1 



672 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1957 



IV.* THE CALCULATION OF QUANTIZING ERROR 



A. Lo(iarithmic Commanding in the Absence of ^'DC Bias" 



As previously noted, we consider the effect of uniformly quantizing a 

 compressed signal. If we designate the uniform output voltage step size 

 by (Av), then 



(A.) = ?^ (10) 



since the full voltage range between — V and +F, of extent 2V, is to be 

 divided into A^ equal steps. For a number of levels, N, which is suffi- 

 ciently large to justify the substitution of the differentials df andc?e for the 

 step sizes Av and Ae, differentiation of (8a) yields 



(Av) 

 V 



where k = 1/log (1 + n). 



^[rrwrj^-^ 



Combining (10) with (11) and the counterpart of the latter in the 

 domain of (8b), we find 



and 



where 



Substitution of (12) into (6) yields 



a = (a7l2)[F' + mV + 2mF I"^ ] (14) 



where the c^uantity | e | is introduced by the difference in sign in (12a) 

 and (12b). For ordinary compandor applications, we may write 



= 2 f eP{e) de (15) 



•'0 



since the symmetry of the input signal provides that P{ — e) = P{e) and 



6 = 0. 



* This passage contains mathematical details which may be omitted, in a first 

 reading, without loss of continuity. 



