Gaussian class multivariate Weibull distributions: theory and applications in fading channels

Ascertaining on the suitability of the Weibull distribution to model fading channels, a theoretical framework for a class of multivariate Weibull distributions, originated from Gaussian random processes, is introduced and analyzed. Novel analytical expressions for the joint probability density function (pdf), moment-generating function (mgf), and cumulative distribution function (cdf) are derived for the bivariate distribution of this class with not necessarily identical fading parameters and average powers. Two specific distributions with arbitrary number of correlated variates are considered and studied: with exponential and with constant correlation where their pdfs are introduced. Both cases assume equal average fading powers, but not necessarily identical fading parameters. For the multivariate Weibull distribution with exponential correlation, useful corresponding formulas, as for the bivariate case, are derived. The presented theoretical results are applied to analyze the performance of several diversity receivers employed with selection, equal-gain, and maximal-ratio combining (MRC) techniques operating over correlated Weibull fading channels. For these diversity receivers, several useful performance criteria such as the moments of the output signal-to-noise ratio (SNR) (including average output SNR and amount of fading) and outage probability are analytically derived. Moreover, the average symbol error probability for several coherent and noncoherent modulation schemes is studied using the mgf approach. The proposed mathematical analysis is complemented by various evaluation results, showing the effects of the fading severity as well as the fading correlation on the diversity receivers performance.     

Harmonic mean and end-to-end performance of transmission systems with relays

Closed-form expressions for the statistics of the harmonic mean of two independent and identically distributed gamma variates are presented. The probability density function of the harmonic mean of two F variates is also derived. These statistical results are then applied to study the performance of wireless communication systems with nonregenerative relays over flat Nakagami fading channels. More specifically, outage probability formulas for noise-limited systems as well as systems affected by interference are obtained. Furthermore, general expressions for average bit-error rates are also derived. Finally, comparisons between regenerative and nonregenerative systems are presented. Numerical results show that the former systems clearly outperform the latter ones for low average signal-to-noise ratios (SNRs). They also show that the two systems have similar performance at high average SNRs.     

An investigation of Gaussian tail and Rayleigh tail densityfunctions for importance sampling digital communication systemsimulation

The gains achieved by the use of importance sampling in communication system simulation are strongly influenced by the choice of the biased input noise distribution. The Rayleigh tail and the Gaussian tail distributions are investigated for use as biased distributions in importance sampling. The robustness of these schemes with respect to the estimate of the unknown error probability is examined. It is shown that the Gaussian tail distribution previously considered to be theoretically optimum is not optimum in this regard. Methods of generating the biased random variates are presented     

Generalization of Bonnet’s theorem

Bonnet’s theorem relates the expectation of a nonlinear function of normally distributed variates to their expectations. It is shown that the theorem applies to a general class of distributions which includes the normal distribution as a special case.     

The Distribution of Sums of Path Gains in the IEEE 802.15.3a UWB Channel Model

When multipath arrival times and gains are specified by the IEEE 802.15.3a ultra-wideband (UWB) channel model, formulas are derived for the characteristic function of the sum of gains arriving in a given time window. These formulas are first used to determine structural properties of the corresponding distributions and densities. The formulas are then used to compute the distribution and density functions by numerically inverting the characteristic function. For small time windows, the densities of the path-gain sums are bimodal and therefore highly non-Gaussian     

Lifetime of Combined $k$-out-of-$n$ , and Consecutive $k_{c}$ -out-of-$n$ Systems

A combined k-out-of-n:F(G) & consecutive k_c -out-of-n :F(G) system fails (functions) iff at least k components fail (function), or at least fcc consecutive components fail (function). Explicit formulas are given for the lifetime distribution of these combined systems whenever the lifetimes of components are exchangeable, and have an absolutely continuous joint distribution. The lifetime distributions of the aforementioned systems are represented as a linear combination of distributions of order statistics by using the concept of Samaniego's signature. Formulas for the mean lifetimes are given. Some numerical results are also presented.     

Unified Laguerre Polynomial-Series-Based Distribution of Small-Scale Fading Envelopes

We derive new unified probability density function (pdf) formulas based on the generalized Laguerre polynomial series expansion that cover a wide range of small-scale fading distributions in wireless communications. The unified envelope pdf covers many known Laguerre polynomial-series-based pdf's and small-scale fading distributions, which include the multiple-waves-plus-diffuse-power (MWDP) fading, kappa-mu, Nakagami-m, Rician (Nakagami-n), Nakagami-q (Hoyt), Rayleigh, Weibull, and alpha-mu (Stacy) distributions as special cases. We highlight that the unified pdf formulas are general and applicable to characterize the distribution of any random variable defined on the positive range. We then propose two truncated forms of the pdf and prove that the original pdf and both truncated forms satisfy the properties of a valid pdf. A new cumulative distribution function and a moment-generating function of this pdf are derived. We also analyze the truncation error for the case when only a finite number of terms are kept in the series expansion.     

The α-μ Distribution: A Physical Fading Model for the Stacy Distribution

This paper introduces a fading model, which explores the nonlinearity of the propagation medium. It derives the corresponding fading distribution-the alpha-mu distribution-which is in fact a rewritten form of the Stacy (generalized Gamma) distribution. This distribution includes several others such as Gamma (and its discrete versions Erlang and central Chi-squared), Nakagami-m (and its discrete version Chi), exponential, Weibull, one-sided Gaussian, and Rayleigh. Based on the fading model proposed here, higher order statistics are obtained in closed-form formulas. More specifically, level-crossing rate, average fade duration, and joint statistics (joint probability density function, general joint moments, and general correlation coefficient) of correlated alpha-mu variates are obtained, and they are directly related to the physical fading parameters     

The multivariate α-μ distribution

An infinite series formulation for the multivariate α-μ joint probability density function with arbitrary correlation matrix and non-identically distributed variates is derived. The expression is exact and general and includes all of the results previously published in the literature concerning the distributions comprised by the α-μ distribution. The general expression is then particularized to an indeed very simple, approximate closed-form solution. In addition, a multivariate joint cumulative distribution function is obtained, again in simple, closed-form manner. As an application example, the exact and approximate performances of the selection combining scheme given in terms of the outage probability is shown. Approximate and exact results are very close to each other for small as well as medium values of correlation.     

End-to-end performance of transmission systems with relays over Rayleigh-fading channels

End-to-end performance of two-hop wireless communication systems with nonregenerative relays over flat Rayleigh-fading channels is presented. This is accomplished by deriving and applying some new closed-form expressions for the statistics of the harmonic mean of two independent exponential variates. It is shown that the presented results can either be exact or tight lower bounds on the performance of these systems depending on the choice of the relay gain. More specifically, average bit-error rate expressions for binary differential phase-shift keying, as well as outage probability formulas for noise limited systems are derived. Finally, comparisons between regenerative and nonregenerative systems are presented. Numerical results show that the former systems clearly outperform the latter ones for low average signal-to-noise-ratio (SNR). They also show that the two systems have similar performance at high average SNR.     

An Inference Methodology for Life Tests With Complete Samples or Type-II Right Censoring

We present a methodology for gaining statistical inference during the actual conduct of a life test experiment that can reduce time on test and cost. The method relies on properties of conditional order statistic distributions to convert censored data into i.i.d. uniform variates. A secondary result presents a new test for uniformity based on the convolution of these i.i.d. uniform random variables that is higher in power than the benchmark Anderson-Darling statistic in certain cases.      

电磁波对导体绕射问题的渐近解

本文应用萨温(Г.Н.Савин)扰动形式的映射函数处理电磁波绕射导体的二维问题。提出了渐近求解这一问题的一般方法及相应的渐近公式,特别是基于圆柱形导体的渐近,文中给出了0级及1级渐近解的具体算式。     

Tail of the distribution of sums of log-normal variates

It is shown that the asymptotic behavior of the tail of the sum distribution of a finite number of individually log-normal distributed variates displays the log-normal character of those variates with maximum logarithmic variance. A quantitative definition of the sum-distribution "tail" is established in terms of upper bounds on the relative error or deviation from the asymptotic log-normal property.     

Nakagami-m approximation to the sum of M non-identical independent Nakagami-m variates

A Nakagami-m approximate distribution to the sum of M non-identical independent Nakagami-m variates is derived. Comparisons between exact and approximate sum distributions show excellent agreement. This new approach finds applicability in important communications issues such as equal-gain combining, signal detection, linear equalisers, outage probability, intersymbol interference and phase jitter.     

The Criterions with LCR and AFD of Dual-Branch SC Diversity over Specified Wireless Radio Fading Channels

In this paper we proposed the results of average LCR (level crossing rate) and AFD (average fading duration) criterions applied to evaluate the performance of dual-branch SC (selection combining) reception in the specified fading channels characterized as statistical distributions with correlated-Rayleigh and correlated-Rice models. Moreover, in order to unify and clarify the criterions of performance formulas with average LCR and AFD for SC diversity over different kinds of fading models, include such as Rayleigh, Rice, Nakagami-m, and Weibull distributions etc., almost all of the ever researched and published results from discussing about the LCR and AFD of SC diversity are comprehensively collected in this report. On the other hand, for the purpose of comparison, there are a large number of LCR and AFD performance formulas for SC diversity and the generalized fading statistic models are extracted and tabulated together, in which the cases of correlated and independent proprieties between diversity branches are taken into consideration too (some of the formulas are illustrated by the assumption of dual-branch SC diversity).

Some probabilistic properties of the line-of-sight angle error to a remote object (Corresp.)

Closed-form expressions are derived for the distribution and the expectation of the error of the line-of-sight angle to a remote object when the rectangular position coordinates are independent identically distributed normal variates. Generalized results also are provided both for the cumulative distribution function for a hyperspace ofNdimensions and for a spherically symmetric probability density function in three-dimensional space.     

Studies in Cascade Reliability?I

An n-Cascade system is defined as a special type of standby system with n components. A component fails if the stress on it is not less than its strength. When a component in cascade fails, the next in standby is activated and will take on the stress. However, the stress on this component will be a multiple k times the stress that acted on its predecessor. The system fails if due to an initial stress, each of the components in succession fails. The stress is random and the component strengths are independent and identically distributed variates, with specified probability functions; k is constant. Expressions for system reliability are obtained when the stress and strength distributions are exponential. Reliability values for a 2-cascade system with Gamma and Normal stress and strength distributions are computed, some of which are presented graphically.     

Oversampling in Shift-Invariant Spaces With a Rational Sampling Period

It is well known that, under appropriate hypotheses, a sampling formula allows us to recover any function in a principal shift-invariant space from its samples taken with sampling period one. Whenever the generator of the shift-invariant space satisfies the Strang–Fix conditions of order $r$, this formula also provides an approximation scheme of order $r$ valid for smooth functions. In this paper we obtain sampling formulas sharing the same features by using a rational sampling period less than one. With the use of this oversampling technique, there is not one but an infinite number of sampling formulas. Whenever the generator has compact support, among these formulas it is possible to find one whose associated reconstruction functions have also compact support.      

On multivariate Rayleigh and exponential distributions

In this paper, expressions for multivariate Rayleigh and exponential probability density functions (PDFs) generated from correlated Gaussian random variables are presented. We first obtain a general integral form of the PDFs, and then study the case when the complex Gaussian generating vector is circular. We consider two specific circular cases: the exchangeable case when the variates are evenly correlated, and the exponentially correlated case. Expressions for the multivariate PDF in these cases are obtained in integral form as well as in the form of a series of products of univariate PDFs. We also derive a general expression for the multivariate exponential characteristic function (CF) in terms of determinants. In the exchangeable and exponentially correlated cases, CF expressions are obtained in the form of a series of products of univariate gamma CFs. The CF of the sum of exponential variates in these cases is obtained in closed form. Finally, the bivariate case is presented mentioning its main features. While the integral forms of the multivariate PDFs provide a general analytical framework, the series and determinant expressions for the exponential CFs and the series expressions for the PDFs can serve as a useful tool in the performance analysis of digital modulation over correlated Rayleigh-fading channels using diversity combining.     

Perturbation formulas for microstrip patch arrays and elements

One-term perturbation formulas are derived to characterize periodic microstrip arrays with nonuniform planar distributions of dielectric and magnetic materials. Analogous results are also obtained for single-patch elements. The formulas are generally applicable to other types of antennas