ith the largest amplitude. If we follow this rule, then the diversity random amplitude is the random variable given by the largest of the N random amplitudes: R = Max {R1,R2,...,RN}- (3A.19) The probability distribution for this is Fn(r) =Pr[R<r] (3A.20) Diversity is not useful unless the signal amplitudes are independent. Assuming R1, R2, , RN are independent and identically distributed, we have the result that Fn(r) = F(r)N (3A.21) where F(r) is the probability distribution of each amplitude. Assuming the same normalized Rayleigh random variables where F(r) = 1 - e r 2/2/2 (3A.22) for all r > 0, we get the distribution for the diversity random variable as < (2)N (3A.23) Noting that = 10=/10 (3A.24) where x is the threshold level in decibels, we get the following table for N = 1, 2, 3, and 4. PULSE GENERATOR MOBILE TRANSMITTER 1 KHz SPECTRUM SWITCH ANALYZER PLOTTER Figure 3A.4. Experiment with diversity antennas. [R1, R2] Min [R , R2] SECONDS Figure 3A.5. Diversity antennas. Commercial Applications Probability Bound Fade Threshold, dB N = 2 N = 3 10-12 10-12 10-16 10-10 10-15 10-20 Recall that before, with no diversity (N = 1), we saw that a deep fade threshold of -40 dB averaged no more than one fade every 16 minutes. With dual diversity (N = 2), the same average occurrence would have a fade threshold of -20 dB. This means that to avoid more than one deeper fade every 16 minutes, power control circuits need to cover fades of only 20 dB down from the average signal power level. Antenna Diversity Figure 3A.4 shows the experimental set for Cylink's diversity antenna mea- surements with a 1 M chip-per-second spread-spectrum transmitter signal. The pulse generator switches between the two antennas at a 1-kHz rate. This 1-MHz chip rate is close to the chip rate considered by the Qualcomm cel- lular and Cylink PCS voice systems. The resulting graph shows the two anten- nas' received signal levels but does not indicate which antenna has the stronger signal. Figure 3A.5 shows a typical measured signal power of a spread-