MATHEMATICAL AXALVSIS OF RAX DOM NOISE 79 



where x is x(/6 — /«) r. When .v is a multiple of tt, Ri and Ri are independent 

 random variables. When .r is zero Ri and R2 are equal. Hence we may 

 say, roughly, that the period of fluctuation of R is the time it takes x to in- 

 crease from to X or (/& — /„)" . This is related to tlie result given in the 

 next section, namely that the expected number of maxima of the envelope 

 is .641 (fb — fa) per second. 



3.8 Maxima of R 



Here we wish to study the distribution of the maxima of R* Our work 

 is based upon the expression, cf. (3.6-1), 



-dRdi f p{R,0,R")R" dR" (3.8-1) 



for the probability that a maximum of R falls within the elementary rec- 

 tangle dR dt. p{R, R', R'.') is the probability density for the three dimen- 

 sional distribution of R, R', R" where the primes denote differentiation with 

 respect to /. 



We shall determine p{R, R', R") from the probability density of Ic , Is , 

 Ic , Ts , Ic , Is , which we shall denote by Xi , X2 , • • • x^ . The interchange 

 of Is and Ic is suggested by the later work. It is convenient to introduce 

 the notation 



bn = (2t)" [ wiDU -UTdf 



*'o (3.8-2) 



bo — 4^0 



where /m is the mid-band frequency, i.e., the frequency chosen in the defini- 

 tion of the envelope R. bn is seen to be analogous to the derivatives of 

 4/{t) at t = 0. 



From the definitions (3.7-3) of Ic and /« we obtain the second moments 



Xi = Ic = 4^0 = bo 



Xi = Is — Do 



3 = /? = Z w{f,,)Af47r\fn - Uf = h 

 1 



^"5 = Ic = bi 

 Xz = Ic = O4 

 X& = Is = Oi 



* Incidentally, most of the analysis of this section was originally developed in a study 

 of the stability of repeaters in a loaded tele])hone transmission line. The envelope, R, 

 was associated with the "returned current" produced by reflections from line irregularities. 

 However, the stud\- fell short of its object and the onl>- results which seemed worth sal- 

 vaging at the time were given in reference^* cited in Section 3.3. 



