Outage Probability with Correlated Lognormal Interferers using Log Shifted Gamma Approximation

The computation of outage probability in cellular radio system has been extensively studied. The Signal-to-Interference-plus-Noise Ratio (SINR) distribution involves the sum of lognormal distributions due to dominant effect of shadowing in both the signal and interference components. Since no closed-form expression can be found for the sum of lognormal distributions, many approximation methods and bounds were proposed in the past. In this paper, Log Shifted Gamma (LSG) approximation is proposed to calculate the sum of correlated lognormal random variables (RVs), hence the outage probability, accurately with a wide range of dB spreads, number of interferers M, correlation coefficients r among interference components, and noise power N. Overall, LSG approximation shows consistent accuracy due to its flexibility over the classical lognormal approximation, especially with small correlation coefficients r and/or large dB spreads.     

Approximation for a Sum of on-off Lognormal Processes With Wireless Applications

In this paper, a lognormal approximation is proposed for the sum of lognormal processes weighted by binary processes. The analytical approach moves from the method early proposed by Wilkinson for approximating first-order statistics of a sum of lognormal components, and extends to incorporate second-order statistics and the presence of both time-correlated random binary weights and cross-correlated lognormal components in moments' matching. Since the sum of weighted lognormal processes models the signal-to-interference-plus-noise ratio (SINR) of wireless systems, the method can be applied to evaluate in an effective and accurate way the outage occurrence rate and outage duration for different wireless systems of practical interest. In a frequency-reuse-based cellular system, the method is applied for various propagation scenarios, characterized by different shadowing correlation decay distances and correlations among shadowing components. A further case of relevant interest is related to power-controlled wideband wireless systems, where the random weights are binary random variables denoting the activity status of each interfering source. Finally, simulation results are used to confirm the validity of the analysis.     

Bounds on the Distribution of a Sum of Correlated Lognormal Random Variables and Their Application

The cumulative distribution function (cdf) of a sum of correlated or even independent lognormal random variables (RVs), which is of wide interest in wireless communications, remains unsolved despite long standing efforts. Several cdf approximations are thus widely used. This letter derives bounds for the cdf of a sum of 2 or 3 arbitrarily correlated lognormal RVs and of a sum of any number of equally-correlated lognormal RVs. The bounds are single-fold integrals of readily computable functions and extend previously known bounds for independent lognormal summands. An improved set of bounds are also derived which are expressed as 2-fold integrals. For correlated lognormal fading channels, new expressions are derived for the moments of the output SNR and amount of fading for maximal ratio combining (MRC), selection combining (SC) and equal gain combining (EGC) and outage probability expressions for SC.     

An optimal lognormal approximation to lognormal sum distributions

Sums of lognormal random variables occur in many problems in wireless communications because signal shadowing is well modeled by the lognormal distribution. The lognormal sum distribution is not known in the closed form and is difficult to compute numerically. Several approximations to the distribution have been proposed and employed in applications. Some widely used approximations are based on the assumption that a lognormal sum is well approximated by a lognormal random variable. Here, a new paradigm for approximating lognormal sum distributions is presented. A linearizing transform is used with a linear minimax approximation to determine an optimal lognormal approximation to a lognormal sum distribution. The accuracies of the new method are quantitatively compared to the accuracies of some well-known approximations. In some practical cases, the optimal lognormal approximation is several orders of magnitude more accurate than previous approximations. Efficient numerical computation of the lognormal characteristic function is also considered.     

Highly accurate simple closed-form approximations to lognormal sum distributions and densities

Sums of lognormal random variables occur extensively in wireless communications, in part, because a shadowing environment is well modeled by a lognormal distribution. A closed-form expression does not exist for the sum distribution and, furthermore, it is difficult to numerically calculate the distribution. Numerous approximations exist that are based on approximating a sum of lognormal random variables as another lognormal random variable. A new paradigm to calculate an approximation to the lognormal sum distribution, based on curve fitting on lognormal probability paper, is introduced in this letter. Highly accurate, simple closed-form approximations to lognormal sum distributions are presented.     

Outage probabilities in the presence of correlated lognormalinterferers

Several approaches that can be used to compute the distribution of a sum of correlated lognormal random variables (RVs) are investigated. Specifically, Wilkinson's approach (Schwartz and Yeh, 1982), an extension to Schwartz and Yeh's (1982) approach, and a cumulants matching approach (Schleher, 1977) are studied. The aim is to determine which method is best for computing the complementary distribution function (CDF) of a sum of correlated lognormal RVs considering both accuracy and computational effort. Then, using these techniques, the authors compute the outage probability of a desired lognormal shadowed signal in the presence of multiple correlated lognormal cochannel interferers. The outage results are presented as a function of the reuse factor. The reuse factor is defined as the distance between the centers of the two nearest cells using the same frequencies divided by the cell radius. It is a key parameter in the design of any frequency reuse system. Simulation results are used for verification and comparison. Overall, the results obtained show that among the three methods considered Wilkinson's approach may be the best method to compute the CDF of sums of correlated lognormal RVs (and hence the outage probability in correlated lognormal shadowed mobile radio environments). This is due to both its accuracy and computational simplicity over the range of parameters valid for practical applications     

Outage analysis in wireless channels with multiple interferers subject to shadowing and fading using a compound pdf model

Wireless communication systems are subject to short- and long-term fading of the channel. Instead of the commonly used Nakagami–lognormal model to account for the conditions existing in these shadowed fading channels, a compound probability density function (pdf) model is used to evaluate the performance of wireless systems. While the Nakagami–lognormal lacks a closed-form solution to the pdf of the received power in shadowed fading channels, the compound pdf has an analytical expression for the pdf of the received signal power. The synergy between these two models for the analysis of wireless systems is explored by calculating the bit error rate in a DPSK modem as well as the outage probability in a wireless system in a shadowed fading channel. This is followed by the computation of the outage probability in the general case where both the desired and cochannels are subject to shadowing and fading. The analyses were carried out for both fixed number of cochannels and random number of cochannels. Results demonstrate the usefulness of the compound pdf model for the performance analyses of wireless systems in shadowed fading channels.     

复合衰落信道下分布式MIMO系统下行中断概率分析

针对复合衰落信道模型下分布式MIMO系统下行中断概率问题,首先建立了一个综合考虑Nakagami-m衰落、路径损耗和阴影衰落的复合衰落信道模型;然后针对存在中心基站的分布式MIMO系统,提出了一种更符合实际应用环境的准均匀小区移动台分布模型;接着利用矩生成函数、Gauss-Hermite积分公式和Simpson积分公式等数学手段推导了任意移动台分布下、单小区内系统平均下行中断概率闭合表达式;最后将准均匀移动台分布模型应用到该中断概率闭合表达式,并通过MATLAB仿真证明了所推导得到的闭合表达式及小区分布模型的合理性。     

Capacity of wireless optical communications

We consider the ergodic capacity and capacity-versus-outage probability of direct-detection optical communication through a turbulent atmosphere using multiple transmit and receive apertures. We assume shot-noise-limited operation in which detector outputs are doubly stochastic Poisson processes whose rates are proportional to the sum of the transmitted powers, scaled by lognormal random fades, plus a background noise. In the high and low signal-to-background ratio regimes, we show that the ergodic capacity of this fading channel equals or exceeds that for a channel with deterministic path gains. Furthermore, knowledge of these path gains is not necessary to achieve capacity when the signal-to-background ratio is high. In the low signal-to-background ratio regime, path-gain knowledge provides minimal capacity improvement when using a moderate number of transmit apertures. We also develop expressions for the capacity-versus-outage probability in the high and low signal-to-background ratio regimes, by means of a moment-matching approximation to the distribution for the sum of lognormal random variables. Monte Carlo simulations show that these capacity-versus-outage approximations are quite accurate for moderate numbers of apertures.     

M-ary symbol error outage over Nakagami-m fading channels in shadowing environments

This letter addresses the problem of finding a tractable expression for the symbol error outage (SEO) in flat Nakagami-m fading and shadowing channels. We deal with M-ary phase shift keying (M-PSK) and quadrature amplitude modulation (M-QAM) which extends our previous results on BPSK signaling. We propose a new tight approximation of the symbol error probability (SEP) holding for M-PSK and M-QAM signals which is accurate over all signal to noise ratios (SNRs) of interest. We derive a new generic expression for the inverse SEP which facilitates the derivation of a tight approximation of the SEO in a lognormal shadowing environment.     

分布式天线系统混合SC/MRC分集中断概率性能

在频率选择性复合Nakagami/lognormal衰落信道条件下,研究了层叠分布式天线系统(Cascade Distributed Antenna System--CDAS)采用混合SC/MRC分集的中断概率性能.针对中断概率性能,分析了多径密度函数(Multipath Intensity Profile MIP)衰减因子、空时分集维数、活动用户数、每比特平均接收信噪比以及衰落因子对系统性能的影响.仿真表明CDAS系统可以有效对抗衰落,提高系统容量,削弱对数正态阴影衰落影响.     

分布式天线系统中的系统上行中断概率分析

中断概率是无线通信系统的一个重要指标。该文针对分布式天线系统中的系统中断概率问题进行了研究。首先,建立了包含路径损耗、阴影衰落和瑞利衰落的复合信道模型;然后,在该信道模型下推导出给定移动台位置时的信噪比;接着,基于选择合并策略,分析出给定移动台位置时系统上行传输的中断概率表达式;最后,考虑到小区内移动台任意分布特点,文章给出系统上行中断概率闭合理论表达式。仿真结果表明,所提的系统上行中断概率理论表达式与仿真结果十分吻合。     

分布式天线系统中的系统下行中断概率分析

本文针对分布式天线系统的系统下行中断概率问题展开研究.文章首先建立了包含路径损耗、阴影衰落和瑞利衰落的复合信道模型.接着,在接收端采用最大比合并的方式接收信号,并运用矩生成函数推导出输出信噪比的概率密度函数.然后,对分布式天线采用选择传输策略,并分析出给定移动台位置时的中断概率的表达式.最后,考虑小区内移动台任意分布特...     

Simple and accurate methods for outage analysis in cellular mobileradio systems-a unified approach

Two unified expressions for computing the refined outage criterion (which considers the receiver noise) in cellular mobile radio systems are derived using the Laplace and Fourier inversion formulas. Since these expressions do not impose any restrictions on the signal statistics while being easy to program, they provide a powerful tool for outage analysis over generalized fading channels. We also assess compatibility and applicability of previously published approaches that treat noise as cochannel interference (noise-limited model) or consider a minimum detectable receiver signal threshold and receiver noise. The outage probability in an interference-limited case can be evaluated directly by setting the minimum power threshold to zero. The analysis of correlated interferers is presented. Results are also developed for a random number of interferers. Several new closed-form expressions for the outage probability are also derived. Some previous studies have suggested approximating Rician desired signal statistics by a Nakagami-m (1960) model (with positive integer fading severity index) to circumvent the difficulty in evaluating the outage in Rician fading. The suitability of this approximation is examined by comparing the outage performance under these two fading conditions. Surprisingly, some basic results for Nakagami-m channel have been overlooked, which has led to misleadingly optimistic results with the Nakagami-m approximation model. However, similar approximation for the interferer signals is valid     

Approximate Outage Analysis of Land Mobile Satellite Systems in Lognormally Shadowed Rician Channels

Among the proposed models for land mobile satellite (LMS) channels, the shadowed Rice model proposed originally by Loo, has found wide applications in different frequency bands. In Loo’s model, it is assumed that the received signal is affected by nonselective Rice fading with lognormal shadowing on the direct component only, while the diffuse scattered component has constant average power level. The resulting composite probability density function (PDF) includes an infinite-range integral and is not available in closed-form, thereby making the performance evaluation of LMS communication links in these channels cumbersome. To bypass this problem, in this paper an approximation method is developed which makes it possible to describe the envelope PDF as a sum of weighted Rice’s PDFs. Therefore, in contrast with Loo’s PDF, the proposed method leads to an easy-to-use, closed-form approximate expression for the PDF and also for the most statistical characteristics, such as cumulative distribution function and moments of the signal envelop in shadowed Rice channels. Based on the derived expressions, the performance analysis of a single receiver operating over lognormally shadowed Rice channel is investigated in terms of the outage probability. Numerically evaluated results show the good accuracy of the proposed approximation method.     

Performance analysis of lognormally shadowed generalized Gamma fading channels

A new composite channel model is proposed for the performance analysis of shadowed fading channels. This model is represented as a mixture of generalized Gamma (GG) multipath fading and lognormal shadowing. GG distribution includes the Rayleigh, Nakagami, and Weibull as special cases; hence the presented model, which is referred to as GG‐L, is a generic model that covers many well‐known composite fading models, including the Rayleigh–lognormal (R‐L), Nakagami–lognormal (N‐L), and Weibull–lognormal (W‐L). The main drawback of the lognormal‐based composite models is that the composite probability density function (PDF) is not in closed form, thereby making the performance evaluation of communication links in these channels cumbersome. To bypass this problem, an approximation method is developed which makes it possible to derive a closed‐form, analytical expression for GG‐L composite distribution. The proposed method only needs the mean and the variance of the underlying lognormal distribution, and hence, bypasses the required complicated integration needed to calculate the PDF of the received signal envelope in GG‐L channel. Based on this method, the most statistical characteristics, such as cumulative density function (CDF) and moments of the GG‐L composite distribution, are derived and used for the performance analysis of a single receiver operating over GG‐L fading channel. Copyright © 2010 John Wiley & Sons, Ltd.     

Adaptive orthogonal signaling with diversity in a frequency-nonselective Rayleigh fading channel

Power and rate adaptations with diversity are considered for M-ary orthogonal signals in a frequency-nonselective Rayleigh fading channel. We derived the minimum required E~/sub b//N/sub 0/ for a given probability of error, the maximum average data rate, and the outage probability, when maximal-ratio combining is performed with an imperfect channel estimation. We analyzed the effects of channel estimation errors on the required E~/sub b//N/sub 0/ and outage probability, as a function of diversity order and adaptation methods. The relative benefits of rate and power control techniques are further presented in terms of channel estimation error, diversity order, and outage probability. The minimum required E~/sub b//N/sub 0/ with Reed-Solomon codes is also presented for comparison purposes.     

Power Minimization of Central Wishart MIMO Block-Fading Channels

In this correspondence, a central Wishart multiple input multiple-output (MIMO) K-block fading channel with an information outage probability constraint is studied. Under this block-limited channel condition, we aim to minimize the transmit power required for attaining a given outage probability based on the statistical channel information at the transmitter (SCIT). Using Gaussian approximation to express the probability density function (pdf) of the instantaneous mutual information and by deriving analytically the mean and variance of the mutual information of the MIMO channel, the optimal power allocation can be obtained numerically by a simple one-dimensional sampling method such as dividing rectange (DIRECT).     

BPSK Bit Error Outage over Nakagami-m Fading Channels in Lognormal Shadowing Environments

In this letter, we address the problem of finding a tractable expression for the bit-error outage (BEO) defined as the probability to observe a given average bit error rate (BER) over a fading channel in a shadowing environment. Our contribution is two-fold: (1) a simple yet tight approximation of the bit error probability (BEP) for binary phase shift keying (BPSK) over a frequency-flat Nakagami-m fading channel is derived, which (2) facilitates the derivation of a tight lower bound of the BEO in presence of lognormal shadowing in closed form. Theoretical results are corroborated by means of simulation results, confirming the tightness of the bounds.     

干扰受限的两跳多接入信道的中断概率分析

在频谱资源重复利用的背景下,干扰受限的两跳多接入信道的中断性能是亟待解决的难题之一.针对目的端干扰受限的情形,本文给出了该信道在放大转发模式下的中断概率闭式解.根据中断事件的定义将中断概率分解为两部分:一部分是中继节点发生中断的概率,另一部分是中继节点未发生中断,但中继节点到目的节点的传输发生了中断的概率.通过对第二部分中断概率进行合理的近似,推导出整个系统中断概率的闭式解,为系统的实际设计提供了理论基础.仿真结果表明,所给出的估计闭式解与蒙特卡洛仿真结果相吻合,验证了此估计解的精确性.