Complementary Cumulative Density Function Research Articles

Design of a Test for Detecting the Presence of Impulsive Noise.

This paper proposes a new test method of detecting the presence of impulsive noise based on a complementary cumulative density function (CCDF). Impulsive noise severely degrades performance of communication systems and the conventional Kolmogorov–Smirnov (K–S) test may not perform well, because the test does not consider the characteristics of impulsive noise. In order to detect the presence of impulsive noise reliably, the CCDF of measurement samples is analyzed and compared with the CCDF of additive white Gaussian noise to find the difference between those CCDFs. Due to the nature of heavy-tails in impulsive noise, only the maximum difference may not be sufficient for the accurate detection of impulsive noise. Therefore, the proposed method applies the test hypothesis using the weighted sum of all the differences between those CCDFs. Simulation results justify that the proposed test is more robust and provides lower miss detection probability than the K–S test in the presence of impulsive noise.

Statistical Analysis of Lifetime in Wireless Cognitive Sensor Network for Multi-Channel Cooperative Spectrum Sensing

This paper deals with the issue of reducing energy consumption and evaluating the network lifetime in a wireless cognitive sensor network (WCSN) composed of randomly-distributed sensors to perform multi-channel cooperative spectrum sensing (MCCSS). More precisely, the paper presents a probabilistic approach for analyzing the WCSN lifetime whereas the detection quality is also considered. To this end, it is assumed that both the number of sensors and primary users in the area follow two-dimensional Poisson distributions, and accordingly, the detection probability and false alarm probability for the sensors are calculated, statistically. Next, the number of required sensors for MCCSS is minimized under the global probability of detection (GDP) and the global probability of false alarm constraints. Then, the tail probability of the lifetime by individual sensors is found based on the average sensors' energy consumption. After that, a tail probability for the network lifetime is calculated through statistical analysis, and it is used to obtain an analytical expression for the complementary cumulative density function (CCDF) of the network lifetime. Using the CCDF, one can find the probability that the network lifetime extends until a given time if the minimum number of sensors is used for satisfying the detection quality and the sensors are distributed randomly. Finally, computer simulations using the Monte-Carlo method are used to verify the achieved results.

Energy Harvesting in Unmanned Aerial Vehicle Networks With 3D Antenna Radiation Patterns

In this paper, an analytical framework is provided to analyze the energy coverage performance of unmanned aerial vehicle (UAV) energy harvesting networks with clustered user equipments (UEs). Locations of UEs are modeled as a Poisson Cluster Process (PCP), and UAVs are assumed to be located at a certain height above the center of user clusters. Hence, user-centric UAV deployments are addressed. Two different models are considered for the line-of-sight (LOS) probability function to compare their effects on the network performance. Moreover, antennas with doughnut-shaped radiation patterns are employed at both UAVs and UEs, and the impact of practical 3D antenna radiation patterns on the network performance is also investigated. Initially, the path loss of each tier is statistically described by deriving the complementary cumulative distribution function and probability density function. Following this, association probabilities with each tier are determined, and energy coverage probability of the UAV network is characterized in terms of key system and network parameters for UAV deployments both at a single height level and more generally at multiple heights. Through numerical results, we have shown that cluster size and UAV height play crucial roles on the energy coverage performance. Furthermore, energy coverage probability is significantly affected by the antenna orientation and number of UAVs in the network.

Unified Analysis of HetNets Using Poisson Cluster Processes Under Max-Power Association

Owing to its flexibility in modeling real-world spatial configurations of users and base stations (BSs), the Poisson cluster process (PCP) has recently emerged as an appealing way to model and analyze heterogeneous cellular networks (HetNets). Despite its undisputed relevance to HetNets—corroborated by the models used in the industry—the PCP’s use in performance analysis has been limited. This is primarily because of the lack of analytical tools to characterize the performance metrics, such as the coverage probability of a user connected to the strongest BS. In this paper, we develop an analytical framework for the evaluation of the coverage probability, or equivalently the complementary cumulative density function (CCDF) of signal-to-interference-and-noise ratio ( SINR ), of a typical user in a $K$ -tier HetNet under a $\max $ power-based association strategy, where the BS locations of each tier follow either a Poisson point process (PPP) or a PCP. The key enabling step involves conditioning on the parent PPPs of all the PCPs, which allows us to express the coverage probability as a product of sum-product and probability generating functionals (PGFLs) of the parent PPPs. In addition to several useful insights, our analysis provides a rigorous way to study the impact of the cluster size on the ${\it SINR}$ distribution, which was not possible using the existing PPP-based models.

Downlink Analysis in Unmanned Aerial Vehicle (UAV) Assisted Cellular Networks With Clustered Users

The use of unmanned aerial vehicles (UAVs) operating as aerial base stations (BSs) has emerged as a promising solution especially in scenarios requiring rapid deployments (e.g., in the cases of crowded hotspots, sporting events, emergencies, and natural disasters) in order to assist the ground BSs. In this paper, an analytical framework is provided to analyze the signal-to-interference-plus-noise ratio (SINR) coverage probability of UAV assisted cellular networks with clustered user equipments (UEs). Locations of UAVs and ground BSs are modeled as Poison point processes, and UEs are assumed to be distributed according to a Poisson cluster process around the projections of UAVs on the ground. Initially, the complementary cumulative distribution function and probability density function of path losses for both UAV and ground BS tiers are derived. Subsequently, association probabilities with each tier are obtained. SINR coverage probability is derived for the entire network using tools from stochastic geometry. Finally, area spectral efficiency (ASE) of the entire network is determined, and SINR coverage probability expression for a more general model is presented by considering that UAVs are located at different heights. Via numerical results, we have shown that UAV height and path-loss exponents play important roles on the coverage performance. Moreover, coverage probability can be improved with smaller number of UAVs, while better ASE is achieved by employing more UAVs and having UEs more compactly clustered around the UAVs.

Reply to “Comment on ‘Revisiting the 1894 Omori Aftershock Dataset with the Stretched Exponential Function’ by A. Mignan” by S. Hainzl and A. Christophersen

ABSTRACT Hainzl and Christophersen (2016) recently suggested that the approach used by Mignan (2016) to visualize the Omori law in a log–log plot using the complementary cumulative density function (CCDF) was misleading. They showed that one should use a temporal upper bound t max =max( t obs ), with t obs the occurrence time of a set of observed aftershocks, instead of t max →∞, as used by Mignan. They found that both the Omori law and stretched exponential function (SEF) are undistinguishable on a corrected CCDF log–log plot but that the Omori law is preferred based on maximum‐likelihood estimations (MLEs). I first clarify the rationale for using the CCDF log–log plot for comparison of the SEF with a power law but verify that the comparison indeed becomes misleading when the data sequence is incomplete. However, I then show that MLEs obtained for the Omori law are ambiguous, because this function may win versus the SEF, even if the true relaxation process follows an SEF (subject to early data incompleteness). This debate highlights that statistics alone may not be enough to choose between the stretched exponential and the Omori law and that physical considerations should be added to the discussion.

Comment on “Revisiting the 1894 Omori Aftershock Dataset with the Stretched Exponential Function” by A. Mignan

ABSTRACT Mignan (2016) claimed that a stretched exponential function describes Omori’s original aftershock data of the 1891 Great Nobi earthquake better than the well‐known Omori law, which is a power law. Besides his preference for the stretched exponential function based on general physical reasoning, he proposed that the Omori law does not hold when the proper visualization method is used; that is, the complementary cumulative density function (CCDF) in a log–log plot. However, his proposed plot is misleading, because it compares data of a finite observation interval with functions integrated over infinite periods. Using the same data as Mignan (2016), we find that an appropriate comparison leads to visually indistinguishable fits for this dataset. The Omori law is preferred based on maximum‐likelihood values. Moreover, the extrapolation shows that it also fits the long‐term data until the centennial anniversary of the Nobi event significantly better than the stretched exponential.

Revisiting the 1894 Omori Aftershock Dataset with the Stretched Exponential Function

In 1894, Fusakichi Omori published his landmark article on aftershocks in which he described the temporal decay rate of aftershocks as a power law. In this article, I show that the expression proposed by Omori does not hold when the proper visualization method is used (i.e., complementary cumulative density function in log–log plot) and that the choice of a stretched exponential function should have been preferred on physical grounds. I reproduce the emblematic aftershock sequence of the 1891 Great Nobi earthquake used by Omori and show that a stretched exponential expression describes the data better than a power law.

Analysis of Peak to Average Power Ratio Reduction Techniques in Sfbc Ofdm System

Orthogonal Frequency Division Multiplexing (OFDM) has become the most popular modulation echnique for high speed data transmission. But the great disadvantage of the OFDM technique is its high Peak to Average Power Ratio (PAPR). In this paper, the Selected Mapping (SLM) technique and Clipping and Differential Scaling is applied to Space Frequency Block Coded (SFBC) OFDM systems to reduce the PAPR with Alamouti coding scheme. In SLM technique, different representations of OFDM symbols are generated by rotation of the original OFDM frame by different phase sequences, and the signal with minimum PAPR is selected and transmitted. To compensate for the effect of the phase rotation at the receiver, it is necessary to transmit the index of the selected phase sequence as side information (SI). Additionally, a suboptimum detection method that does not need SI is introduced at the receiver side. In Clipping and Differential Scaling technique, the amplitude of complex OFDM signal is clipped and then scaled in such a way so that the PAPR is reduced without causing much degradation in bit error rate (BER). The threshold values for clipping and scaling is determined using Monte Carlo Simulations. Simulation results show that the SLM method and Clipping and Scaling method effectively reduce the PAPR. Keywords: Orthogonal frequency division multiplexing (OFDM), peak to average power ratio (PAPR), selected mapping (SLM), space frequency block coded (SFBC), Side information (SI), high power amplifiers (HPA), complementary cumulative density function (CCDF), inter-symbol interference (ISI).

Reduction of PAPR of OFDM Signals Using Nonlinear Companding Transform

In this paper, we propose and evaluate a novel nonlinear companding transform (NCT) scheme for peak-to-average power ratio (PAPR) reduction in orthogonal frequency division multiplexing (OFDM) systems. The key idea of the proposed scheme is to transform the original Gaussian-distributed OFDM signals into a specific statistics form, whose target probability density function is defined by a piecewise function with an inflexion point. By properly choosing the transform parameters, this scheme can enable more flexibility and freedom in the companding form so that a favorable tradeoff between PAPR reduction and bit error rate (BER) performance can be achieved. Moreover, compared to existing NCT techniques, this scheme dramatically decreases the impact of companding distortion on the BER performance to reach a given PAPR level. The analytical expressions regarding the achievable transform gain in PAPR, complementary cumulative density function, attenuation coefficient, and selection criteria of transform parameters are derived. Simulation results justify the significance and accuracy of the analytical expressions presented.

Blackouts risk evaluation by Monte Carlo Simulation regarding cascading outages and system frequency deviation

In this paper, a blackout risk assessment method is proposed considering the effects of cascading outages, the active power and frequency response of the system. Here, the Monte Carlo Simulation (MCS) is used to analyze the risk of blackouts by generating rare event scenarios. In the proposed model the cascading outages are considered as the result of transmission equipments overloading and hidden failure of the protection system, which they can increase the probability of large blackouts. The primary frequency control of the generators and the frequency sensitivity of the loads are also modeled in each step of cascading outages. A DC OPF model is used as the operator remedial action to minimize the lost load. The obtained lost load data are used to calculate Expected Load Not Served (ELNS) and Complementary Cumulative Density Function (CCDF) to analyze the risk of blackout. The CCDF of the lost load data is directly calculated from the lost load data and Gaussian Mixture Method (GMM) is applied to estimate the Probability Distribution Function (PDF) and the smoothed CCDF of the lost load data. Based on the proposed method, the power law distribution of the blackout size is studied by looking at the IEEE 118 bus test case.

Performance Evaluation of Generalized Selection Combiners Over Slow Fading with Estimation Errors

In this paper we derive closed-form expressions for the single-user adaptive capacity of generalized selection combining (GSC) system, taking into account the effect of imperfect channel estimation at the receiver. The channel considered is a slowly varying spatially independent flat Rayleigh fading channel. The complex channel estimate and the actual channel are modelled as jointly Gaussian random variables with a correlation that depends on the estimation quality. Three adaptive transmission schemes are analyzed: (1) optimal power and rate adaptation; and (2) constant power with optimal rate adaptation, and (3) channel inversion with fixed rate. In addition to deriving an exact expression for the capacity of the aforementioned adaptive schemes, we analyze the impact of channel estimation error on the capacity statistics and the symbol error rate for GSC systems. The capacity statistics derived in this paper are the moment generating function, complementary cumulative distribution function and probability density function for arbitrary number of receive antennas. Moreover, exact closed-form expressions for M-PAM/PSK/QAM employing GSC are derived. As expected, the channel estimation error has a significant impact on the system performance.

Spectral efficiency evaluation for selection combining diversity (SCD) scheme over slow fading

In this paper we derive closed-form expressions for the single-user capacity of selection combining diversity (SCD) system, taking into account the effect of imperfect channel estimation at the receiver. The channel considered is a slowly varying spatially independent flat Rayleigh fading channel. The complex channel estimate and the actual channel are modelled as jointly Gaussian random variables with a correlation that depends on the estimation quality. Three adaptive transmission schemes are analysed: 1) optimal power and rate adaptation opra; 2) constant power with optimal rate adaptation ora; and 3) channel inversion with fixed rate cifr. Furthermore, we derive in this paper analytical results for capacity statistics including moment generating function (MGF), complementary cumulative distribution function (CDF) and probability density function (PDF). These statistics are valid for arbitrary number of receive antennas. Our numerical results show the effect of Gaussian channel estimation error on the achievable spectral efficiency.