098 THE BELL SYSTEM TECHNICAL JOUKXAL, MAY 1957 



iiig, might he employed in coiijunetioii with volume regulation is com- 

 pletely consistent with Goodall's experimental results.^" 



VII. CONCLUSIONS 



An effective process for choosing the proper combination of the num- 

 ber of digits per code group and companding characteristic for quantized 

 speech communication sj^stems has been formulated. Under typical con- 

 ditions, the calculated companding improvement for the iveakest signals 

 proves to be equivalent to the addition of about 4 to digits per code 

 group, i.e., to an increase in the number of quantizing steps by a factor 

 between 2 =16 and 2^ = 64. 



Although a precise application of the results requires a more detailed 

 knowledge of the subjective nature of the quantizing impairment of 

 speech than is presently available, the assumption of reasonably typical 

 S3^stem requirements yields conclusions in good agreement with existing 

 exnerimental evidence. 



ACKNOW^LEDGMENTS 



Frequent references in the text attest to the indebtedness of the author 

 to the writings of Bennett and Panter and Dite. It is also a pleasure to 

 acknowledge stimulating conversations on certain aspects of the problem 

 with J. L. Glaser, D. F. Hoth, B. McAIillan, and S. 0. Rice. 



Appendix 



THE minimization OF QUANTIZING ERROR POWER 



In spite of the demonstrated utilit\^ of the ^-characteristics, one can- 

 not avoid speculating about the possibility of achieving substantially 

 more companding improvement by using a characteristic which differs 

 from (8). We shall therefore outline a study of the actual minimiza- 

 tion of quantizing error power without regard to the relative treatment 

 of various amplitudes in the signal. The results will confirm that a signifi- 

 cant reduction of the quantizing error power beyond that attainable with 

 logarithmic companding is self-defeating — for it not only imposes the 

 risk of diminished naturalness, but also implies a compandor too "vol- 

 ume-selective" for the applications envisioned herein. 



1. The Variational Problem and Its Formal Solution 

 Equation (6) may l)e (expressed in the form 



^ -%^[ (dv/de)-'P(e) de (A-1) 



