590 



BELL SYSTEM TECHNICAL JOURNAL 



It has been generally accepted that fading, commonly experienced 

 in the nighttime reception of programs from a distant station, is due 

 to the arrival of the signals along at least two different paths. In the 

 mathematical analysis of this problem it will be convenient to represent 

 each portion of the carrier which arrives at the receiving point via an 

 independent path as a vector of constant amplitude and random 

 phase variation. It will then be possible to represent the fading signal 

 received from a single station as the sum of at least two such vectors. 

 It is then logical to assume that the signal received from two distant 

 stations operating on approximately the same frequency is the summa- 

 tion of at least four of these vectors of constant amplitude and random 

 phase relation. This assumption of random phase relation is valid for 

 any of the common frequency broadcast systems now being developed 

 commercially either here or abroad. If the carriers are derived directly 

 from a reference frequency transmitted via wire line circuits to the 

 several stations, the slight phase variations caused by temperature and 

 humidity changes are sufficient to cause the phases of the derived car- 

 riers of the different stations to vary in a fortuitous manner. Further- 

 more, even if the carriers of two stations were held exactly in phase at 

 their respective antennas, or at some point midway between the trans- 

 mitters, the variations in the path-lengths of the waves arriving at any 

 given distant point would be sufficient to cause a random phase varia- 

 tion. It is helpful in the mathematical analysis to assume also that 

 these vectors are of equal amplitude. While this is not strictly true in 

 all cases, our field observations have shown that it is the limit which 

 tends to be approached as the distance from the stations is increased. 



With these assumptions it can be shown mathematically ^° that the 

 probability P2, that the ratio of the sum of two vectors to their absolute 

 sum will be less at any instant than a given value X, is given exactly 

 by the expression 



P2 = ^sin-iX. 



For larger values of "n," the exact expression is difficult to evaluate 

 but a close approximation to the probability P„ for "n" vectors is af- 

 forded by the expression given below : 



^" = Un '" 24 ' 



+ 72 ^ 288 ^ 



12„5 _ 90n^ 4- 350n^ _ \2n' - U6n'' -f 646;^^ ^^, 



"T 1440 8640 



'" See Lord Rayleigh's, "Scientific Papers," Vol. 6 section on "Plights in 1, 2, and 

 3 Dimensions," and also section on "Random Unit Vibrations." 



