Accurate closed-form approximations to Ricean sum distributions and densities

The statistical distribution of a sum of Ricean random variables occurs extensively in wireless communications. A closed-form expression does not exist for the sum distribution and, furthermore, the Ricean random variable does not have a closed-form characteristic function. For these reasons, it is somewhat difficult to numerically calculate the sum distribution. Highly accurate, closed-form approximations for the sum distributions and densities are presented. These approximations are valid for a wide range of argument values, Rice factors and number of summands.     

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.     

Accurate simple closed-form approximations to Rayleigh sum distributions and densities

Sums of Rayleigh random variables occur extensively in wireless communications. A closed-form expression does not exist for the sum distribution and consequently, it is often evaluated numerically or approximated. A widely used small argument approximation for the density is shown to be inaccurate for medium and large values of the argument. Highly accurate, simple closed-form expressions for the sum distributions and densities are presented. These approximations are valid for a wide range of argument values and number of summands.     

Mixture Lognormal Approximations to Lognormal Sum Distributions

In wireless communication, co-channel interference is usually characterized by a sum of lognormal random variables. Since calculating the exact distribution of a lognormal sum has a lot of challenges, lognormal distributions are often used to approximate lognormal sum distributions. However, it has been shown that lognormal approximations can only capture a certain part of the body of a lognormal sum distribution, which implies that to accurately approximate a lognormal sum distribution, one has to resort to non-lognormal approximations. In this paper we propose to use a two-component mixture lognormal model to approximate lognormal sum distributions. Numerical examples are provided to compare the proposed mixture lognormal approximation with the existing ones.     

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.     

Bounds on the distribution of a sum of independent lognormal randomvariables

The distribution function of a sum of lognormal random variables (RVs) appears in several communication problems. Approximations are usually used for such distribution as no closed form nor bounds exist. Bounds can be very useful in assessing the performance of any given system. In this letter, we derive upper and lower bounds on the distribution function of a sum of independent lognormal RVs. These bounds are given in a closed form and can be used in studying the performance of cellular radio and broadcasting systems     

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.     

Lognormal Sum Approximation with Type IV Pearson Distribution

In this paper, the Type IV Pearson distribution is proposed to approximate the distribution of the sum of lognormal random variables, and the parameters of the Type IV Pearson distribution are derived through matching the mean, variance, skewness and kurtosis of the two distributions. Numerical simulations show that the Type IV Pearson distribution can accurately approximate distribution of the sum of lognormal random variables in a wide probability range.     

A Simple Polynomial Approximation to the Gaussian Q-function and Its Application

A simple polynomial approximation to the Gaussian Q-function is proposed, based on the observation that a Gaussian random variable can be well approximated by a sum of uniform random variables. The approximation can be used to obtain accurate explicit approximations to problems that otherwise do not have explicit solutions or approximate explicit solutions. As an example, an explicit expression for the average symbol error rate of M-ary pulse amplitude modulation in lognormal channels is derived using the new approximation, and the approximate symbol error rate is shown to be very close to the exact value.     

Error performance of a pulse amplitude and position modulated ultra-wideband system over lognormal fading channels

In this paper, the error performance of an ultra-wideband (UWB) system with a hybrid pulse amplitude and position modulation (PAPM) scheme over indoor lognormal fading channels is analyzed. In the PAPM UWB system, input data is modulated onto both the pulse amplitudes and pulse positions. The receiver employs a RAKE to combine energy contained in the resolvable multipath components. Derivation of closed-form error rate expressions of the system in lognormal fading channels is based on approximating a sum of independent lognormal random variables (RV) as another lognormal RV using the Wilkinson method. Given the same delay spread of the channel, the proposed PAPM scheme can provide a higher throughput than the binary pulse amplitude or pulse position modulation scheme.     

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.     

Lognormal Sum Approximation with a Variant of Type IV Pearson Distribution

In this paper, a variant of the Type IV Pearson distribution is proposed to approximate the distribution of the sum of lognormal random variables. Numerical and computer simulations show that independent of the statistical characteristics of the lognormal sum distribution, the Type IV Pearson variant outperforms the standard Type IV Pearson distribution and the normal variant distribution in accurately approximating the lognormal sum distribution for a whole probability range.     

Accurate Approximations to the Sum of Generalized Random Variables and Applications in the Performance Analysis of Diversity Systems

Accurate closed-form approximations to the sum of independent identically distributed η-μ and κ-μ random variables are provided. The proposed approximations turn out to be simple, precise, and useful for obtaining important performance metrics of communications systems where sums of variates arise. In particular, average bit error rate and level crossing rate of multibranch equal-gain combining receivers are attained to illustrate the applicability of the approximations. Some sample examples show that the intricate exact solution and the simple approximate expressions yield results that are almost indistinguishable from each other.     

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.     

Summing received signal powers with arbitrary probability density functions on a logarithmic scale

A technique is presented to calculate the probability density function (pdf) for a sum of random variables that have pdf's on a logarithmic scale. In mobile radio it is often necessary to calculate the pdf of the total received signal power, which is the power sum of a number of simultaneously received signal powers. When the signal powers are given on a linear scale (e.g. Watts) probability density functions (pdf's) of the individual signals can be convolved to give the pdf for the received power of all the signals together. When, as is usual, the signal powers are given on a logarithmic scale (e.g. dBs) this is not possible. The simple convolution for the linear domain must now be replaced by a convolution for the logarithmic domain, which is not straightforward to compute. In this paper, a method is presented to compute the pdf of a power sum of two random variables, the logarithmic convolution. The results are not in closed form, numerical integration is necessary to find the resulting pdf. The method can be applied recursively to give results for power sums of more than two random variables. Although methods exist that give solutions in a closed form, they mainly use approximations and are valid only for specific distributions. The method presented in this paper yields exact results for arbitrary distributions. The results of the logarithmic convolution are verified by Monte-Carlo simulations. Even for large numbers of random variables the power sum results are shown to be correct.     

A statistical basis for lognormal shadowing effects in multipathfading channels

Empirical justifications for the lognormal, Rayleigh and Suzuki (1977) probability density functions in multipath fading channels are examined by quantifying the rates of convergence of the central limit theorem (CLT) for the addition and multiplication of random variables. The accuracy of modeling the distribution of rays which experience multiple reflections/diffractions between transmitter and receiver as lognormal is quantified. In addition, it is shown that the vector sum of lognormal rays, such as in a narrow-band signal envelope, may best be approximated as being either Rayleigh, lognormal or Suzuki distributed depending on the fading channel conditions. These conditions are defined statistically     

Estimation of typical sum of lognormal random variables using log shifted gamma approximation

The sum of lognormal distributions is a well-known problem that no closed-form expression exists and it is difficult to evaluate numerically. In this paper, log shifted gamma (LSG) approximation method is proposed to represent the sum of lognormal distributions and to derive a closed-form expression of the typical value of the sum. Illustrative results show that the LSG model provides much more accurate approximation than other previous methods for a wide range of lognormal variances.     

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     

一种有长拖尾特性的G-分布海杂波模型

针对雷达海杂波概率密度函数的长拖尾特征,在K一分布杂波模型基础上提出一种新的描述海杂波的长拖尾分布G-分布.该分布是由二元Raylei#独立积随机变量和广义x分布的随机变量进行级联而生成的三元独立积.使用Mel-lin变换方法得到G-分布概率密度函数的解析表达式.仿真结果表明G-分布的长拖尾特性比K-分布强.最后使用实验数据验证G-分布描述海杂波的有效性,并给出一种产生相关G-分布随机变量的方法,产生所需的相关随机变量.     

Generalized moment matching for the linear combination of lognormal RVs: application to outage analysis in wireless systems

Moving from the need for a simple and versatile method for outage computation in various contexts of interest in wireless communications, in this paper we propose a lognormal approximation for the linear combination of a set of lognormal random variables (RV) with one-sided random weights. The approximation is based on a generalization of the well known moment matching approximation (MMA) for the sum of lognormal RVs, and it allows quite simple handling of the power sum of interfering signals even in rather complicated scenarios. Specifically, composite multiplicative channel models with unequal parameters can be handled, and generic (unequal) correlation patterns for some channel components can be handled with reference to any pair of signals. At this stage of the computation, only moments of the random weights are required. The probability density function of the random weight for the useful signal component may be required in computing outage probability, and numerical methods may be only required to solve a single integral at this second stage. The suitability of the approximation is examined by evaluating outage performance for various values of system parameters in some contexts of interest, namely spread spectrum systems and typical reuse-based systems with composite Rayleigh-lognormal and Nakagami-lognormal channels.