Two-dimensional Poisson Point Process Research Articles

Interference Analysis for UAV Radar Networks With Guard Zones Based on Stochastic Geometry

Due to the increasing number of aerial radars and joint communication/sensing technologies, interference from uncoordinated radars will limit the target detection and ranging performance in the future. In this paper, we investigate the interference behavior in an aerial radar network for sensing ground targets. We consider that the radars mounted on unmanned aerial vehicles (UAVs) that fly at a certain altitude are randomly distributed according to a two-dimensional homogeneous Poisson point process (HPPP), and that the propagation is modeled using a probabilistic line-of-sight (LoS) channel model. For such a sensing network, we derive the distribution of the radar interference using a stochastic geometry based analysis. In particular, when Swerling I model is considered for radar cross-section area (RCS) for the target, we derive the Laplace transform of the radar interference. To avoid a strong interference between neighboring radars, a guard zone is introduced within which the UAV radar transmission around the permitted active radar is inhibited. As the radar performance metric, we derive the successful ranging probability (SRP) of a given radar by exploiting the Laplace transform of radar interference. Using the analytic SRP, we show that we can optimize the radar network parameters such as the radius of the guard zone and the density of the active radars. In addition, we also discuss how the analytic SRP gives an insight into the spectrum utilization strategy for the UAV radar networks with the guard zones.

Interference Characterization in Cellular-Assisted Vehicular Communications With Jamming

The automobile industry has been evolving with rapid growth over the last two decades. Conventionally, the narrow band dedicated short-range communications is used for the wireless connectivity of vehicular networks in the automobile industry. However, due to its shorter wireless range, higher latency, and lower capacity; cellular-assisted vehicular communications can be adopted to improve networks performance by disrupting the nearby infrastructure. The performance of cellular-assisted vehicular communications is typically compromised if jammers are present in their vicinity. Thus, it is important to characterize the interference in cellular-assisted vehicular communications with jammers. In this paper, we investigate jamming interference due to clustered-jammers for vehicular networks by considering that vehicles, road-side units, and pedestrians are modeled using an independent one-dimensional Poisson line process while cellular base stations are modeled using an independent two-dimensional homogeneous Poisson point process. The results show that jamming disrupts the performance of the vehicular networks and that this performance is severely compromised by increasing the density of jamming clusters and their transmission power. Therefore, the prime focus of the system designers should be to introduce anti-jamming scenario, when the transmission power and the number of jamming clusters in a network increase.

Statistical analysis of the best relay location in a random two‐way relay network with multi‐slope path loss

This paper presents an exact statistical analysis for the best relay location in a random two-way relay network whose channels are subject to multi-slope path loss and Rayleigh fading. In this network, relays are distributed randomly as a two-dimensional Poisson point process, and the best relay is selected according to a max–min criterion. The midpoint between the two sources is adopted as a reference point and the relays are indexed such that the closer to the reference point, the smaller is the index. An exact analytical expression is derived for the probability that the kth nearest relay to the reference point is the best relay. Then, a detailed statistical analysis for the location of the best relay with respect to the reference point is presented. In each case, a low-complexity approximation is also derived and its accuracy is verified by numerical simulation. The proposed analysis demonstrates the counter-intuitive fact that the best relay in a random two-way relay network is quite unlikely to be located on, or very close to the reference point. Numerical results also suggest that the proposed analysis can be used to restrict the search region for finding the best relay.

A Stochastic Geometry Approach to EMF Exposure Modeling

Downlink exposure to electromagnetic fields due to cellular base stations in urban environments is studied using the stochastic geometry framework. A two-dimensional Poisson Point Process is assumed for the base station distribution. Mathematical expressions of statistics of exposure are derived from a simple propagation model taking into account the height of the base stations. The error on exposure made by taking a limited number of base stations, instead of the whole set, is quantified. The relative impact of the model parameters on the statistics of exposure is highlighted. The method is then applied and the model parameters are calibrated using experimental data obtained by drive-tests in two Brussels municipalities, in Belgium, for the 2100 MHz and 2600 MHz frequency bands. It is shown that the proposed model fits experimental values, paving the way to a new methodology to assess general public exposure to electromagnetic fields, for any frequency band. An insight is given on how to apply the methodology to a real case without access to experimental data.

The Performance Analysis of Spectrum Sharing Between UAV Enabled Wireless Mesh Networks and Ground Networks

Unmanned aerial vehicle (UAV) has the advantages of large coverage and flexibility, which could be applied in disaster management to provide wireless services to the rescuers and victims. When UAVs forms an aerial mesh network, line-of-sight (LoS) air-to-air (A2A) communications have long transmission distance, which extends the coverage of multiple UAVs. However, the capacity of UAV is constrained due to the multiple hop transmissions in aerial mesh networks. In this paper, spectrum sharing between UAV enabled wireless mesh networks and ground networks is studied to improve the capacity of UAV networks. Considering two-dimensional (2D) and three-dimensional (3D) homogeneous Poisson point process (PPP) modeling for the distribution of UAVs within a vertical range $\Delta \textit h$ , stochastic geometry is applied to analyze the impact of the height of UAVs, the transmit power of UAVs, the density of UAVs and the vertical range, etc., on the coverage probability of ground network user and UAV network user. Besides, performance improvement of spectrum sharing with directional antenna is verified. With the object function of maximizing the transmission capacity, the optimal altitude of UAVs is obtained. This paper provides a theoretical guideline for the spectrum sharing of UAV enabled wireless mesh networks, which may contribute significant value to the study of spectrum sharing mechanisms for UAV enabled wireless mesh networks.

Physical Layer Security Enhancement Using Artificial Noise in Cellular Vehicle-to-Everything (C-V2X) Networks

The secure transmission of confidential information in cellular vehicle-to-everything (C-V2X) communication networks is vitally important for user's personal safety. However, for C-V2X there have not been much studies on the physical layer security (PLS). Since artificial noise (AN) and secure beamforming are popular PLS techniques for cellular communications, in this paper we investigate the potential of these PLS techniques for enhancing the security of C-V2X networks. In particular, leveraging stochastic geometry, we study the PLS of an AN assisted C-V2X network, where the locations of legitimate vehicular nodes, malicious vehicular nodes and road side units (RSUs) are modeled by Cox processes driven by a common Poisson line process (PLP), and the locations of cellular base stations (BSs) are modeled by a two-dimensional (2D) Poisson point process (PPP). Based on the maximum signal-to-interference-ratio (SIR) association scheme, we calculate the coverage probability of the network. We also derive bounds on the secrecy probability, which are validated by simulation results. Moreover, we obtain an analytical result of the effective secrecy throughput for characterizing the reliability and security of wiretap channels. Simulation results are given to validate the analytical result, and provide interesting insights into the impact of network parameters on the achievable secrecy performance. Simulation results show that a larger array antenna can provide a better robustness of the secure transmission strategy, and the optimal power allocation ratio between confidential information and AN remains almost unchanged for different numbers of antennas.

A spatial open-population capture-recapture model.

A spatial open-population capture-recapture model is described that extends both the non-spatial open-population model of Schwarz and Arnason and the spatially explicit closed-population model of Borchers and Efford. The superpopulation of animals available for detection at some time during a study is conceived as a two-dimensional Poisson point process. Individual probabilities of birth and death follow the conventional open-population model. Movement between sampling times may be modeled with a dispersal kernel using a recursive Markovian algorithm. Observations arise from distance-dependent sampling at an array of detectors. As in the closed-population spatial model, the observed data likelihood relies on integration over the unknown animal locations; maximization of this likelihood yields estimates of the birth, death, movement, and detection parameters. The models were fitted to data from a live-trapping study of brushtail possums (Trichosurus vulpecula) in New Zealand. Simulations confirmed that spatial modeling can greatly reduce the bias of capture-recapture survival estimates and that there is a degree of robustness to misspecification of the dispersal kernel. An R package is available that includes various extensions.

Covert Wireless Communication With Artificial Noise Generation

Covert communication conceals the transmission of the message from an attentive adversary. Recent work on the limits of covert communication in additive white Gaussian noise (AWGN) channels has demonstrated that a covert transmitter (Alice) can reliably transmit a maximum of $\mathcal{O}\left(\sqrt{n}\right)$ bits to a covert receiver (Bob) without being detected by an adversary (Warden Willie) in $n$ channel uses. This paper focuses on the scenario where other friendly nodes distributed according to a two-dimensional Poisson point process with density $m$ are present in the environment. We propose a strategy where the friendly node closest to the adversary, without close coordination with Alice, produces artificial noise. We show that this method allows Alice to reliably and covertly send $\mathcal{O}(\min\{{n,m^{\gamma/2}\sqrt{n}}\})$ bits to Bob in $n$ channel uses, where $\gamma$ is the path-loss exponent. Moreover, we also consider a setting where there are $N_{\mathrm{w}}$ collaborating adversaries uniformly and randomly located in the environment and show that in $n$ channel uses, Alice can reliably and covertly send $\mathcal{O}\left(\min\left\{n,\frac{m^{\gamma/2} \sqrt{n}}{N_{\mathrm{w}}^{\gamma}}\right\}\right)$ bits to Bob when $\gamma >2$, and $\mathcal{O}\left(\min\left\{n,\frac{m \sqrt{n}}{N_{\mathrm{w}}^{2}\log^2 {N_{\mathrm{w}}}}\right\}\right)$ when $\gamma = 2$. Conversely, we demonstrate that no higher covert throughput is possible for $\gamma>2$.

Carrier to interference ratio analysis in shotgun cellular systems over a generalized shadowing distribution

Shotgun cellular systems (SCSs) are wireless communication systems with randomly placed base stations (BSs) over the entire plane according to a two-dimensional Poisson point process. Such a system can model a dense cellular or wireless data network deployment, where the BS locations end up being close to random due to constraints other than optimal coverage. SCSs have been studied by considering path-loss and independent shadowing paths between BS to mobile station (MS) pairs in the channel models. In this paper, we consider correlated shadowing paths between BS to MS pairs as a most important factor, and analyze the carrier to interference ratio (CIR), in a SCS over this correlation, and determine an expression for distribution of CIR, and obtain the tail probability of the CIR.

Exact throughput analysis of random cooperative medium access control networks in the presence of shadowing

The throughput performance of a random cooperative medium access control (CoopMAC) network in the presence of shadowing and path loss is considered. The nodes are assumed to be distributed as a homogeneous two-dimensional Poisson point process with constant intensity. The helpers are divided into several tiers each having a distinct operating region and a distinct cooperative throughput. Then, the conditions under which a helper in a particular tier can improve the transmission rate between a given pair of nodes are examined. Based on these conditions, an exact analytical expression is derived for the average cooperative throughput of a Poisson CoopMAC network that is subject to path loss and shadowing. The expression is then used to investigate the effects of shadowing, intensity of helpers and distance between source and destination nodes on the average cooperative throughput of the network.

On the security of K-tier heterogeneous cellular networks

Motivated by recent developments in heterogeneous cellular networks and physical-layer security, we aim to characterize the fundamental limits of secure communication in networks. Based on a general model in which both transmitters and receivers are randomly scattered in space, we model the locations of K-tier base stations, users, and potential eavesdroppers as independent two-dimensional Poisson point processes. Using the proposed model, we analyze the achievable secrecy rates for an arbitrarily located mobile user. Assuming that the cell selection is based on achievable-secrecy-rate threshold, we obtain approximations for: (a) secrecy coverage probability and (b) average secrecy load per tier. We also investigate how the network performance is affected by secrecy rate threshold, eavesdropper density, and different access strategies are analyzed, respectively. Finally, our theoretical claims are confirmed by the numerical results.

Opportunistic Jamming for Enhancing Security: Stochastic Geometry Modeling and Analysis

This correspondence studies the secrecy communication of the single-input single-output multi-eavesdropper (SISOME) channel with multiple single-antenna jammers, where the jammers and eavesdroppers are distributed according to the independent two-dimensional homogeneous Poisson point process (PPP). For enhancing the physical layer security, we propose an opportunistic multiple jammer selection scheme, where the jammers whose channel gains to the legitimate receiver less than a threshold, are selected to transmit independent and identically distributed (\emph{i.i.d.}) Gaussian jamming signals to confound the eavesdroppers. We characterize the secrecy throughput achieved by our proposed jammer selection scheme, and show that the secrecy throughput is a quasi-concave function of the selection threshold.

A tractable approach to analyzing the physical-layer security in K-tier heterogeneous cellular networks

This study proposes a tractable approach to analyze the physical-layer security in the downlink of a multi-tier heterogeneous cellular network. This method is based on stochastic geometry, has low computational complexity, and uses the two-dimensional Poisson point process to model the locations of K-tier base stations and receivers, including those of legitimate users and eavesdroppers. Then, the achievable secrecy rates for an arbitrary user are determined and the upper and lower bounds of secrecy coverage probability derived on the condition that cross-tier interference is the main contributor to aggregate interference. Finally, our analysis results reveal the innate connections between information-theoretic security and the spatial densities of legitimate and malicious nodes.

Communication Neighbors Comparison in a Poisson Field of Nodes

The statistics of the ratio between the propagation losses of a given node and its $i$ -th and $j$ -th least-propagation-loss neighbors are obtained assuming that the spatial distribution of the neighboring nodes follows a two-dimensional homogeneous Poisson point process. It is shown that these statistics, unlike those for each individual propagation loss, are independent of the environmental fading type as well as the density of the neighbor nodes. The analytical results are validated and illustrated through numerical results and computer simulations. Some applications of the derived statistics are also presented.

Uncoordinated Jammer Selection for Securing SIMOME Wiretap Channels: A Stochastic Geometry Approach

This paper studies a single-input multi-output multi-eavesdropper (SIMOME) wiretap channel with multiple friendly single-antenna jammers. We consider random networks where the jammers and the eavesdroppers are distributed according to independent two-dimensional homogeneous Poisson point processes (PPP). We propose an opportunistic jammer selection scheme for the physical layer security enhancement, where the jammers whose channels are nearly orthogonal to the channel direction information (CDI) of the legitimate channel are selected to transmit independent and identically distributed (i.i.d.) Gaussian jamming signals to confound the eavesdroppers. The proposed scheme does not require a centralized design and signal coordinations among multiple jammers are not needed anymore. Furthermore, we analyze both the achievable secrecy throughput and the ergodic secrecy rate of the proposed jammer selection scheme. Based on the analysis results, we optimize the selection threshold for the secrecy throughput maximization and ergodic secrecy rate maximization. Simulation results show that the proposed jammer selection scheme can achieve a substantial performance gain.

Physical Layer Security in Downlink Multi-Antenna Cellular Networks

In this paper, we study physical layer security for the downlink of cellular networks, where the confidential messages transmitted to each mobile user can be eavesdropped by both (i) the other users in the same cell and (ii) the users in the other cells. The locations of base stations and mobile users are modeled as two independent two-dimensional Poisson point processes. Using the proposed model, we analyze the secrecy rates achievable by regularized channel inversion (RCI) precoding by performing a large-system analysis that combines tools from stochastic geometry and random matrix theory. We obtain approximations for the probability of secrecy outage and the mean secrecy rate, and characterize regimes where RCI precoding achieves a nonzero secrecy rate. We find that unlike isolated cells, the secrecy rate in a cellular network does not grow monotonically with the transmit power, and the network tends to be in secrecy outage if the transmit power grows unbounded. Furthermore, we show that there is an optimal value for the base station deployment density that maximizes the secrecy rate, and this value is a decreasing function of the signal-to-noise ratio.

Probabilistic relay assignment strategy for cooperation networks with random relays

A new approach in fairness is proposed by using a probabilistic relay assignment strategy for cooperative communication systems. In this approach, relays are allocated to sources based on a probabilistic relay assignment function, which is characterised by the outage probability. To examine the proposed relay assignment strategy, an exact expression for the outage probability of an opportunistic coded cooperation diversity system is derived. It is assumed that the relays in this system are distributed according to a two-dimensional homogeneous Poisson point process, and the transmission channels undergo Rayleigh fading as well as path loss. It is observed that the proposed strategy reduces the number of requests for using the relays and, thus, utilises the relays more efficiently. Furthermore, compared to the conventional opportunistic relay assignment strategy, the total power consumed by the relays decreases, while the quality of service requirements are still satisfied.

Generalized Analysis of Dual-Hop DF Opportunistic Relaying with Randomly Distributed Relays

The performance of dual-hop opportunistic relaying with randomly distributed decode-and-forward relays is investigated. The relays are assumed to be distributed according to a two-dimensional homogeneous Poisson point process, in contrast to most of the existing literature where the locations of relays are assumed to be known and deterministic. An exact expression for the outage probability of this cooperative system is derived for general fading environments. The expression is then specialized to the case where the transmission channels are modeled as Rayleigh fading channels.

Physical Layer Security in Cellular Networks: A Stochastic Geometry Approach

This paper studies the information-theoretic secrecy performance in large-scale cellular networks based on a stochastic geometry framework. The locations of both base stations and mobile users are modeled as independent two-dimensional Poisson point processes. We consider two important features of cellular networks, namely, information exchange between base stations and cell association, to characterize their impact on the achievable secrecy rate of an arbitrary downlink transmission with a certain portion of the mobile users acting as potential eavesdroppers. In particular, tractable results are presented under diverse assumptions on the availability of eavesdroppers' location information at the serving base station, which captures the benefit from the exchange of the location information between base stations.

Performance Analysis of Opportunistic Relaying in a Poisson Field of Amplify-and-Forward Relays

The performance of opportunistic relaying in a dual-hop amplify-and-forward relay network where the relaying nodes are distributed according to a homogeneous two-dimensional Poisson point process with fixed density is analyzed. Largely ignored in the literature, this assumption leads to more realistic results, as in a typical practical scenario neither the number of relays nor their distances from the source and destination are known. An exact expression for the outage probability of this cooperative system for general fading environments and different types of relaying is derived. The expression is then {simplified under an assumption that the radius of the relay deployment region is sufficiently large}, and specialized to additive white Gaussian noise (no fading), Rayleigh fading and severe (half-Gaussian) fading environments. Simple and tight upper and lower bounds on the outage probability are also derived. Simulation results validate the theoretical analyses and verify the accuracies of the proposed bounds.