STATISTICAL ENERGY-FREQUENCY SPECTRUM 379 



If, on the other hand, we suppose that the impulses, instead of 

 systematically alternating in sign, are equally likely to be positive or 

 negative, the double summation term of (9) vanishes and 



R(^) = i: f" o-qia)da • f" |/(/a;, \)\'p(\)d\. (15) 



^ J-oo Jo 



This follows from the fact that the amplitude a is equally likely to 

 be positive or negative. Consequently the integration with respect 

 to da must be extended from - oo to + =o and, since by hypothesis 



g(_ a) = g(a), it follows that 



f 



aq(a)da = 0. 



To apply the preceding formulas to actual calculations, it is necessary 

 to know the function /(^co, X) and in addition the probability functions 

 involved. These latter may be supposed known from statistical data 

 or calculable on theoretical assumptions. For example, if we assume 

 that the times of incidence of the elementary disturbances are dis- 

 tributed entirely at random, the application of well-known probability 

 theory gives ^(X) = ve'"^. 



A third case is of interest. Here, instead of postulating that the 

 termination of one impulse coincides with the start of the next (i.e. 

 Wi = ^m + Xm), we suppose that the times of incidence are entirely 

 unrelated, and that the amplitudes are equally likely to be positive 

 or negative. For this case the formula for R(u)) is formally identical 

 with (15). 



Ill 



The foregoing analysis will now be applied to deriving what repre- 

 sents more or less accurately the statistical energy-frequency spectrum 

 of telegraph signals. To this end we shall suppose that the elementary 

 disturbance may have any one of three possible values (all equally 

 probable), characterized by durations Xi, X2, X3 and amplitudes 

 ai, az, as. The corresponding spectra of the elementary disturbances 

 are then determined by the equations, 



Mio:) = f ' Me^'^'dt, (16) 



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Miu:) = f ' ct>(t)e''^'dt. 

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