100 BELL SYSTEM TECHNICAL JOURNAL 



Sometimes the distribution of the envelope of 



I = Pcospt -{- Im (3.10-9) 



is of interest. Here we have replaced Wp by p and have set >fp to zero. By 

 the envelope we mean R{t) given by 



R\t) ^ r' ^ {P + I,f + 7^ (3.10-10) 



where Ic is the component of I^ "in phase" with cos pt and Is is the com- 

 ponent "in phase"' with sin pt: 



Ic = ^ Cn cos [{(Jin — P)t — ^Pn] 



/j = 2^ c„ sin [(co„ — p)t — ^„] 



^ In = Ic cos pt — Is sin pt 



In = I'c ^ I's ^ ypo 



Since I c and /« are distributed normally about zero with a variance of 

 i/'o , the probability densities of the variables 



are 



(27n/'o) 

 (27n/'o)' 



respectively. Setting 



X = R cos d 



y = Rsmd 



and using these distributions shows that the probability of a point {x, y) 



(3.10-11) 



