exponentials. This also allows for establishing a trade-off between absolute and relative error by controlling weights assigned to the error functions' extrema. We further extend the proposed procedure to derive approximations and bounds for any polynomial of the Q-function, which in turn allows approximating and bounding many functions of the Q-function that meet the Taylor series conditions, and consider the integer powers of the Q-function as a special case. In the numerical results, other known approximations of the same and different forms as well as those obtained directly from quadrature rules are compared with the proposed approximations and bounds to demonstrate that they achieve increasingly better accuracy in terms of the global error, thus requiring significantly lower number of sum terms to achieve the same level of accuracy than any reference approach of the same form.