MATHEMATICAL AX A LYSIS OF RANDOM NOISE 75 



SO that when / is large and positive 



^-Af 11/2/21 A/ 1 ^^ ^-/2/2^0 



Also, in these circumstances the 1 + erf is nearly equal to two. Thus re- 

 taining only the important terms and using the definitions of the M's gives 

 the approximation to (3.6-5): 



\^T--' 





From this it follows that the expected number of maxima per second lying 

 above the line 7 = /i is approximatel}' when 7i is large, 



27rL 'Ao J 



(3.6-11) 

 _ ^-iiiHo y i[the expected number of zeros of / per second] 



It is interesting to note that the approximation (3.6-11) for the expected 

 number of maxima above /i is the same as the exact expression (3.3-14) for 

 the expected number of times I will pass through /i with positive slope. 



3.7 Results on the Envelope or the Noise Current 



The noise current flowing in the output of a relatively narrow band pass 

 filter has the character of a sine wave of, roughly, the midband frequency 

 whose amplitude fluctuates irregularly, the rapidity of fluctuation being 

 of the order of the band width. Here we study the fluctuations of the 

 envelope of such a wave. 



First we define the envelope. Let fm be a representative midband fre- 

 quency. Then if 



03m = 2irfrn (3.7-1) 



the noise current may be represented, see (2.8-6), by 



I = Z^ Cn cos (Uni — OOmt — (fn -\- C>^mi) 



n=l 

 = Ic COS (i}mi ~ la sin COto^ 



where the components Ic and /« are 



(3.7-2) 



7c = X/ ^n cos (cOnt — Oimi — S^n) 

 n=l 



N 



Is = Z2 Cn sin (cO„ t — COmt — (p„) 



(3.7-3) 



^ This expression agrees with an estimate made by V. D. Landon, Froc. I. R. E., 29 

 (1941), 50-55. He discusses the number of crests exceeding four times the r.m.s. value 

 of /. This corresponds to I\ = IGiZ-o . 



