76 BELL SYSTEM TECHNICAL JOURNAL 



The envelope, R, is a function of t defined by 



R = [fc-\- fr (3.7-4) 



It follows from the central limit theorem and the definitions (3.7-3) of /« 

 and Is that these are two normally distributed random variables. They are 

 independent since IJt = 0. They both have the same standard deviation, 

 namely the square root of 



7! = 7! = 7 = r w{f) df = ,^0 (3.7-5) 



Jo 



Consequently, the probabiHty that the point (/c , h) lies within the ele- 

 mentary rectangle dicdis is 



die dl. 



linpi 



-expf-^^n (3.7-6) 



L 2^0 J 



In much of the following work it is convenient to introduce another ran- 

 dom variable 6 where 



Ic = R cos e 



(3.7-7) 

 I^ = R sin 6 



Since Ic and /^ are random variables so are R and 6. The dififerentials are 

 related by 



dIcdIs = RdBdR (3.7-8) 



and the distribution function for R and 6 is obtainable from (3.7-6) when 

 the change of variables is made: 



dd RdR-R2i2^^ (3.7-9) 



lir \p, 



Since this may be expressed as a product of terms involving R only and 6 

 only, R and d are independent random variables, d being uniformly dis- 

 tributed over the range to Iv and R having the probability density 



^0 



e " ''*" (3.7-10) 



Expression (3.7-10) gives the probability density for the value of the en- 

 velope. Like the normal law for the instantaneous value of I, it depends 

 only upon the average total power 



^0 = f wif) df 

 Jq 



JO 



^ See V. D. Landon and K. A. Norton, LR.E. Proc, 30 (1942), 425-429. 



