ss than 10 nulls per second. Typical measurements taken by Cylink in an office building with a sin- gle antenna are shown in Figure 3A.3. Here, the transmitter is a spread- spectrum signal of 1 M chips per second. These measurements confirm that this rate of 10 multipath nulls per second is somewhat high. Even in these short measurements, deep nulls of 20-dB drop from the average can be observed. As the spread-spectrum signal becomes wider, the effects of frequency diversity begin to appear, and signals will show fewer deep nulls than a CW or narrowband signal. In fact, a wider signal can be thought of as consisting of many different CW components, which make for frequency-selective mul- tipath fading. That is, cancellation of multipath signals at one frequency does not mean a cancellation of signals at another frequency. Although spread- spectrum signals may show fewer deep nulls, frequency diversity effects will not be too evident with the 2-MHz bandwidth signals in these measurements except to reduce the depth of the nulls. If we must accept some loss of signal, we need to assume some statisti- cal dynamics for the fading process. Based on the discussion above, sup- pose that we have a statistically independent sample of R(t) once every 100 msec. This corresponds to the maximum of 10 fades per second. In 1 minute's time, we estimate that there will be 600 independent amplitude random variables while a person walks at 5 km/hr. Suppose we let these independent random Rayleigh variables be R(tk) for k = 1, 2, What is the probability that in 1 minute the fade drops to X dB at least once? This is the event that {R(tk) < r} for one or more independent sam- SECONDS Figure 3A.3. Single antenna. Commercial Applications ple times given by indices k = 1,2, 600. We use the simple union bound for this: = 600F(r) VI 600 10x/10 (3A.18) Note that for x = -30 dB, the probability is less than 0.6 that such a fade will occur in 1 minute. For x = -40 dB, the probability is less than 0.06 or, on average, no more than once every 16