674 THE BELL SYSTEM TECHNICAL JOITRNAL, MAY 1957 



present eel to the arrtiy <»f ([iiaiitizinji; steps with its (iui(;scent \tilue dis- 

 placed by an amount c^ horn the center of the voltage interval (—V to 



+n. 



Such an effect, regardless of its oi'igin, may formally be described by 

 considering the composite input voltage 



E = e + e„ (20) 



where e is the previously considered symmetrical speech signal and Co 

 is the superimposed constant voltage. 



Substitution of E for e in (8) and (12) yields 



aE - (a/12)[V' + ^l'E' + 2^iV \E\] (21) 



where the subscript E is introduced to distinguish this result from (14). 

 Note that the value [e]E=o = —eo now separates the domain of applica- 

 bility of (8a) and (12a) from that of (8b) and (12b), so that (15) is re- 

 placed by 



yWl = [ \-E)P{e) de + ( EP{e) de (22) 



J—V •'— eo 



which reduces to 



n^ = |7[ + 2eo f " P{c) de - 2 r eP{e) de (23) 



Since e = 0, and eo = const., we also find 



^2 = ^ -f eo (24) 



C. Application to Speech as Represented by a Negative Exponential Dis- 

 tribution of Atnplitudes 



It is necessary to assume an explicit function for P(e) in (15) and (23) 

 before applying the general results which have thus far been deduced. 

 We shall assume, as a simple but adec^uate first approximation, that the 

 distribution of amplitudes in speech at constant volume^ may be repre- 

 sented by 



Pie) = Gexp i-Xe) for e ^ (25) 



where P(-e) = P(e), G = X/2, and X" = 2/7\ The values of G and X 



