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.