On The Distribution Function and Moments of Power Sums With Log-NormalComponents

The power sum with K independent components K PK = 10 logio X 10-Y*/10 _ k--i (1) is a random variable which appears in many areas of communications. With Xk Gaussian, the quantity Lk = 10'Y*/10 (2) is called a log-normal variate. The characterization of the sum of LK is * Professor Schwartz is currently with the Department of Electrical Engineering and Computer Science, Princeton University. 1441 of importance in multihop scatter systems,1 log-normal shadowing environments,2,3 target detection in clutter,4,5 and the general problem of propagation through a turbulent medium.* Thus, the distribution and moments of PK are quantities of considerable importance. Unfortunately, these quantities do not appear to be expressible in simple analytical formulae and, as a consequence, approximate procedures have been investigated for some time. Of particular interest is the Wilkinson approach which uses a normal approximation for the distribution of PK. The problem of characterizing the distribution function then reduces to finding the first two moments of the power sum. The Wilkinson approach is consistent with an accumulated body of evidence indicating that, for the values of K that are of interest, the distribution of the sum of a finite number of log-normal random variables is well-approximated, at least to first-order, by another lognormal distribution.1'9"12* The central question, then, is how to estimate the mean and variance of the approximately Gaussian variate PK, i.e., the power sum.