litude random variable is given by R=VX2+Y2 (3A.3) where X and Y are often approximated as Gaussian random variables. If X or Y or both have non-zero mean values, then R is called a Rician random Appendix 3A. Multipath and Diversity X COS Wot + Y sin Wot R=VX'+Y Figure 3A.1. Rayleigh fading. variable (after Steve Rice of Bell Laboratories, who first used this model around 1940). With zero mean values for X and Y, then R is known as a Rayleigh random variable. The worst case assumes that R is a Rayleigh random variable. Therefore, if there are no strong direct components in X and Y, we can assume that they are zero-mean Gaussian random variables. Without loss in generality, we nor- malize by assuming that X and Y are independent zero-mean Gaussian ran- dom variables with unit variance. Then R is the classical Rayleigh random variable with probability density f(r) = re-212 (3A.4) and the probability distribution function F(r) = 1 - e-r/2 (3A.5) for all r > 0. For this normalized case, we have the expected value of R2/2 given by (3A.6) Note that this corresponds to the case of normalizing the average power of the channel output signal - FIR (3A.7) Commercial Applications A cos R COS [w.1 + 0 ] X - N ( 0,02 ) Figure 3A.2. Log normal fading. Another model sometimes considered is the log normal fading statistics for the output amplitude of a propagation channel. In Figure 3A.2, a signal is assumed to consist of many independent amplitude variations, with the resultant signal amplitude a product of many random terms. The product of such terms forms the total random amplitude R=IIA (3A.8) in which each term has the form Ak = Aoe (3A.9) where Zk is a random variable. We then have (3A.10) Again, the Central Limit Theorem is applied, this time to the exponent term: x = Z Z (3A.11) This Gaussian approximation for the random amplitude exponent results in the log normal model. The log normal model tends to be more accurate for optical communica- tions with narrow beams. Long radio paths with many reflections and focussed anten