The Statistical Energy-Frequency Spectrum of 

 Random Disturbances 



By JOHN R. CARSON 



A mathematical discussion of the statistical characteristics of Random 

 Disturbances in terms of their "energy-frequency spectra" with applica- 

 tions to such typical disturbances as telegraph signals and " static ". 



IN a paper entitled "Selective Circuits and Static Interference" 

 {B. S. T. J., April, 1925) the writer discussed the "energy- 

 frequency spectrum" (hereinafter precisely defined) of irregular 

 random disturbances extending over a long interval of time. In 

 view of our lack of even statistical information regarding static or 

 atmospheric disturbances the specification of the energy-frequency 

 spectrum, denoted by R(co), was necessarily qualitative, and it was 

 merely postulated that 



" R{co) is a continuous finite function of co which converges to zero 

 at infinity and is everywhere positive. It possesses no sharp maxima 

 or minima and its variation with respect to aj(co = Irf), where it 

 exists, is relatively slow." 



In a paper entitled "The Theory of the Schroteffekt," ^ T. C. Fry 

 deals with a similar problem, namely, the energy or "noise" absorbed 

 in a vacuum tube from a stream of electrons with random time dis- 

 tribution. His method of attack is widely different from that of the 

 present paper. In a more recent paper on "The Analysis of Irregular 

 Motions with Applications to the Energy-Frequency Spectrum of 

 Static and of Telegraph Signals" (Phil. Mag., Jan., 1929), G. W. 

 Kenrick, by making certain hypotheses regarding the wave-form of 

 the elementary disturbances whose aggregate is supposed to represent 

 static interference, and by applying probability analysis, arrives at 

 explicit formulas for the "statistical" or "expected" value of R{co) 

 for a number of different cases. 



I 



In the present paper the statistical or "expected" energy-frequency 

 spectrum i?(co) of random disturbances is investigated by a method 

 which is believed to be somewhat more general and direct than that of 

 Kenrick.2 The results are applicable to the Schroteffekt, telegraph 



^ Jour. Franklin Inst., Feb., 1925. 



' Kenrick's analysis is based on a formula derived originally by N. Wiener instead 

 of proceeding directly from the Fourier integral. 



374 



