MATHEMATICAL ANALYSIS OF RANDOM NOISE 55 



For an ideal band pass filter whose pass band extends f roni /« to /& the 

 expected number of zeros per second is 



Uf» -/J 



r3 3-12) 



When /a is zero this becomes 1.155 fi, and when /"„ is very nearly equal to 

 fb it approaches fb-\-fa. 



In a recent paper M. Kac" has given a result which, after a slight gene- 

 ralization, leads to 



e-^'i'^o ^\ -ipi'\u (3J-13) 



27r L i^uJ 



for the probability that the noise current will pass through the value / 

 with positive slope during the interval /, / + (//. The expected number of 

 such passages per second is 



e~' '''^° X [h the expected number of zeros per second] (3.3-14) 



The expression (3.3-13) may also be derived from analogue of (3.3-5) 

 obtained by replacing the zero in p{0, r/; .Ti) b}' y. 

 In some cases the integral 



^;' = -^r [ fu'if) df 



Jo 



does not converge. 



An example occurs when we apply a broad band noise voltage to a re- 

 sistance and condenser in series. The power spectrum of the voltage across 

 the condenser is of the form 



"^'^f^ = tAt-" ^^-5.3-1 5) 



Although \po is infinite, i^o is finite and equal to ir/2a. A straightforward 

 substitution in our formula (3.3-11) gives infinity as the expected number 

 of zeros per second. 



Some light is thrown on this breakdown of our formula when we consider 

 a noise current consisting of two bands of noise. One band is confined to 

 relatively low frequencies, and its power spectrum will be denoted by 

 u'lif). The other band is very narrow and is centered at the relatively high 

 frequency fo . The complete power spectrum of our noise is then 



w(f) = w,(f) + A'8(f - U) 



2^ On the Distribution of Values of Trigonometric Sums with Linearly Independent 

 Frequencies, Amtr. Jour. Math., Vol. LXV, pp 609-615, (1943). 



