676 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1957 



such an impairment maj^ easily be understood in terms of the physical 

 interpretation of (6), as discussed in connection with Fig. 1. 



Equations (27) and (30) also reveal that the penalty inflicted by a 

 finite Co is largely determined by the ratio {tx/B). If (m/-S) « 1, the pres- 

 ence of eo will be unimportant. At the other extreme, if {ii/B) ^ 1, 

 (1 + n/Bf -^ {y,/Bf and 



^/lNDE -> log (1 + m) 



(32) 

 •[1 + {C/Bf + (V2C/m) exp {-V2C/B)Y 



which proves to be relatively insensitive to changes in ^u for the values of 

 ju, C and B considered herein. In this case B largely usurps the algebraic 

 role previously assigned to ju in (27). 



D. Uniform Quantization: pt = 



The mean sciuare cjuantization voltage error in the absence of com- 

 panding, corresponding to direct, uniform quantization of the input sig- 

 nal, follows immediately from (7) and (10) since Aw = Ae under these 

 conditions. Thus 



cro = (Ay)7l2 = V^ZN' 



whence 



Do = {<Joh'f = C/V3iV (33) 



This inverse proportionality of Dq and A'' is well known." ' 



Equation (33) may also be deduced by letting ju approach zero in the 

 expressions for D and De , since (8) implies that v approaches e as /x 

 approaches zero. The fact that Do = (De)^^o reveals that, in the absence 

 of companding, the addition of Sq does not change the quantizing error 

 power. This conclusion was anticipated in the discussion of Fig. 1. 



V. DISCUSSION OF GENERAL RESULTS 



Since the nature of a companded signal depends on a rather large 

 number of variables, it is appropriate to consider their respective roles 

 in general terms before discussing detailed system requirements. This 

 general discussion will, however, emphasize those particular modes of 

 operation which are suggested by existing proposals for application of 

 PCM.^'^'^ Thus, we shall consider common channel companding of 



