Two Useful Bounds Related to Weighted Sums of Rayleigh Random Variables with Applications to Interference Systems

Weighted sums of Rayleigh random variables occur in diverse problems in wireless, and particularly in interference systems. Previous work has reported upper bounds on the cumulative distribution function of weighted Rayleigh sums. New lower bounds to the cumulative distribution function of weighted Rayleigh sums are derived. The new lower bounds to the cumulative distribution function are used as an intermediate result in deriving a new upper bound on the ratio of a Rayleigh random variable to a weighted sum of Rayleigh random variables shifted by a nonnegative constant. Special cases of this ratio occur in the context of cognitive radio systems and synchronization components. Novel approximations, that are tighter than any known bounds, to the cumulative distribution function of weighted Rayleigh sums are also presented.