270 BELL SYSTEM TECIIXICAL JOURXAL 



(4) While the interference artually appercei\ed either \isuall\' or 

 by car will certainly depend upon and increase with the energ\- ab- 

 sorbed from static, it is not at all certain that it increases lincarK- 

 therewith. Consequently, it is believed that the additional retine- 

 ment of formula (7) as compared with formula (6) is not justified 

 by our present knowledge and that the representation of the receiving 

 device as a pure constant resistance is sufficiently accurate for present 

 purposes. It will be understood, however, that throughout the 

 following argument and formulas, P of formula {!) may be sub- 

 stituted for /- of (G), when the additional refinement seems justified. 

 The theory is in no sense limited to the idea of a pure constant resist- 

 ance receiver, although the simplicity of the formulas and their ease 

 of computation is considerabK' cnhanred ihcrebN'. 



The problem of random interference, as formulated by equations 

 (6) and (7) was briefly discussed by the writer in "Transient Oscilla- 

 tions in Electric Wave Filters" ' and a number of general conclusions 

 arrived at. That discussion will be briefly summarized, after which a 

 more detailed analysis of the problem will be given. 



Referring to formula (6), since both numerator and denominator 

 of the integrand are everywhere ^o, it follows from the mean value 

 theorem that a value w of w exists such that 



TT .'11 



z{/a>) r 



(8) 



The apijroximale location ol a; on the fre(itieni\' scale is ba.-^ed on the 

 following considerations : 



(a) In the case of efficient selective circuits designed to select a 

 continuous finite range of frequencies in the interval toi^co^ojo, 

 the important contributions to the integral (0) are confined to a finite 

 continuous range of frequencies which includes, but is not greatly 

 in excess of, the range which the circuit is designed to select. This 

 fact is a consequence of the impedance characteristics of selective 

 circuits, and the following properties of the spectrum R (oj) of random 

 interference, which are discussed in detail subsecjuently. 



(b) R (ci)) is a continuous finite function of co which converges to 

 zero at infinity and is everywhere positiv'e. It possesses no sharp 

 maxima or minima, and its variation with respect to oj, where it exists, 

 is relatively slow. 



On the basis of these considerations it will be assumed that w lies 

 within the band wi^u^toj and that without serious error it may be 



