na beams sometimes show this behavior. For indoor applica- tions with radios at 1 GHz, the Rayleigh fading model is more commonly used and is used in the remainder of this appendix. Appendix 3A. Multipath and Diversity Upper Bound on F(r) It is easy to show that, in general, 1 - e-irs (3A.12) for all r > 0. This bound is tightest when r is close to zero. The proof of this bound is seen by showing that the function (3A.13) has a minimum value at = = o of g(0) = 0. This bound is useful for evaluating the probability of deep fades owing to this Rayleigh fading model. We normalized the fade term to (3A.14) and now we want to know the probability of a deep fade, specifically, the probability that (3A.15) By converting to decibels, we have (3A.16) Thus, the desired probability is 10x/10 (3A.17) By using this bound, the following table gives the probability of deep fades with a single Rayleigh random variable. Fade Threshold x, dB Probability Probability Bound 6.321 x 10-1 9.516 X 10-2 9.950 X 10-3 9.995 X 10-4 9.999 X 10-5 Commercial Applications This shows that starting with an average signal channel output power of 0 dB, the probability of getting into deep fades decreases as shown here. The simple bound is accurate for the deeper-fade cases of interest, as summa- rized in this table. Dynamics of Fading Note that choosing the mathematical model of a Rayleigh random variable for the amplitude owing to multipath always gives a non-zero probability of hitting a deep fade level. Consequently, one must accept some level of fade, either a "hit" or loss of the signal at the receiver. The only question is, How often can we accept such hits? The answer depends on the time dynamics of the fading process. In a study of fading, we should model the stochastic process for R(t), the time-varying Rayleigh fading amplitude. Typically, a person walks at a rate less than 5 km/hr. With a carrier frequency of 1 GHz, the carrier signal wave length is 30 cm. Thus, a person roaming and talking with a mobile unit will typically experience le