TABLE m. Computed fade-out statistics. 



X during the interval be less than X itself. In terms of 

 statistical averages, we require 



Now our assumptions (7), (8), (9) imply 



(x') = ln^\{x') = in^' , 



and therefore the condition (25) becomes 

 |7rV<l . 



i€7,"'QM = {tjT) 



(28) 



(25) 

 (26) 

 (27) 



The condition is barely satisfied with € = 0. 1, and so 

 we assume in the following discussion that e sO. 1 

 (fa 20 dB). 



The computed fade-out statistics (Table m) for 10- 

 dB fades do not satisfy this condition. Further, the 5- 

 min averages in the observations will suppress most 

 of the 20 dB and a good fraction of the 10-dB fade-outs. 

 Thus, there is little left for a quantitative comparison. 

 The computed durations are consistent with the observa- 

 tion that for midstation 92% (61%) and for Bermuda 100% 

 (90%) of the 20- (10-) dB fade-outs consist of single 

 terms, that is, the duration is less than 5 min. We ex- 

 pect to miss most of the 20-dB fade-outs, and many of 

 the 10-dB fade-outs, particularly at Bermuda. In tact, 

 97 (567) were observed at midstation compared to 275 

 (860) computed, and 34 (375) at Bermuda compared to 

 371 (1125) computed. All one can say is that the re- 

 sults do not contradict the computations, but for ade- 

 quate studies one will need to sample at least once per 

 minute. 



VI. RANDOM WALK AND SPECTRA 



We picture the movement of the multipath signal 

 {X, Y) as a two-dimensional zig-zag random walk, 

 shown schematically in Fig. 6. The track is composed 

 of discrete straight segments of mean duration t^ (to be 

 estimated). We assume that the motion in each segment 

 is uniform and that the tracks in different segments are 

 uncorrelated. Then the behavior of the multipath phase 

 (j) is defined if we assign a probability distribution 

 Q(e)de for finding a phase change A(^ in the range [6, 

 e + de] in a given segment of track. Values of Acji close 

 to ± IT are associated with fade-outs. 



The zig-zag walk model is not intended as a quantita- 

 tive representation of reality but only as a guide to the 

 analysis of observations. In particular, it does not 

 make sense to try to compute the distribution function 

 Q(e) exactly. We have two pieces of information about 



Q(e). 



(1) The probability of a fade-out in any one segment 

 of the track is 



where T is the interval between fade-outs given by Eq. 

 (24). We thus obtain the estimate 



QM = (2Tr^r'i't, 



(29) 



(2) The average phase change per segment is related 

 to the mean value of i I which we obtain from Eqs. (11) 

 and (26): 



<A0) = | \e\Q{e)de=(\ip\)t,^vt, 



(30) 



We assume for Q(6) the simple form 



Q(e) = (2;r)"' (1+6 cose) , (31) 



and use the two conditions (29) and (30) to determine 

 the two parameters t^ and b. The result is 



b = n\2Tr^-iy^ = 0.Q3 , 



vt^=ii{l-b)=l.n . 



The mean -square phase change per segment is 



(32) 



{(^4>f)= [ e^Q{e)de=U^-Zb = 2. 04 



(33) 



This means that the root-mean-square phase change 

 per segment of track is 1.43 rad or 82°. Over a time 

 t long compared with t^ the mean-square phase wander 

 is 





vt. 



3;r''-12 



= 1.74 



(34) 



The model is of course very crude; from a numerical 

 experiment (Sec. VII) we find 



([<^(/)-(|)(0)f) = 2.78 W . 



Over a month's duration the expected random phase walk 



FIG. 6. Random-walk model of multipath signal in the (X, Y) 

 plane. A fade-out occurs when the track crosses the small 

 circle of radius cm. 



240 



