Lognormal Sum Distribution Research Articles

RSS-Based Multiple Co-Channel Sources Localization With Unknown Shadow Fading and Transmitted Power

Multi-source localization based on received signal strength has drawn great interest in wireless sensor networks. However, the shadow fading term caused by obstacles cannot be separated from the received signal, which leads to severe error in location estimate. In this paper, we approximate the log-normal sum distribution through Fenton-Wilkinson method to formulate a non-convex maximum likelihood (ML) estimator with unknown shadow fading factor and transmitted power. In order to overcome the difficulty in solving the non-convex problem, we propose a novel algorithm to estimate the locations of sources. Specifically, the region is divided into <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> grids firstly, and the multi-source localization is converted into a sparse recovery problem so that we can obtain the sparse solution. Then we utilize the k-means clustering method to obtain the rough locations of the off-grid sources as the initial feasible point of the ML estimator. Finally, an iterative refinement of the estimated locations is proposed by dynamic updating of the localization dictionary. The proposed algorithm can efficiently approach a superior local optimal solution of the ML estimator. It is shown from the simulation results that the proposed method has a promising localization performance and improves the robustness for co-channel multi-source localization in shadow fading environments. Moreover, the proposed method provides a better computational complexity from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(K^{3}\,N^{3})$</tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(N^{3})$</tex-math></inline-formula> .

Microphysical Structure of the Marine Boundary Layer under Strong Wind and Sea Spray Formation as Seen from a 2D Explicit Microphysical Model. Part III: Parameterization of Height-Dependent Droplet Size Distribution

AbstractThis paper completes a series of studies using a 2D hybrid Lagrangian–Eulerian model to investigate the effect of sea spray on the thermodynamics and microphysics of the hurricane mixed layer. The evolution of the mixed layer was simulated by mimicking the motion of an air volume (in a Lagrangian sense) toward a tropical cyclone eyewall along a background airflow. During the radial motion, sea surface temperature, pressure, background wind speed, sea spray production rate, and turbulence intensity were altered according to the available observations. Analysis of the interaction between the hurricane mixed layer and the upper layers in terms of entrainment heat and moisture fluxes gives a new insight into the role of sea spray in the thermodynamics and microphysics of the mixed layer. The evaporation of sea spray leads to an increase in the relative humidity by 10%–15% and to a decrease in temperature by about 1–1.5 K, as compared to cases where sea spray is excluded. Sea spray leads to formation of drizzling clouds with the cloud base at the height of about 250 m. Taking the sea spray effect into account provides a good agreement between the thermodynamics of a simulated mixed layer and the observation data.A parameterization of droplet mass and size distributions as functions of height and wind speed is proposed. The horizontally averaged size distributions are approximated by a sum of lognormal distributions. The moments of size distributions and other integral properties are parameterized as functions of 10-m wind speed by means of simple polynomial expressions.

Channel Modeling and Performance Analysis of Diversity Reception for Implant UWB Wireless Link

SUMMARY This paper aims at channel modeling and bit error rate (BER) performance improvement with diversity reception for in-body to on-body ultra wideband (UWB) communication for capsule endoscope application. The channel characteristics are firstly extracted from 3.4 to 4.8 GHz by using finite difference time domain (FDTD) simulations incorporated with an anatomical human body model, and then a two-path impulse response channel model is proposed. Based on the two-path channel model, a spatial diversity reception technique is applied to improve the communication performance. Since the received signal power at each receiver location follows a lognormal distribution after summing the two path components, we investigate two methods to approximate the lognormal sum distribution in the combined diversity channel. As a result, the method matching a short Gauss-Hermite approximation of the moment generating function (MGF) of the lognormal sum with that of a lognormal distribution exhibits high accuracy and flexibility. With the derived probability density function (PDF) for the combined diversity signals, the average BER performances for impulse-radio (IR) UWB with non-coherent detection are investigated to clarify the diversity effect by both theoretical analysis and computer simulation. The results realize an improvement around 10 dB on Eb/No at BER of 10−3 for two-branch diversity reception.

A Low-Complexity Approximation to Lognormal Sum Distributions via Transformed Log Skew Normal Distribution

Sums of lognormal random variables (RVs) occur in many important problems in wireless communication. The lognormal sum distribution is known to have no closed form and is difficult to numerically compute. Several methods have been proposed to approximate the lognormal sum distribution. In this paper, we first propose a low-complexity approximation method called log skew normal (LSN) approximation to model and approximate the lognormal sum distributed RVs. For typical lognormal sum cases in wireless communication, the proposed LSN method has high accuracy in most of the region of the cumulative distribution function (cdf), particularly in the lower region. The closed-form probability density function (pdf) and cdf of the resulting LSN RV are presented, and its parameters are derived from those of the individual lognormal RVs by using a moment-matching technique. However, the LSN approximation has a restriction for the skewness of samples in the logarithm domain. To overcome this drawback, a transformed LSN (TLSN) approximation method is proposed, which uses another parameter to control the skewness of samples in the transform logarithm domain. Simulation results on the pdf and cdf of lognormal sum RVs confirm the effectiveness of the TLSN approximation method.

Signal detection for optical communications through the turbulent atmosphere

The probability of a miss in the detection of a signal in an optical communications system through the turbulent atmosphere using intensity modulation is studied. The turbulence of the atmosphere causes scintillation of the received signal intensity which is treated as a lognormal random process. The received background radiation and electronic noise in the receiver is treated as additive white Gaussian noise (AWGN). A Chernoff bound is derived for the lognormal sum distribution. An approximation for the lognormal sum distribution is investigated for its utility in calculating the probability of miss. For practical values of the signal-to-noise power ratio (SNR), a series solution for the characteristic function of the lognormal random variable is used to find the probability of miss. Simulation results agree with theoretical results. The method developed in this paper can be used by the system designer to choose the proper signal length and meet the system specifications for signal detection.

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.

Approximating Lognormal Sum Distributions With Power Lognormal Distributions

In wireless communications, cochannel interference is usually characterized by a sum of lognormal random variables. Since the characteristic function of a lognormal distribution lacks explicit expression, and numerical calculation of a lognormal sum distribution is very challenging, lognormal distributions are often used to approximate lognormal sum distributions. However, it has been shown that a lognormal distribution can only capture a certain part of the body of a lognormal sum distribution. To improve the accuracy of approximation of lognormal sum distributions, one must resort to non-lognormal approximations. In this paper, we propose the use of power lognormal distributions to approximate lognormal sum distributions. To illustrate the superiority of the proposed model, some numerical experimental results are provided.

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.

Log-Shifted Gamma Approximation to Lognormal Sum Distributions

This paper proposes the log-shifted gamma (LSG) approximation to model the sum of M lognormally distributed random variables (RVs). The closed-form probability density function of the resulting LSG RV is presented, and its parameters are directly derived from those of the M individual lognormal RVs by using an iterative moment-matching technique without the need for curve fitting of computer-generated distributions. Simulation and analytical results on the cumulative distribution function (cdf) of the sum of M lognormal RVs in different conditions indicate that the proposed LSG approximation can provide better accuracy than other lognormal approximations over a wide cdf range, especially for large M and/or standard deviation.

Approximating a Sum of Random Variables with a Lognormal

A simple, novel, and general method is presented in this paper for approximating the sum of independent or arbitrarily correlated lognormal random variables (RV) by a single lognormal RV. The method is also shown to be applicable for approximating the sum of lognormal-Rice and Suzuki RVs by a single lognormal RV. A sum consisting of a mixture of the above distributions can also be easily handled. The method uses the moment generating function (MGF) as a tool in the approximation and does so without the extremely precise numerical computations at a large number of points that were required by the previously proposed methods in the literature. Unlike popular approximation methods such as the Fenton-Wilkinson method and the Schwartz-Yeh method, which have their own respective short-comings, the proposed method provides the parametric flexibility to accurately approximate different portions of the lognormal sum distribution. The accuracy of the method is measured both visually, as has been done in the literature, as well as quantitatively, using curve-fitting metrics. An upper bound on the sensitivity of the method is also provided.

Least Squares Approximations to Lognormal Sum Distributions

In this paper, the least squares (LS) approximation approach is applied to solve the approximation problem of a sum of lognormal random variables (RVs). The LS linear approximation is based on the widely accepted assumption that the sum of lognormal RVs can be approximated by a lognormal RV. We further derive the solution for the LS quadratic (LSQ) approximation, and our results show that the LSQ approximation exhibits an excellent match with the simulation results in a wide range of the distributions of the summands. Using the coefficients obtained from the LSQ method, we present the explicit closed-form expressions of the coefficients as a function of the decibel spread and the number of the summands by applying an LS curve fitting technique. Closed-form expressions for the cumulative distribution function and the probability density function for the sum RV, in both the linear and logarithm domains, are presented

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.

Highly accurate range-adaptive lognormal approximation to lognormal sum distributions

The distribution of the sum of independent non-identical lognormal variates is approximated by a lognormal distribution the parameters of which are suitably varied over the range of probabilities. The new method is highly accurate and keeps track of the exact sum distribution, outperforming previous lognormal approximations.

Stochastic Analysis of Turbo Decoding

This paper proposes a stochastic framework for dynamic modeling and analysis of turbo decoding. By modeling the input and output signals of a turbo decoder as random processes, we prove that these signals become ergodic when the block size of the code becomes very large. This basic result allows us to easily model and compute the statistics of the signals in a turbo decoder. Using the ergodicity result and the fact that a sum of lognormal distributions is well approximated using a lognormal distribution, we show that the input-output signals in a turbo decoder, when expressed using log-likelihood ratios (LLRs), are well approximated using Gaussian distributions. Combining the two results above, we can model a turbo decoder using two input parameters and two output parameters (corresponding to the means and variances of the input and output signals). Using this model, we are able to reveal the whole dynamics of a decoding process. We have discovered that a typical decoding process is much more intricate than previously known, involving two regions of attraction, several fixed points, and a stable equilibrium manifold at which all decoding trajectories converge. Some applications of the stochastic framework are also discussed, including a fast decoding scheme

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.

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.

Towards a realistic description of the contribution of primary and secondary aerosols to ambient particle number and mass distributions

The commonly employed concept of describing measured size-dependent particle number concentrations of ambient aerosols as the sum of lognormal distributions is analysed critically. The starting point is the idea that it is generally desirable to assign a single distribution function to each source of aerosol particulate matter. The problems encountered in determining the most appropriate function are illustrated by considering the very pronounced changes in shape produced by converting number concentration distributions n( D) of particles with diameter D to mass concentration distributions m( D). It is shown that m( D) can exhibit a pronounced peak even if n( D) is a function that decreases monotonically with increasing particle size. The basic considerations are applied to generate much more realistic fit functions for the contribution of the accumulation and the dust mode to measured particle number concentrations. Depending on the particle size, the improved fit functions may differ from the lognormal fit by up to several orders of magnitude. The contribution of fugitive dust to the mass concentration can be made to agree with predicted concentrations as well as with PIXE data for the concentration of crustal elements in samples collected with impactors.

Vertical distribution of atmospheric aerosol size distribution over south-central New Mexico

The vertical distribution of background atmospheric aerosols was measured over south-central New Mexico as a part of the Atmospheric Lidar Verification Experiment (ALIVE) during four research field periods in the summers and winters of 1989 and 1990. Aerosol size distribution was measured from the surface up to 4500 m above sea level (asl) over the particle size range 0.1∼32 μm, using two Particle Measuring Systems (PMS) probes mounted on the wings of the NOAA King Air research aircraft. Vertical profiles of aerosol number concentrations of both fine- (0.1–2.0 μm) and coarse- (>2.0 μm) particle modes show seasonal differences, with higher number concentrations and higher mixed layer heights during summers. The measured aerosol size distribution data of each ALIVE intensive were averaged for boundary layer and free troposphere regions. These data mostly exhibit bi-modal distributions, typical for the continental atmospheric aerosols. Exceptions were the free troposphere size distributions measured during December 1989 (ALIVE III) and June 1990 (ALIVE IV), which resemble Junge's power-law distribution. Each of the averaged aerosol size distributions was approximated by the sum of log-normal distributions. Different characteristics of aerosol size distribution were observed between the two summer measurements of 1989 and 1990. Back-trajectory analysis revealed that decreased aerosol concentrations were observed during June 1990 when the air mass was transported from the southwestern U.S.A.

High-volume screen diffusion batteries and α-spectroscopy for measurement of the radon daughter activity size distributions in the environment

An experimental set-up is described which facilitates the measurement of radon daughter activity size distributions in the diameter size range 0.5–2000 nm at radon activity concentrations down to 5 Bq m −3. The radioactive aerosol particles of different sizes are separated simultaneously by means of a set of high-volume screen diffusion batteries (DB) operating at volumetric flow rates of ∼ 2 m 3 h −1. The activities are collected before and after each DB on membrane filters. During the sampling period and after the end of sampling the α-activities are measured with surface barrier detectors. Using a minimization method (SIMPLEX), the activity size distributions of RaA ( 218Po) and RaC/RaC′ ( 214Pb/ 214Po) are calculated by comparing the measured penetration values with simulated data. This least square method considers the penetration curves of the DBs, and the size distributions are approximated by a sum of log-normal distributions. Initial measurements in homes yield activity size distributions which have up to three modes with median diameters near 1, 20–80 and 100–300 nm.