. The Bell System technical journal . om the stations is increased. With these assumptions it can be shown mathematically ^° that theprobability P2, that the ratio of the sum of two vectors to their absolutesum will be less at any instant than a given value X, is given exactlyby the expression P2 = ^sin-iX. For larger values of n, the exact expression is difficult to evaluatebut 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 Rayleighs, Scientific Pape
. The Bell System technical journal . om the stations is increased. With these assumptions it can be shown mathematically ^° that theprobability P2, that the ratio of the sum of two vectors to their absolutesum will be less at any instant than a given value X, is given exactlyby the expression P2 = ^sin-iX. For larger values of n, the exact expression is difficult to evaluatebut 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 Rayleighs, Scientific Papers, Vol. 6 section on Plights in 1, 2, and3 Dimensions, and also section on Random Unit Vibrations. COMMON FREQUENCY BROADCASTING 591 This probability of the sum of w vectors being less than any givenpercentage of the absolute sum of w vectors has been computed bymeans of these expressions for the cases corresponding to the distantreception of 1, 2, 3, and 5 stations. The results of these computationshave been plotted in Fig. 20 30 40 50 60 70 PERCENTAGE OF TIME 100 pig_ 9—Curves showing the probability that the instantaneous sum of the signalsfrom a number of distant stations will be less than any given percentage, X, of theirabsolute sum. There are two aspects of these curves which are of especial interestin connection with this problem. First it will be noticed that as thenumber of stations is increased the percentage of time that the signalfades below a small value such as 5 per cent of its maximum should bedecreased. Thus the percentage of time that bad quality will be re-ceived due to the elimination of the carrier should be noticeably reducedas the number of stations is increased. Also it will be noted that arapid reduction in the percentage of time that the signal approaches themaximum should occur as the number of stations is increased. Thisserves to emphasize the second aspect of the problem, , the level ofthe signal received should remain near the mean
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