On the Multivariate Gamma-Gamma (ΓΓ\Gamma \Gamma) Distribution with Arbitrary Correlation and Applications in Wireless Communications

The statistical properties of the multivariate Gamma-Gamma ($\Gamma \Gamma$) distribution with arbitrary correlation have remained unknown. In this paper, we provide analytical expressions for the joint probability density function (PDF), cumulative distribution function (CDF) and moment generation function of the multivariate $\Gamma \Gamma$ distribution with arbitrary correlation. Furthermore, we present novel approximating expressions for the PDF and CDF of the sum of $\Gamma \Gamma$ random variables with arbitrary correlation. Based on this statistical analysis, we investigate the performance of radio frequency and optical wireless communication systems. It is noteworthy that the presented expressions include several previous results in the literature as special cases.Comment: 7 pages, 6 figures, accepted by IEEE Transactions on Vehicular Technolog

Performance Analysis of Selection Combining Over Correlated Nakagami-m Fading Channels with Constant Correlation Model for Desired Signal and Cochannel Interference

A very efficient technique that reduces fading and channel interference influence is selection diversity based on the signal to interference ratio (SIR). In this pa¬per, system performances of selection combiner (SC) over correlated Nakagami-m channels with constant correlation model are analyzed. Closed-form expressions are obtained for the output SIR probability density function (PDF) and cumulative distribution function (CDF) which is main contribution of this paper. Outage probability and the average error probability for coherent, noncoherent modulation are derived. Numerical results presented in this paper point out the effects of fading severity and cor¬relation on the system performances. The main contribu¬tion of this analysis for multibranch signal combiner is that it has been done for general case of correlated co-channel interference (CCI)

An efficient approximation to the correlated Nakagami-m sums and its application in equal gain diversity receivers

There are several cases in wireless communications theory where the statistics of the sum of independent or correlated Nakagami-m random variables (RVs) is necessary to be known. However, a closed-form solution to the distribution of this sum does not exist when the number of constituent RVs exceeds two, even for the special case of Rayleigh fading. In this paper, we present an efficient closed-form approximation for the distribution of the sum of arbitrary correlated Nakagami-m envelopes with identical and integer fading parameters. The distribution becomes exact for maximal correlation, while the tightness of the proposed approximation is validated statistically by using the Chi-square and the Kolmogorov-Smirnov goodness-of-fit tests. As an application, the approximation is used to study the performance of equal-gain combining (EGC) systems operating over arbitrary correlated Nakagami-m fading channels, by utilizing the available analytical results for the error-rate performance of an equivalent maximal-ratio combining (MRC) system

Distribution of Gaussian Process Arc Lengths

We present the first treatment of the arc length of the Gaussian Process (GP) with more than a single output dimension. GPs are commonly used for tasks such as trajectory modelling, where path length is a crucial quantity of interest. Previously, only paths in one dimension have been considered, with no theoretical consideration of higher dimensional problems. We fill the gap in the existing literature by deriving the moments of the arc length for a stationary GP with multiple output dimensions. A new method is used to derive the mean of a one-dimensional GP over a finite interval, by considering the distribution of the arc length integrand. This technique is used to derive an approximate distribution over the arc length of a vector valued GP in $\mathbb{R}^n$ by moment matching the distribution. Numerical simulations confirm our theoretical derivations.Comment: 10 pages, 4 figures, Accepted to The 20th International Conference on Artificial Intelligence and Statistics (AISTATS

Aproximações estatísticas para somas de variáveis aleatórias correlacionadas dos tipos Rayleigh e exponencial com aplicação a esquemas de combinação de diversidade

Orientador: José Cândido Silveira Santos FilhoDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Somas de variáveis aleatórias são amplamente aplicadas em sistemas de comunicação sem fio. Exemplos incluem equalização linear, detecção de sinais, fenômenos de interferência e esquemas de combinação de diversidade. No entanto, a formulação exata para as funções estatísticas dessas somas, como a função densidade de probabilidade e a função distribuição acumulada, requer em geral um tratamento matemático complicado, o que tem motivado a busca por soluções aproximadas mais simples. Apesar de haver várias propostas de aproximação disponíveis na literatura, muitas das quais obtidas usando-se a tradicional técnica de casamento de momentos, elas não oferecem um bom ajuste em regime de alta relação sinal-ruído. Sabe-se, porém, que essa é uma região primordial para a análise de desempenho de sistemas de comunicação em termos de métricas importantes como taxa de erro de bit e probabilidade de interrupção. Mais recentemente, com o intuito de contornar essa limitação, foi proposta uma nova técnica promissora conhecida como casamento de assíntotas, capaz de fornecer aproximações para estatísticas de somas de variáveis aleatórias positivas com um ótimo ajuste em regime de alta relação sinal-ruído. Ainda assim, essa técnica foi inicialmente implementada apenas para o caso de somas de variáveis independentes, não sendo até então aplicável para somas de variáveis correlacionadas. Neste trabalho, uma nova análise assintótica é proposta, a partir da qual é possível generalizar o uso do casamento de assíntotas para o caso correlacionado. A análise proposta é ilustrada para somas de variáveis Rayleigh e somas de variáveis exponenciais com correlação e parâmetros de desvanecimento arbitrários. Além disso, deduzem-se expressões assintóticas em forma fechada com o intuito de obter novas aproximações simples e precisas em regime de alta relação sinal-ruído. Como exemplos de aplicação, esquemas práticos de combinação de diversidade são abordados, quais sejam, combinação por ganho igual e combinação por razão máxima. Por fim, resultados numéricos mostram o excelente desempenho das aproximações propostas em comparação com as aproximações obtidas via casamento de momentosAbstract: Sums of random variables are widely applied to wireless communications systems. Examples include linear equalization, signal detection, interference phenomena, and diversity-combining schemes. However, the exact formulation for the statistical functions of these sums, such as the probability density function and the cumulative distribution function, requires in general a complicated mathematical treatment, which has motivated the search for simple approximate solutions. Although there are several approximate proposals available in the literature, many of which obtained through the traditional moment-matching technique, they do not offer a good fit under the regime of high signal-to-noise ratio. It is well-known that this regime is a paramount region for the performance analysis of communications systems in terms of important metrics such as bit error rate and outage probability. More recently, in order to circumvent this limitation, a new promising technique known as asymptotic matching was proposed, capable of providing approximations for statistics of the sum of random variables with an excellent fit under the regime of high signal-to-noise ratio. Even so, this technique was initially proposed for the sum of mutually independent variables only, and thus it has not been applicable to sums of correlated variables. In this work, a novel asymptotic analysis is proposed, from which it is possible to generalize the application of asymptotic matching to the correlated case. The proposed analysis is illustrated for sums of Rayleigh and sums of exponential variables with arbitrary correlation and arbitrary fading parameters. Furthermore, closed-form asymptotic expressions are derived in order to obtain new simple and precise approximations under the regime of high signal-to-noise ratio. As application examples, practical diversity-combining schemes are addressed, namely, equal-gain combining and maximal-ratio combining. Finally, numerical results show the excellent performance of the proposed approximations in comparison to the approximations obtained via moment matchingMestradoTelecomunicações e TelemáticaMestre em Engenharia ElétricaCAPE

The κ - μ shadowed fading model with arbitrary intercluster correlation

In this paper, we propose a generalization of the well-known κ-μ shadowed fading model. Based on the clustering of multipath waves as the baseline model, the novelty of this new distribution is the addition of an arbitrary correlation for the scattered components within each cluster. It also inherits the random fluctuation of the dominant component, which is assumed to be the same for all clusters. Thus, it unifies a wide variety of models: Rayleigh, Rician, Rician shadowed, Nakagami- m, κ-μ and κ-μ shadowed as well as multivariate Rayleigh, Rician and Rician shadowed. The main statistics of the newly proposed model, i.e. moment generating function, probability density function and cumulative density function, are given in terms of exponentials and powers, and some numerical results are provided in order to analyze the impact of the arbitrary intercluster correlation.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tec