276 BELL SYSTEM TECIIXIC.U. JOrRX.-IL 



durations. Rough calculations of r (oj), based on their results, are 

 in support of the hypothesis made in this paper, at least in the radio 

 frequency range. In addition, the writer has made calculations 

 based on a number of reasonable assumptions regarding variations 

 of wa\e form among the individual disturbances, all of which resulted 

 in a spectrimi R (a;) of negligible fluctuations over a frequency range 

 necessary to justify equation (27) for efficient selective circuits. 

 Howe\-er the problem is not theoretically solvable by pure mathe- 

 matical analysis, so that the rigorous %'erification of the theory of 

 selecti\ity developed in this paper must be based on e.\perimental 

 e\idencc. On the other hand, it is submitted that the hypothesis 

 introduced regarding static interference is not such as to \-itiate 

 the conclusions, qualitatively considered, or in general to introduce 

 serious quantitative errors. Furthermore, even if it were admitted 

 for the sake of argument that the figure of merit 5 was not an accurate 

 measure of the ratio of mean square signal to interference current, 

 nevertheless, it is a true measure of the e.vcellence of the circuit in 

 e.xcluding interference energy outside the necessary frequency range. 



W 



The practical applicalioii;, of the foregoing analysis depend upon 

 ihe fornuilas 



-, R(w„) C^ do, 



ir Jn Zilui) r 



(11) 



c_ 1 /*" ' dw /*" do) _ 1 ^ ,.^. 



W2 — Wl.(j| |Z(fa))P' Jo \Z(ioj)\^ 0)2 — wi p 



which contain all the information which it is possible to deduce in the 

 case of purely random interference. They are based on the prin- 

 cii)le that the effect of the interference on the signaling system is 

 measured by the mean square interference current in the receiving 

 branch, and that the efficiency of the selective circuit is measured 

 by the ratio of the mean square signal and interference currents. As 

 stated above, in the case of random interference results must be 

 expressed in terms of mean values, and it is dear that either the mean 

 square current or the mean energy is a fundamental and logical 

 criterifjn. 



Referring to formula (11), the following important proposition is 

 deducible. 



