680 



THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1957 



which yields 



SC^ + ficAiS - l)C 



Mc- 



= 



(36)^ 



where *S = [(1 + /x,.) log (1 + M':)/m<-] — 1, when appHed t(3 (17). 



The graph of (36) in Fig. 7 may be used to determine numerical values 

 of Mc without repeated recourse to the equation. 



The curve labeled \/3A^Z)^_min iu Fig. 6 was determined by substi- 

 tuting values of Hc , obtained from Fig. 7, into (27). Each curve in Fig. 6 



A-c 



I 



10 



c=v// 



10' 



6 8 



10^ 



Fig. 7 — Critical compression parameters, /xc , required to minimize the quan- 

 tizing error power as a function of relative signal strength, as determined by 

 equation (36). Each point on the curve defines a compandor in the optimum com- 

 pandor ensemble. It must be understood that such an ensemble provides the best 

 performance consistent with equation (8) rather than the absolute minimum quan- 

 tizing error discussed in the Appendix. 



* A similar equation, with A = corresponding to the previously noted errone- 

 ous identification of A with e rather than | e \, has been deduced by Panter and 

 Dite.^ Their definition of C as a "crest factor" changes the significance of what we 

 have called D^i-min and does not lead to the ensemble interpretation of nc ■ 



