Closed-form Probability Density Function Research Articles

Instantaneous Frequency and Chirp Rate Estimation for Noisy Quadratic FM Signals by CNN

Deep learning and machine learning are widely employed in various domains. In this paper, Artificial Neural Network (ANN) and Convolution Neural Network (CNN) are used to estimate the Instantons Frequency (IF), Linear Chirp Rate (LCR), and Quadratic Chirp Rate (QCR) for Quadratic Frequency Modulated (QFM) signals under Additive White Gaussian (AWG) noise and Additive Symmetric alpha Stable (ASαS) noise. SαS distributions are impulsive noise disturbances except for a few circumstances, lack a closed-form Probability Density Function (PDF), and an infinite second-order statistic. Geometric SNR (GSNR) is used to determine the impulsiveness of mixture noise for Gaussian and SαS noise. ANN is a machine learning classifier with few layers that reduce FE, LCRE, and QCRE complexity and achieve high accuracy. CNN is a deep learning classifier that is built with multiple layers of FE, LCRE, and QCRE. CNN is more accurate than ANN when dealing with large amounts of data and determining optimal features. The results reveal that SαS noise is substantially more damaging to FE, LCRE, and QCRE than Gaussian noise, even when the magnitude is modest, and it is less damaging when alpha is greater than one. After training DCNN for FE, LCRE, and QCRE estimation of QFM signals. The 2D-CNN model accuracy achieved 98.7603 and 1D-CNN is 75.8678 for ten epochs. ANN model accuracy achieved 37.5 for 1000 epochs. The accuracy of TFD (spectrogram & pspectrum) for frequency estimation of QFM signals was 38.4254 by spectrogram and 38.6746 by pspectrum.

Instantaneous Frequency Estimation of FM Signals under Gaussian and Symmetric α-Stable Noise: Deep Learning versus Time–Frequency Analysis

Deep learning (DL) and machine learning (ML) are widely used in many fields but rarely used in the frequency estimation (FE) and slope estimation (SE) of signals. Frequency and slope estimation for frequency-modulated (FM) and single-tone sinusoidal signals are essential in various applications, such as wireless communications, sound navigation and ranging (SONAR), and radio detection and ranging (RADAR) measurements. This work proposed a novel frequency estimation technique for instantaneous linear FM (LFM) sinusoidal wave using deep learning. Deep neural networks (DNN) and convolutional neural networks (CNN) are classes of artificial neural networks (ANNs) used for the frequency and slope estimation for LFM signals under additive white Gaussian noise (AWGN) and additive symmetric alpha stable noise (SαSN). DNN is composed of input, output, and two hidden layers, where several nodes in the first and second hidden layers are 25 and 8, respectively. CNN is the content input layer; many hidden layers include convolution, batch normalization, ReLU, max pooling, fully connected, and dropout. The output layer consists of a fully connected softmax and classification layers. SαS distributions are impulsive noise disturbances found in many communication environments such as marine systems, their distribution lacks a closed-form probability density function (PDF), except for specific cases, and infinite second-order statistics, hence geometric SNR (GSNR) is used in this work to determine the effect of noise in a mixture of Gaussian and SαS noise processes. DNN is a machine learning classifier with few layers for reducing FE and SE complexity. CNN is a deep learning classifier, designed with many layers, and proved to be more accurate than DNN when dealing with big data and finding optimal features. Simulation results show that SαS noise can be much more harmful to the FE and SE of FM signals than Gaussian noise. DL and ML can significantly reduce FE complexity, memory cost, and power consumption as compared to the classical FE based on time–frequency analysis, which are important requirements for many systems, such as some Internet of Things (IoT) sensor applications. After training CNN for frequency and slope estimation of LFM signals, the performance of CNN (in terms of accuracy) can give good results at very low signal-to-noise ratios where time–frequency distribution (TFD) fails, giving more than 20 dB difference in the GSNR working range as compared to the classical spectrogram-based estimation, and over 15 dB difference with Viterbi-based estimate.

Novel Approximate Distribution of the Sum of Lognormal-Rician Turbulence Channels With Pointing Errors and Applications in MIMO FSO Links

In this paper, an approximate closed-form probability density function expression for the sum of lognormal-Rician turbulence channels with Rayleigh pointing errors is developed. The results of Kolmogorov-Smirnov goodness-of-fit statistical tests show that the proposed approximation is highly accurate across a wide range of channel conditions. Also, the analysis of approximation error is presented in detail, and it indicates that a more efficient approximation can be achieved for larger coherence parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$r$</tex-math></inline-formula> and smaller variance <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma _{z}^{2}$</tex-math></inline-formula> . To reveal the importance of proposed approximation, the closed-form expressions for the ergodic capacity, outage probability, and bit-error rate are derived in terms of Meijer’s G-function. The performance of multiple-input multiple-output (MIMO) free-space optical (FSO) systems with equal gain combining (EGC) diversity technique are analyzed in detail under different scenarios, including the number of transmit and receive apertures, turbulence channels, and presence of pointing errors. It is observed that MIMO technology can offer a significant improvement in FSO performance when compared with the single-input single-output (SISO) systems. The ergodic capacity and BER performance at high signal-to-noise ratio are also obtained to provide further insights. Numerical results demonstrate the accuracy of the proposed approach.

Performance Analysis of FSO Systems Over a Lognormal-Rician Turbulence Channel With Generalized Pointing Errors

In this paper, an approximate closed-form probability density function expression for the Lognormal-Rician atmospheric turbulence with generalized pointing errors model is developed by using a series representation. The Kolmogorov-Smirnov (KS) goodness-of-fit statistical tool is employed to verify the accuracy of the proposed approximation. We demonstrate that a more efficient approximation can be achieved for the smaller jitter variances and nonzero boresight displacements. To reveal the importance of the proposed approximation, the approximate analytic expressions for the ergodic capacity and outage performance of single-input single-output free-space optical (FSO) systems are derived. It is observed that the pointing errors significantly degrade the performance of FSO links. By using the obtained asymptotic expressions, the impact of pointing errors on the ergodic capacity is considerably mitigated by selecting the optimum beam width at the transmitter side. We find that the optimum beam width is independent of atmospheric turbulence and the Pearson correlation coefficient in high signal-to-noise ratio conditions. All of the derived analytical results are confirmed by Monte Carlo simulation results.

Performance of Vertical Underwater Wireless Optical Communications With Cascaded Layered Modeling

This paper investigates the performance of vertical underwater wireless optical communication (UWOC) systems in the presence of air bubbles and temperature gradients. We consider a generalized UWOC channel model that contains <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> layers, each having the same distribution but with different parameters to consider the vertically inhomogeneous nature of the underwater environment. To capture the effects of air bubbles and temperature gradients on channel statistics, we model each layer with a mixture exponential-generalized Gamma distribution. A closed-form probability density function (PDF) is presented for the end-to-end signal-to-noise ratio. Based on the PDF, we derive the average bit-error-rate (BER) and the ergodic capacity. Moreover, we provide asymptotic BER and ergodic capacity results in simple forms. The performance and behavior of vertical UWOC systems are thoroughly analyzed, and all the derived expressions are verified via Monte Carlo simulations.

On the Distribution of SINR for Widely Linear MMSE MIMO Systems With Rectilinear or Quasi-Rectilinear Signals

Although the widely linear least mean square error (WLMMSE) receiver has been an appealing option for multiple-input-multiple-output (MIMO) wireless systems, a statistical understanding on its pose-detection signal-to-interference-plus-noise ratio (SINR) in detail is still missing. To this end, we consider a WLMMSE MIMO transmission system with rectilinear or quasi-rectilinear (QR) signals over the uncorrelated Rayleigh fading channel and investigate the statistical properties of its SINR for an arbitrary antenna configuration with $N_t$ transmit antennas and $N_r$ receive ones. We first derive an analytic probability density function (PDF) of the SINR in terms of the confluent hypergeometric function of the second kind, for WLMMSE MIMO systems with an arbitrary $N_r$ and $N_t=2, 3$. For a more general case in practice, i.e., $N_t>3$, we resort to the moment generating function to obtain an approximate but closed form PDF under some mild conditions, which, as expected, is more Gaussian-like as $2N_r-N_t$ increases. The so-derived PDFs are able to provide key insights into the WLMMSE MIMO receiver in terms of the outage probability, the symbol error rate, and the diversity gain, all presented in closed form. In particular, its diversity gain and the gain improvement over the conventional LMMSE one are explicitly quantified as $N_r-(N_t-1)/2$ and $(N_t-1)/2$, respectively. Finally, Monte Carlo simulations support the analysis.

The Polarimetric Detection Optimization Filter and its Statistical Test for Ship Detection

Ship detection via synthetic aperture radar (SAR) has been demonstrated to be very useful as polarimetric information helps discriminate between targets and sea clutter. Among the available polarimetric detectors, optimal polarimetric detection (OPD) theoretically provides the best detection performance under the assumption that the fully developed speckle hypothesis stands. This study proposes a polarimetric detection optimization filter (PDOF). The target clutter ratio (TCR) over the speckle variation was maximized using a matrix transform to derive the PDOF. The objective function based on a matrix transform instead of a vector transform is optimized to obtain synthetic effects by combining a polarimetric whitening filter (PWF) and a polarimetric matched filter (PMF). Subspace form of the PDOF (SPDOF) is also proposed, which gives performance comparable to the PDOF. Assuming a Wishart distribution, the exact and approximate expressions of the closed-form probability density function (PDF) of the PDOF are derived. The probability of false alarm (PFA) was derived in a closed-form expression, which allows obtaining the PDOF threshold analytically. Moreover, the gamma model is extended to a generalized gamma distribution ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{G}\Gamma \text{D}$ </tex-math></inline-formula> ) to adapt complicated resolutions and sea states. Experiments with simulated and real data validate the correctness and effectiveness of the results. The PDOF detector achieves the best performance in most virtual and real-world environments, especially in cases where the target statistics and clutter are not Wishart-distributed.

Robust Frequency Estimation Under Additive Mixture Noise

In many applications such as multiuser radar communications and astrophysical imaging processing, the encountered noise is usually described by the finite sum of -stable variables. In this paper, a new parameter estimator is developed, in the presence of this new heavy-tailed noise. Since the closed-form PDF of the -stable variable does not exist except and , we take the sum of the Cauchy () and Gaussian () noise as an example, namely, additive Cauchy-Gaussian (ACG) noise. The probability density function (PDF) of the mixed random variable, can be calculated by the convolution of the Cauchy's PDF and Gaussian's PDF. Because of the complicated integral in the PDF expression of the ACG noise, traditional estimators, e.g., maximum likelihood, are analytically not tractable. To obtain the optimal estimates, a new robust frequency estimator is devised by employing the Metropolis-Hastings (M-H) algorithm. Meanwhile, to guarantee the fast convergence of the M-H chain, a new proposal covariance criterion is also devised, where the batch of previous samples are utilized to iteratively update the proposal covariance in each sampling process. Computer simulations are carried out to indicate the superiority of the developed scheme, when compared with several conventional estimators and the Cramér-Rao lower bound.

Distance distribution between two random points in arbitrary polygons

Distance distributions are a key building block in many subfields in mathematics, science and engineering. In this paper, we propose a novel framework for analytically computing the closed form probability density function (PDF) of the distance between two random points each uniformly randomly distributed in respective arbitrary polygon regions. The proposed framework is based on measure theory and uses polar decomposition for simplifying and calculating the integrals to obtain closed form results. We validate our proposed framework by comparison with simulations and published closed form results in the literature for simple cases. We illustrate the versatility and advantage of the proposed framework by deriving closed form results for a case not yet reported in the literature. Finally, we also develop a Mathematica implementation of the proposed framework which allows a user to define any two arbitrary (concave or convex) polygons, with or without holes, which may be disjoint or overlap or coincide and determine the distance distribution numerically.

Performance analysis of MIMO FSO systems with EGC diversity receiver over lognormal-Rician turbulence

Two approximate closed-form probability density function expressions for the sum of lognormal-Rician random variates (RVs) and the sum of square root of lognormal-Rician RVs are obtained. The proposed expressions are shown to provide high accuracy over a wide range of channel conditions according to Kolmogorov–Smirnov goodness-of-fit statistical tests results. Also, the analysis of the approximation error is presented to indicate that a higher approximation accuracy can be achieved for larger r or smaller variance σz2. To reveal the importance of the proposed approximation, new approximate closed-form expressions of the ergodic capacity, average bit error rate for the intensity-modulation direct-detection (IMDD), and coherent multiple-input multiple-output (MIMO) free-space optical (FSO) systems with equal gain combining diversity technique over lognormal-Rician turbulence channels are developed. It is observed that MIMO technology can offer a significant improvement in system performance and the performance of coherent FSO systems outperforms that of IMDD systems. The Monte Carlo simulation results are further utilized to illustrate the accuracy of the proposed approach.

Performance Analysis of Multi-Branch Reconfigurable Intelligent Surfaces-Assisted Optical Wireless Communication System in Environment With Obstacles

Since optical links are easily blocked by obstacles in the environment, they have been considered difficult to directly perform wireless communications. However, reconfigurable intelligent surfaces (RISs), as a new type of digital coding meta-materials, can significantly improve the coverage of optical communication by establishing new links. We propose a controllable multi-branch wireless optical communication system based on optical RISs. By setting up multiple optical RISs in the environment, multiple artificial channels are built to improve the system performance and to reduce the outage probability. In this paper, we investigate three factors affecting channel coefficients, including beam jitter, RIS jitter, and obstruction probability. We derive closed-form probability density function of channel coefficients, asymptotic average bit-error rate (BER), and the outage probability for systems with multiple branches. Our analysis reveals that the probability density function contains an impulse function, which causes irreducible error rate and outage probability floors. Based on our numerical results, the BER and outage probability floor of the multi-branch system are significantly reduced compared with the single direct path system. Therefore, the optical RIS assisted multi-branch wireless communication is a promising solution against obstacles of optical channel.

On the Performance of Downlink Non-Orthogonal Multiple Access Wireless Networks With Directional Beamforming and Limit of the User Number

Non-orthogonal multiple access (NOMA) is an enabling and key wireless technology for the next generation of mobile communications. Generally, previous works focus on the performance in NOMA networks, based on some assumptions. For instance, the number of users in a NOMA group, is two-category: 1) only two users; 2) any number of users without restriction. And just a certain factor of channel gain, e.g., the propagation path loss or the small scale fading, is subjectively considered to be dominant. Besides, there is lack of the closed-form expression for the probability density function (PDF) of ordered channel gain. In this paper, we investigate a downlink cellular NOMA network with the directional beamforming and the upper limit for user number (ULFUN) in a NOMA group. Meanwhile, the path loss and the Rayleigh fading are taken into account as a whole for ordering channel gains. Based on the derived closed-form PDF of ordered channel gain, we analyze the performance for an arbitrary user in the network and derive closed-form expressions for the outage probability, the approximate meta distribution of conditional success probability (CSP) and the ergodic data rate. The provided simulations demonstrate that the proposed analytical results are accurate and reasonable. Moreover, it illustrates that the existence of ULFUN can have a positive impact on the performance gain in low SINR environments, while in terms of the outage probability, ULFUN would become a non-negligible and negative factor in the improvement of performance for high SINR environments.

A Detection Sensitivity Analysis Model for Structural Health Monitoring to Inspect Wall Thinning considering Random Sensor Location

ABSTRACT Structural health monitoring (SHM), which allows the detection of defects at an early stage by attaching sensors to the target, is an effective method of enhancing the reliability and the safety of important engineering structures. One of the practical difficulties of SHM is that usually a large area must be monitored using a limited number of sensors fixed at certain locations. And the sensor placement is a decisive contributor to the detection capability of SHM because measured signals generally depend on the location of a defect with respect to a sensor. In order to quantify the detection sensitivity more reasonably, this study proposes an analytical method based on a closed-form probability density function and a numerical method based on Monte Carlo simulation to quantify the detection sensitivity, taking into account the randomness of sensor location. The effectiveness of the proposed detection sensitivity analysis model has been examined using simulated inspection signals of low frequency electromagnetic monitoring for detecting full circumferential pipe wall thinning.

Novel approximate distribution of the sum of Gamma–Gamma variates with pointing errors and applications in MIMO FSO links

In this work, a novel approximate closed-form probability density function for the sum of Gamma–Gamma random variates with Rayleigh pointing errors is derived by means of the well-known Jensen’s inequality. We employ the Kolmogorov–Smirnov goodness-of-fit statistical tests to verify the accuracy of the proposed approximation. The test results demonstrate that an exact distribution can be efficiently approximated by this derived approximate expression over a wide range of channel conditions. Also, the analysis of the approximation error is presented to indicate that a higher approximation accuracy can be achieved for small pointing errors. The series representation of this approximate expression is also developed, which enables us to avoid dealing with the integration. To reveal the importance of proposed approximation, new approximate closed-form expressions of the average bit error rate (BER) and ergodic capacity are derived. The performance for multiple-input multiple-output (MIMO) free-space optical (FSO) systems with equal gain combining (EGC) diversity technique are analyzed with different parameters, including the number of transmit and receive apertures, strength of atmospheric turbulence, and presence of pointing errors. It is observed that the pointing errors significantly degrade the performance of MIMO FSO systems. The numerical results and simulation are further utilized to illustrate the accuracy of the proposed approach.

Signal Fractions Analysis and Safety-Distance Modeling in V2V Inter-Lane Communications

For vehicular networks, safety distances are important, but existing spatial models fail to characterize this parameter, especially for inter-lane communications. This work proposes a Matérn hard-core processes based framework to appraise the performance of signal fractions (SF), where the hard-core distance is used to depict safety distances. By considering both semicircle and omnidirectional antennas, we derive high-accurate closed-form probability density functions of communication distances to acquire the complementary cumulative distribution function of SF. The derived expressions theoretically demonstrate that the nearest vehicle within the safety distance follows a uniform distribution and there is an upper limit for SF in terms of the transmit power.

Impact of RF I/Q Imbalance on Interference-Limited Mixed RF/FSO TWR Systems With Non-Zero Boresight Error

In this letter, we investigate a generic model assessing the effect of in-phase/quadrature-phase imbalance (IQI) on an asymmetric dual hop radio frequency/free space optical (RF/FSO) two-way relay (TWR) system in the presence of multiple co-channel interferers (CCIs) at the relay. The fading on the RF and FSO links have been modeled using K-distribution and double generalized Gamma (D-GG) turbulence model, respectively. The impact of non-zero boresight pointing error and type of optical demodulation schemes have been considered on the FSO link. To this end, a closed-form probability density function (PDF) has been derived for the FSO link undergoing D-GG irradiance with non-zero boresight pointing error. Furthermore, the exact and high signal-to-noise (SNR) asymptotic expression for outage probability have been presented. To gain insights into the throughput of the system, approximate and asymptotic expressions for the achievable sum rate (ASR) have been derived. The results show dependency of reliability and throughput offered on IQI at the RF front-end and strength of interference for the considered TWR system.

Modeling the cryptocurrency return distribution via Laplace scale mixtures

The search of appropriate models for describing the currency return distribution is one of the main interests not only in finance, but also in the more recent trans-disciplinary econophysics research field. Such a search is recently focusing on cryptocurrencies, due to their proliferation. Although there is no agreement of what theoretical models are the most appropriate, there is a general consensus that the cryptocurrency return distribution is highly-peaked and heavy-tailed, with a large excess kurtosis and the tail distribution decreases as the inverse cubic power-law. With these requirements, the Laplace distribution can be considered as a valid candidate model. However, there are two limitations to be taken into account: 1) the Laplace tail distribution does not decrease as the inverse cubic power-law and 2) the excess kurtosis is fixed at 3. To make the tailedness of the Laplace distribution more flexible, still maintaining its peculiar symmetric shape, we introduce the Laplace scale mixture (LSM) family of distributions. Each member of the family is obtained by dividing the scale parameter of the conditional Laplace distribution by a convenient mixing random variable taking values on all or part of the positive real line and whose distribution depends on a parameter vector θ governing the tail behavior of the resulting LSM. For illustrative purposes, we consider different mixing distributions; they give rise to LSMs having a closed-form probability density function where the Laplace distribution is obtained as a special case under a convenient choice of θ. We describe an EM algorithm to obtain maximum likelihood estimates of the parameters for all the considered LSMs. Interestingly, we show how the influence of observations associated with large scaled absolute distances is reduced (downweighted), with respect to the nested Laplace distribution, in the estimation phase. We fit these models to the returns of 4 cryptocurrencies, considering several classical symmetric distributions for comparison. The analysis shows how the proposed models represent a valid alternative to the considered competitors in terms of AIC, BIC and likelihood-ratio tests, but also in reproducing the larger empirical excess kurtosis and in resembling the empirical inverse cubic power-law decrease of the tail distribution.

McLeish Distribution: Performance of Digital Communications Over Additive White McLeish Noise (AWMN) Channels

The objective of this article is to propose and statistically validate a more general additive non-Gaussian noise distribution, which we term McLeish distribution, whose random nature can model different impulsive noise environments commonly encountered in practice and provides a robust alternative to Gaussian noise distribution. In particular, for the first time in the literature, we establish the laws of McLeish distribution and therefrom derive the laws of the sum of McLeish distributions by obtaining closed-form expressions for their probability density function (PDF), cumulative distribution function (CDF), complementary CDF (C2DF), moment-generating function (MGF) and higher-order moments. Further, for certain problems related to the envelope of complex random signals, we extend McLeish distribution to complex McLeish distribution and thereby propose circularly/elliptically symmetric (CS/ES) complex McLeish distributions with closed-form PDF, CDF, MGF and higher-order moments. For generalization of one-dimensional distribution to multi-dimensional distribution, we develop and propose both multivariate McLeish distribution and multivariate complex CS/ES (CCS/CES) McLeish distribution with analytically tractable and closed-form PDF, CDF, C2DF and MGF. In addition to the proposed McLeish distribution framework and for its practical illustration, we theoretically investigate and prove the existence of McLeish distribution as additive noise in communication systems. Accordingly, we introduce additive white McLeish noise (AWMN) channels. For coherent/non-coherent signaling over AWMN channels, we propose novel expressions for maximum a priori (MAP) and maximum likelihood (ML) symbol decisions and thereby obtain closed-form expressions for both bit error rate (BER) of binary modulation schemes and symbol error rate (SER) of various M-ary modulation schemes. Further, we verify the validity and accuracy of our novel BER/SER expressions with some selected numerical examples and some computer-based simulations.

A New Probability Density Function for Minimizing Geometric Dilution of Precision in Location-Aware Wireless Communications

Cellular site planners have long used propagation models for optimizing an architecture deployment strategy. The foundation of these models are a plethora of theoretical results that all contribute to a probabilistic understanding of this phenomenon. In lower-band frequencies, coverage and propagation phenomena were sufficient considerations for legacy infrastructure deployment; however, as cellular technology probes uncharted millimeter wave spectra and beyond, it is natural to inquire as to whether other metrics could contribute to an informed infrastructure deployment. As the benchmark goals for positioning accuracy grow more ambitious and location-aware communications becomes a reality, we argue that localization accuracy should also play a prominent role in cellular infrastructure deployment planning. To this end, we submit a new closed-form probability density function (PDF) to characterize the angular difference of a pair of base stations and a mobile terminal. The importance of the angular difference is demonstrated by showing that the Cramer-Rao lower bound for localization is solely a function of it and measurement accuracy. Further, we submit a computationally tenable algorithm for producing the required PDF. To demonstrate the power of the density, we show some base station deployments that are guaranteed to yield geometrically favorable environments for positioning. Finally, we demonstrate how this new distribution outperforms numerical analysis when planning wireless network architecture deployment for location-aware communications.

Modeling and Analysis of MmWave V2X Networks With Vehicular Platoon Systems

Due to the low traffic congestion, high fuel efficiency, and comfortable travel experience, vehicular platoon systems (VPSs) become one of the most promising applications in millimeter wave (mmWave) vehicular networks. In this paper, an effective spatial framework for mmWave vehicle-to-everything (V2X) networks with VPSs is proposed by utilizing stochastic geometry approaches. Base stations (BSs) are modeled by a Poisson point process and vehicles are distributed according to multiple type II Matern hard-core processes. To characterize the blockage process caused by vehicles, a closed-form expression is deduced to distinguish line-of-sight (LOS) and non-LOS transmission. This expression demonstrates that LOS links are independent of horizontal communication distances. Several closed-form probability density functions of the communication distance between a reference platoon and its serving transmitter (other platoons or BSs) are derived for analyzing the generated path loss. After designing three practical user association techniques, tractable expressions for coverage probabilities are figured out. Our work theoretically shows that the maximum density of VPSs exists and large antenna scales benefit the networks’ coverage performance. The numerical results illustrate that platoons outperform individual vehicles in terms of road spectral efficiency and the considered system is LOS interference-limited.