The Sum of Log-Normal Probability Distributions in Scatter Transmission Systems

TLDR

Scatter loss in propagation systems follows a log‑normal distribution, so the total noise in multihop or multichannel systems is the sum of log‑normal variables, for which no exact solution is known. The study proposes an approximate solution for summing log‑normal distributions, with a simplified version tailored for tactical multihop scatter systems. The authors model each hop’s noise power as a log‑normal variable and derive an approximate analytical expression, then simplify it for tactical systems to reduce computational effort. An example computation demonstrates the application of the proposed approximation.

Abstract

The long-term fluctuation of transmission loss in scatter propagation systems has been found to have a logarithmicnormal distribution. In other words, the scatter loss in decibels has Gaussian statistical distribution. Therefore, in many important communication systems (e.g., FM), the noise power of a radio jump, or hop, has log-normal statistical distribution. In a multihop system, the noise power of each hop contributes to the total noise. The resulting noise of the system is therefore the statistical sum of the individual noise distributions. In multihop scatter systems and others, such as multichannel speech-transmission systems, the sum of several log-normal distributions is needed. No exact solution to this problem is known. The following discussion presents an approximate solution which is satisfactory in most practical cases. For tactical multihop scatter systems, a further approximation is proposed, which reduces significantly the necessary computation. An example of the computation is given.