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.

[1]  Muhammad Fainan Hanif,et al.  On the statistics of cognitive radio capacity in shadowing and fast fading environments , 2010, IEEE Transactions on Wireless Communications.

[2]  Irving S. Reed,et al.  Average time to loss of lock for an automatic frequency control loop with two fading signals, and a related probability distribution (Corresp.) , 1966, IEEE Transactions on Information Theory.

[3]  W. Rudin.  Principles of mathematical analysis , 1964 .

[4]  George K. Karagiannidis,et al.  A closed-form upper-bound for the distribution of the weighted sum of Rayleigh variates , 2005, IEEE Communications Letters.

[5]  Norman C. Beaulieu,et al.  Accurate simple closed-form approximations to Rayleigh sum distributions and densities , 2005, IEEE Communications Letters.

[6]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[7]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[8]  S. Halpern,et al.  The effect of having unequal branch gains practical predetection diversity systems for mobile radio , 1977, IEEE Transactions on Vehicular Technology.

[9]  Norman C. Beaulieu,et al.  Performance of an AFC Loop in the Presence of a Single Interferer in a Fading Channel , 2010, IEEE Transactions on Communications.

[10]  Abbas Jamalipour,et al.  Wireless communications , 2005, GLOBECOM '05. IEEE Global Telecommunications Conference, 2005..

[11]  Dariush Divsalar,et al.  Trellis-Coded Modulation for Fading Channels , 1987 .

[12]  Norman C. Beaulieu,et al.  An infinite series for the computation of the complementary probability distribution function of a sum of independent random variables and its application to the sum of Rayleigh random variables , 1990, IEEE Trans. Commun..