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.     

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.     

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.     

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 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     

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.     

Intermodulation interference probability distribution in mobile radiocommunication systems

The problem of mobile radio system outage in the presence of two signal third order intermodulation interferences is presented. The probability that the desired signal level exceeds the interference level by a predetermined value is defined, computed and graphically illustrated. For all signals the superposition of lognormal and Rayleigh distributions is assumed.<>     

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.     

Outage Probability in Mobile Telephony Due to Multiple Log-Normal Interferers

The mobile radio channel is characterized by three important factors: path losses larger than free space, fading typically taken as Rayleigh, and shadowing generally characterized as lognormal. For cellular systems, in order to determine acceptable reuse distances between base stations and to compare modulation methods, the probability of unacceptable cochannel interference (outage probability) has to be determined in the realistic situation where both fading and shadowing occur. In this paper, the average outage probability is computed for centrally located base stations when multiple log-normal interferers are present. This is done for both the mobile-to-base and base-to-mobile communication links. An unexpected result of this study is that the outage probabilities for the two cases do not differ in a significant way. Cumulative probability curves of the short-term average-signal-toaverage-interference ratio (SIR) are presented for a variety of system parameters: channel set number, propagation law exponent (γ), and dB spread (σ) of the log-normal distribution for the signal and interferers. An important observation is the large sensitivity of the performance curves to the propagation parameters: for a system with seven channel sets with a 10 dB SIR threshold, the average outage probability varies from 10 percent forgamma = 3.7, sigma = 6dB, to 70 percent forgamma = 3, sigma = 14dB. Alternatively, for a fixed outage objective of 10 percent, the required SIR threshold value ranges from -17 dB to 11 dB, depending on the propagation parameters. These variations make it imperative that accurate measurements of these parameters be obtained for the different service areas. Outage probabilities are also easily related to specific modulation methods and diversity approaches; detailed results are given for several representative cases.     

Power and rate control with outage constraints in CDMA wireless networks

A radio power control strategy to achieve maximum throughput for the up-link of CDMA wireless systems with variable spreading factor is investigated. The system model includes slow and fast fading, rake receiver, and multi-access interference caused by users with heterogeneous data sources. The quality of the communication is expressed in terms of outage probability, while the throughput is defined as the sum of the users' transmit rates. The outage probability is accounted for by resorting to a lognormal approximation. A mixed integer-real optimization problem P1, where the objective function is the throughput under outage probability constraints, is investigated. Problem P1 is solved in two steps: firstly, we propose a modified problem P2 to provide feasible solutions, and then the optimal solution is obtained with an efficient branch-and-bound search. Numerical results are presented and discussed to assess the validity of our approach.     

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.     

Performance analysis of maximum ratio combining with imperfect channel estimation in the presence of cochannel interferences

In this paper, we analyze the performance of maximum ratio combining (MRC) systems with imperfect channel estimation in the presence of cochannel interference (CCI) with an arbitrary power interference-to-noise ratio (INR). The maximum combining weights are the imperfect estimates of the desired user's fading channel coefficients and are assumed to be complex Gaussian distributed. The quantified measure for estimation error is the correlation coefficient between the true fading channel coefficients and their estimates. Exact closedform expressions are derived for the probability density function (pdf) of the signal-to-interference-plus-noise ratio (SINR), as well as performance metrics including outage probability and the average symbol error probability (ASEP) for some modulation formats. Simulation results demonstrate the accuracy of our theoretic 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.     

Outage probability in the presence of correlated lognormal usefuland interfering components

In many modern wireless communications systems (e.g., in OFDM based digital audio broadcasting/digital video broadcasting (DAB/DVB), CDMA) the received signal can be a superposition of multiple useful and interfering lognormal components, and correlation may exist between the components. We propose a simple method for estimating the correlation coefficient between the total useful and total interfering signals as the function of the correlation between the individual components. We show that, by taking into account this correlation factor in the derivation of the distribution of the signal-to-interference ratio, considerable accuracy improvement can be achieved in the outage probability estimation     

Power Allocation for Amplify-and-Forward Relaying with Correlated Shadowing

Power allocation is a key technique to exploit the benefits of cooperative relaying. In this paper, we investigate the effect of shadowing on the power allocation of amplify-and-forward cooperative relaying systems. Considering the joint effects of path loss, correlated shadowing and flat Rayleigh fading, the approximate outage probability at high signal-to-noise ratio (SNR) is first derived. Then we solve the power allocation problem by minimizing the approximate outage probability subject to a total power constraint. It is shown by the analytical results that the correlation coefficients and the standard deviations of shadowing have significant impacts on the power allocation. The simulation results show that the proposed power allocation scheme yields about 2 dB SNR gain compared to the equal power allocation in the high SNR regime.     

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.     

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.     

Ergodic capacity and outage probability optimization for secondary user in cognitive radio networks under interference outage constraint

This paper considers a cognitive radio network where a secondary user (SU) coexists with a primary user (PU). The interference outage constraint is applied to protect the primary transmission. The power allocation problem to jointly maximize the ergodic capacity and minimize the outage probability of the SU, subject to the average transmit power constraint and the interference outage constraint, is studied. Suppose that the perfect knowledge of the instantaneous channel state information (CSI) of the interference link between the SU transmitter and the PU receiver is available at the SU, the optimal power allocation strategy is then proposed. Additionally, to manage more practical situations, we further assume only the interference link channel distribution is known and derive the corresponding optimal power allocation strategy. Extensive simulation results are given to verify the effectiveness of the proposed strategies. It is shown that the proposed strategies achieve high ergodic capacity and low outage probability simultaneously, whereas optimizing the ergodic capacity (or outage probability) only leads to much higher outage probability (or lower ergodic capacity). It is also shown that the SU performance is not degraded due to partial knowledge of the interference link CSI if tight transmit power constraint is applied.     

一种超密集异构网中联合干扰协调方法研究

针对5G超密集异构网中严重的跨层干扰问题,该文提出一种基于现有的增强型小区间干扰协调技术和协同多点传输技术的联合干扰协调方法。运用随机几何理论工具推导了两层超密集异构网下用户的中断概率,频谱效率和网络平均遍历容量表达式。仿真结果表明：该文提出的联合的干扰协调方案,相比于传统协同多点传输技术,不仅降低了协作用户数目,同时使得用户在信干比阈值为0 dB时的中断概率降低了15%;相比较于增强型小区间干扰协调技术,在偏置值为10 dB时,扩展区域的用户频谱效率改善为35%,整个网络平均遍历容量提升了3.4%。

     

On the performance of interference cancellation in wireless ad hoc networks

This letter presents a novel approach for deriving a lower bound on the outage probability of interference cancellation receiver in ad hoc networks for a class of channel fading. Our approach is based on the following observation: under the power decay law model for large-scale fading with an exponent α strictly greater than 2, the sum of the interferences at a receiver has the same order as the strongest signal. This feature still holds when the small-scale fading has a finite 2/α-fractional moment. This fact is used to develop an approximation which is also a compact lower bound on the outage probability.