98 BELL SYSTEM TECHNICAL JOURNAL 



If an exact answer is desired the integrations may be performed. When we 

 assume that/s —fa <<i fb -{■ fa we may obtain approximations for the last 

 two integrals. 



■ = Wo ■ 



iA{t) - A)- = Wo] '- '- tan 



fb — fa .^-1 2ir(fb — fa) 



a 



_ ^ a' + 47r-(/6 - fa? (/, - /„)■ 



47r- a' a- + iir-ifb + fa) 



Furthermore, if 2x(/b — fa)/ci is large we have 



la 

 and the relative r.m.s. l^uctuation is 



'{Ail) - A] 



J 



r.m.s. of 



L2(/6-/a)J 



A 



This result may also be obtained from (3.9-10) and (3.9-11) by assuming 

 a so small that the integral for A{t) may be broken into a great many in- 

 tegrals each extending over an interval T. aT is assumed so small that 

 e""" is substantially constant over each interval. 



3.10 Distribution of Noise Plus Sine Wa\t 



Suppose we have a steady sinusoidal current 



Ip = I pit) = P cos (co,; - <pp) (3.10-1) 



We pick times ti , t2 , • ■ ■ a.t random and note the corresponding values of 

 the current. How are these values distributed? Picking the times at ran- 

 dom in (3.10-1) is the same, statistically, as holding / constant and picking 

 the phase angles tpp at random from the range to 2x. If Ip be regarded as 

 a random variable defined by the random variable ipp , its characteristic 

 function is 



ave. e 



zip _ J_ r izP COS {o:j,i-<f) dv 



Itt Jo 



= J,{Pz) 



(3.10-2) 



and its probability density is 



-^ / e-'"^MPz) dz =\t^ "^ I ^p I ^ ^ (3.10-3) 



^"^ -^-^^ [ Up I > ^ 



In this case it is simpler to obtain the probability density directly from 

 (3.10-1) instead of from the characteristic function. 



