88 BELL SYSTEM TECHNICAL JOURNAL 



where /(/) is a noise current and ti is chosen at random, has been given in a 

 recent article. Here we study this behavior from a somewhat different 

 point of view. 



If we agree to use the representations (2.8-1) or (2.8-6) we may write, as 

 in the paper, the random variable E as 



/r/2 

 I\t) dt (3.9-2) 



r/2 



where the randomness on the right is due either to the a„'s and bnS if (2.8-1) 

 is used or to the <^„'s if (2.8-6) is used. 



The average value of £ is Wj- where, from (3.1-2), 



/r/2 -.T/2 



P{t) dt = / i/'(0) dt = TiPo 

 r/2 J—T/2 



= T [ w{f) df 

 Jo 



(3.9-3) 



Jo 



The second moment of E is 



/r/2 »r/2 



dti / dt2P{ti)Pit2) (3.9-4) 



7-/2 J—T/2 



If, for the time being, we set ^2 equal to /i + t, it is seen from section 3.2 

 that we have an expression for the probability density of I(ti) and /(/i + t) 

 arid hence we may obtain the required average : 



^2 = A f ^^1 f dhlUlexp 



ZtA J-ao •'-00 



A' = 4^1 -rr, h = /(/i), h = m + r) = m) 



The integral may be evaluated by (3.5-6) when we set 



(3.9-5) 



/. = ,..^. /. = .4,^ 



^pT = — 'Ao COS <p 



A = \po sin ^ 



(3.9-6) 



^ "Filtered Thermal Noise — Fluctuation of Energy as a Function of Interval Length" , 

 Jour. Acous. Soc. Am., 14 (1943), 216-227. 



