Generalized Gram-Charlier series with application to the sum of log-normal variates (Corresp.)

A generalized Gram-Charlier series, applicable to non-Gaussian problems, is developed. Expressions are given for the first six error coefficients. The high inherent accuracy of the series is demonstrated by development of the expansion for the sum of independent, identically distributed log-normal variates.     

A closed-form upper-bound for the distribution of the weighted sum of Rayleigh variates

The problem of finding the distribution of the sum of more than two Rayleigh fading envelopes has never been solved in terms of tabulated functions. In this letter, we present a closed-form union upper-bound for the cumulative distribution function of the weighted sum of N independent Rayleigh fading envelopes. Computer simulation results verify the tightness of the proposed bound for several values of N. The proposed bound can be efficiently applied in various wireless applications, such as satellite communications, equal-gain receivers, and radars.     

Power margin quality measures for correlated random variates derived from the normal distribution

The lack of a well-defined and meaningful measure of the quality of computer-generated correlated random variates complicates the design and comparison of generation algorithms for communications system simulation applications. This paper presents quantitative quality measures for computer-generated random variates having a multivariate Gaussian distribution or having a distribution that derives from the multivariate Gaussian distribution. These measures are, in particular, useful for digital communication system simulation applications. A theoretical basis for the definition of the measures is given and it is shown that the measures also possess useful intuitive interpretations. Examples of the use of these quality measures are provided.     

On the distribution of the product and ratio of products of EGK variates with applications

Video streaming over peer-to-peer (P2P) networks is a promising approach for scalable IPTV and online games. Distributing high-quality videos among users who employ numerous channels in multi-channel P2P live streaming systems, however, still suffers from many challenges arisen from the network size and dynamicity, channel resource imbalance, and bandwidth limitations. Other challenges also exist, including instability of peers, the peers’ low participation, large startup and playback delays, low quality of received video, and resource insufficiency in unpopular channels. In this paper, ReDePoly, with some levels of distributed intelligence, is proposed to reduce the bootstrapping delay and to increase the service quality. The key idea is to substitute agents with bootstrapping peers in the channels to model user behaviors and to share the aggregated knowledge among the agents to disseminate the learned models. Accordingly, the agents dynamically predict the behavior of participating peers to pre-assign them to the predicted channels and to parallelize the threads related to viewing behavior and channel switching. In case of congested channels, the new agents are distilled to mitigate the agent responsibilities. Simulation results show that the proposed approach outperforms the other existing methods regarding the channel switching delay, recovery delay, and quality of service.     

Optimal simple step-stress plan for cumulative exposure model using log-normal distribution

Optimal times of changing stress level for simple step-stress plans under a cumulative exposure model using the log-normal distribution are determined for a wide range of values of the parameters in the model. A table of optimal times of changing stress level for various model parameters values is obtained. A formula for optimal time of changing stress level is also estimated from the table. This paper provides an optimal life testing plan which will enable us to accurately estimate the 50th percentile of the life time of a product being tested without having to wait long time for the product to fail.     

Sum and difference of two squared correlated Nakagami variates in connection with the McKay distribution

General formulas for the probability density function of the sum and the difference of two correlated, not necessarily identically distributed, squared Nakagami variates (or equivalently, gamma variates) are derived. These expressions are shown to be in the form of the McKay "Bessel function" distributions. In addition, formulas for the moments of these distributions, in terms of the Gauss hypergeometric function, are provided. An application of these new results relevant to the calculation of outage probability in the presence of self-interference is discussed.     

On the generation of Tikhonov variates

A novel, simple and efficient method for the generation of Tikhonov (a.k.a. von Mises) random variates is proposed. In the proposed method, circular variates of a prescribed Tikhonov distribution p_T(x;alpha,xi) are generated via the transformation of variates selected randomly, on a one-for-one basis, from a bank of K distinct Cauchy and Gaussian generators. The mutually exclusive probabilities of sampling from each of the Cauchy or Gaussian generators, as well as the variance and half-width parameters that specify the latter, are derived directly from the Cauchy, Gaussian and Tikhonov characteristic functions, all of which are either known or given in closed form. The proposed random mixture technique is extremely efficient in that a single pair of uniform random numbers is consumed in the generation of each Tikhonov (or von Mises) sample, regardless of the prescribed concentration and centrality parameters (alpha, xi), all requiring neither the rejection of samples, nor the repetitive evaluation of computationally demanding functions. Additional attractive features of the method are as follows. By construction, the first (dominant) N circular moments of Tikhonov variates generated with the proposed random mixture technique are the ones that best approximate their corresponding theoretical values, with errors measured exactly. The exact distribution of generated Tikhonov variates is determined analytically, and its (Kullback-Leibler) divergence to the exact Tikhonov PDF is shown also analytically to be negligible. Finally, the technique establishes a connection between Tikhonov and Gaussian variates which can be exploited, e.g., in the generation of piecewise-continuous pseudo-random functions with Tikhonov-distributed outcomes.     

Formulas for the distribution of sums of independent exponential random variables

This paper derives a new type of formula for the probability that, among a collection of items with s-independent exponential times to failure, a certain subset of them fails in a given order before a certain time, and all the remaining items survive beyond that time. This formula is in the form of a power series that satisfies a certain constant coefficient linear differential equation with specified initial conditions. This provides an alternative to existing closed-form formulas of the "exponomial" variety, viz., a nonlinear combination of exponential terms, where the coefficients of the exponential terms are polynomials in the mission time. Some results are given which quantify the computation effort required to achieve a specified accuracy using partial sums of the infinite series; a simple example illustrates these results. This approach can be very efficient for system reliability analysis where the product of the mission time and the sum of the failure rates down any path leading to system failure is small. Further work is needed to expand the practical applicability of this approach to cases where some rates are large and/or the mission time is long.     

On the capacity of log-normal fading channels

In this letter we provide an analytical expression for the moments of the capacity for the log-normal fading channel. Since the developed expression involves infinite series, we show that the error that results from the truncation of these series is insignificant. We also analyze in more details the ergodic capacity by giving a simpler expression for the remainder of the truncated series. Relying on the fact that the sum of log-normal Random Variables (RV) is well approximated by another lognormal RV, we further utilize the obtained results to approximate the capacity of diversity combining techniques in correlated lognormal fading channels. The results that we provide in this letter are an important tool for measuring the performance of communication links in a log-normal environment.     

Generation of correlated log-normal clutter samples

A method is described for generating random log-normal vectors with the desired correlation matrix and specified parameters. Such vectors may represent samples of correlated clutter signal. The presented method makes use of the suitable nonlinear transformation of a random normal vector with correlated components.     

Computing the distribution of sums of random sine waves and ofRayleigh-distributed random variables by saddle-point integration

The method of saddle-point integration is applied to calculating the cumulative distribution of sums of independent random variables that either have a Rayleigh distribution or are sine waves with independently random phases in the presence of Gaussian random noise     

Detection of fixed targets embedded in log-normal clutter

Measurements of radar clutter using high-resolution systems show that clutter echoes cannot be modelled as Gaussian-distributed. A log-normal distribution is assumed and single-pulse detection performances are evaluated according to the Marcum-Swerling approach as well as the signal losses with respect to the Gaussian case.     

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.     

Closed-form statistics for the sum of squared Nakagami-m variates and its applications

We present closed-form expressions for the probability density function (PDF) and the cumulative distribution function (CDF) of the sum of non-identical squared Nakagami-m random variables (RVs) with integer-order fading parameters. As it is shown, they can be written as a weighted sum of Erlang PDFs and CDFs, respectively, while the analysis includes both independent and correlated sums of RVs. The proposed formulation significantly improves previously published results, which are either in the form of infinite sums or higher order derivatives of the fading parameter m. The obtained formulas can be applied to the performance analysis of diversity combining receivers operating over Nakagami-m fading channels.     

On the difference of two chi-square variates with application tooutage probability computation

An expression for the cumulative distribution function evaluated at zero of the difference of two chi-square variates with different number of degrees of freedom is derived and applied to the outage probability computation of cellular mobile radio systems in fading channels. In particular, a generic result is developed for this probability which takes on several forms, the simplest of which, is a single integral with finite limits and an integrand composed of elementary (exponential and trigonometric) functions. The results are applicable to cellular systems that are subject to independent identically distributed (i.i.d.) interfering signals and that employ maximal-ratio combining reception over i.i.d. diversity paths. Various fading channel models are assumed for the desired signal and interferers. For each desired signal/interferer combination, the outage probability is expressed in closed form in terms of a set of parameters that characterize the particular scenario. A tabulation of these various scenarios and their associated parameters for channels of practical interest is included     

Performance analysis of DF relaying multi-hop systems over log-normal fading channels

In this paper, we studied the end-to-end average bit error rate (BER) performance of decode-and-forward (DF) relaying multi-hop chain systems (MCSs) over log-normal fading channels. After specifying the distribution of the signal-to-noise rate for each hop, the analytical expression of the end-to-end average BER has been derived. The derived end-to-end average BER model can be used in the case that all the single-hops in the chain have different statistical behaviors, i.e., all links suffer various types of fadings, not limited to one type of fading. The comparisons between Monte Carlo simulation results and analytical results under various conditions verify the accuracy of the proposed model.     

Sum of gamma variates and performance of wireless communicationsystems over Nakagami-fading channels

Capitalizing on the Moschopoulos (1985) single gamma series representation of the probability density function (PDF) of the sum of gamma variates, we provide a PDF-based approach for the performance analysis of maximal-ratio combining and postdetection equal-gain combining diversity techniques as well as cochannel interference of cellular mobile radio systems over Nakagami (1960)-fading channels with arbitrary parameters. Aside from putting under the same umbrella many of the past results obtained via characteristic function (CF) or moment generating function (MGF)-based approaches, the proposed approach also allows the derivation of additional performance measures, which are harder to analyze via CF or MGF-based approaches     

The generation of correlated Rayleigh random variates by inversediscrete Fourier transform

A number of different algorithms are used for the generation of correlated Rayleigh random variates. This paper presents an analysis of the statistical properties of methods based on the inverse discrete Fourier transform (IDFT). A modification of the algorithm of Smith (1975) is presented, the new method requiring exactly one-half the number of IDPT operations and roughly two-thirds the computer memory of the original method. Evaluations of and comparisons between various variate generation methods using meaningful quantitative measures are believed to be lacking. New quantitative quality measures for random variate generation have been proposed that are, in particular, meaningful and useful for digital communication system simulation. This paper presents the application of these measures to the IDFT method and three other methods of correlated variate generation, comparing the algorithms in terms of the quality of the generated samples and the required computational effort