98 BELL SYSTEM TECHNICAL JOURNAL 



The angular width of this interval is {PI — El)~^'^dEs. There are two such 

 intervals in the range < coi < 2ir. Values of Es outside the range —Po to 

 Po do not exist. Hence, the probability that the signal voltage lies in the 

 interval c?£« at any particular Es is given by 



*.(£.)i£. = {^^ If ll >.f-'n ^^j^^^ I £^ I < p] dE. (3) 



Random noise as discussed in this section may be characterized by the 

 fact that the instantaneous ampUtudes are normally distributed in time; that 

 is, if $„ (s) dz is the probabiUty that the noise amplitude lies in the amplitude 

 interval of width dz at z, 



$n(s) = -^ e-''""' (4) 



(TV iTT 



where a is the root mean square noise amplitude. The mean noise power 

 dissipated in unit resistance is given by TT',, = a . The corresponding mean 

 signal power is given by W^ = Pol 2- Let 4>,(2) represent the probability 

 density function of the instantaneous sum of the signal and noise ampli- 

 tudes. Then 



^r{z)dz = dz [ ^M *n(s - \)d\ (5) 



or 



By the substitution X = Po cos 6, we may convert the integral to the form 



$^3) = -^ r r^^-^" ^°^ '''"'" dd (7) 



TTO" v27r J o 



Suppose we insert a half-wave linear rectifier in series with the source of 

 signal and noise, so that the current 7 is given in terms of the resultant 

 instantaneous voltage E by 



fO, £ < 



l-\ (8) 



\aE, £ > 



Then the average value of current flowing in the circuit is 



I = a I z^riz) dz 

 Jo 



= -^ r zdz r,-(-^«-s«w^^ 



TTCrV 217 Jo Jo 



