670 



THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1957 



100 

 80 



60 



40 



20 



10 

 8 



6 



O^ 4 



o 

 o 



1.0 

 0.8 



0.6 

 0.4 



0.2 



0.1 



\ 



Iff 



1-3 



6 8 



la 



-2 



2 4 



(e/v) 



6 8 



10" 



6 8 



Fig. 5 — Pulse sample to step size ratios, as a function of relative sample 

 amplitude, for various degrees of logarithmic companding (i.e.. values of p). The 

 factor (2AY) in the ordinate permits the curves to be drawn without reference to 

 the total number of quantizing steps (.V) ; the factor (100) is included to permit the 

 ordinates directly to convey the proper order of magnitude for (e/Ae), since pres- 

 ent interest will be found to center about values of .V for which 100(2/iV) ~ 1. 

 As noted in the text, the ordinates, which constitute an index of the precision of 

 quantization, approach constancv for (e/F) » yT'^, and vary linearly with abscissa 

 for (e/]') « p-'. 



The essentially logarithmic behavior (e/Ae = const) for large pulse 

 amplitudes is intuitively desirable since it implies an approach to the 

 equitable reproduction of the entire distribution of amplitudes in a spec- 

 ified signal. Although existing experimental evidence indicates that the 

 small amplitudes are not only most numerous,^* but also most .significant 

 for the intelligihilitif^''^ of speech at constant volume, the absence of 

 comparable e\-idence on the properties of naturalness makes it plausible 

 to consider only tho.se compression characteristics \vhi('h give promi.se of 

 providing the .same, acceptabh' small, upper limit on the fractional quan- 

 tizing error for pulse samples of all sizes. 



