8 L — t/'o'Ao J 



MATHEMATICAL ANALYSIS OF RANDOM NOISE 59 



1/22 and If 23 go to zero as r , M22 — M23 as r , and consequently // goes 



to infinity as t~ . The final result is that (3.4-1) approaches 



(4) ,"2- 



dr T ■ '"""■ 



— tAcAo 



as r — ^ 0, assuming \l/ exists. Here the superscript (4) indicates the fourth 

 derivative at r = 0, 



rp'o*' = 16/ f Mf) df (3.4-6) 



For a low pass filter cutting off at/t (3.4-5) is 



dr ^ {lirf.f (3.4-7) 



WTien (3.4-1) is applied to a low pass filter, it turns out that instead of r 

 the variable 



ip = lirfbT, dip = lirfb dr (3.4-8) 



is more convenient to handle. Thus, if we write (3.4-1) as p((p) d<p^ it fol- 

 lows from (3.4-4) and (3.4-7) that 



1 



p{^) -^ ^—71 = .0919 as (^ -> CO 



Pi^) "^ ^ ^^ *^ 



(3.4-9) 



p{ip) has been computed and plotted on Fig. 1 as a function of (p for the 

 range to 9. From the curve and the theory it is evident that beyond 

 9 p{ip) oscillates about 0.0919 with ever decreasing amplitude. 



We may take p{(p) d\p to be the probability that / goes through zero in 

 <p,ip -\- dip, when it is known that I goes through zero at (^ = with a slope 

 opposite to that at ip. p{ip) dip exceeds the probability that / goes through 

 zero at (^ = and in 9?, <^ + dip with no zeros in between. This is because 

 p{ip) dip includes all curves of the latter class and in addition those which 

 may have an even number of zeros between and ip. From this it follows 

 that the curve giving the probability density of the intervals between zeros 

 must be underneath the curve of p{ip). 



A partial check on the curve for p{ip) may be obtained by comparing it 

 with a probability density function obtained experimentally by M. E. 

 Campbell for the intervals between 754 successive zeros. He passed thermal 

 noise through a band pass filter, the lower cutoff being around 200 cps and 

 the upper cutoff being around 3000 cps. The upper cutoff was rather grad- 



