MATHEMATICAL ANALYSIS OF RANDOM NOISE 

 It is convenient to define y as the ratio 



y = 



85 



R 



^ — fsVi^. 



Tft\ ~ TT72 ~ ^^' ^ 

 r.m.s. /(/) ypl'^ 



where R is understood to correspond to a maximum of the envelope. Since 

 the value of R corresponding to a maximum of the envelope selected at 

 random is a random variable, y is also a random variable. Its probability- 

 density is Paiy), where 



pRiy) has been computed and is plotted as a function of y in Fig. 3. 



0.5 1.0 1.5 2.0 y 25 3.0 3.5 4.0 



Fig. 3 — Distribution of maxima of envelope of noise current. Noise through ideal band- 

 pass filter. 



"^^^ dR = probability that a maximum of R selected at random lies between R and 

 R + dR. 



The distribution function P(i?,nax < y\^\po) defined by 



P(/2max < y^^^o) = I Paiy) dy 

 Jo 



and which gives the probability that a maximum of the envelope selected 

 at random is less than a specified value y\/^Q = R, is plotted in Fig. 4 to- 

 gether with other curves of the same nature. 



