664 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1957 



miiiiimim acceptable rate (slightly in excess of twice the top signal fre- 

 quency^) in order to inv^oke Bennett's results, which tell us that, for this 

 sampling rate, the cjuantizing error power' in the signal band and the 

 total quantizing error power are identical.- Thus, sampling at the mini- 

 mum rate is assumed throughout. 



3. Number of Quantizing Steps 



As already remarked, the present results are based on the assumption 

 that A'' is not small, inasmuch as we assume a probability density which, 

 although varying from step to step, remains effectively constant within 

 each quantizing step; indeed the step sizes will be treated as differential 

 quantities. 



Experimental evidence^' "^ ■ ^° (as well as the analysis to follow) argues 

 against the consideration of fewer than five digits (i.e., 2^ = 32 quantizing 

 steps) for high quality transmission of speech. Numerical estimates indi- 

 cate that the present approximation should be reasonable for five or 

 more digits per code group. These estimates are confirmed by the con- 

 sistency of actual measurements of cjuantizing error power with calcula- 

 tions based on the same approximation (see Fig. 8 of reference 2 for 5, 6, 

 and 7 digit data obtained with an input signal consisting of thermal 

 noise instead of speech) . 



Further indication of the adequacy of this approximation is provided 

 by the knowledge that Sheppard's corrections (see Section II-B) appear 

 adeciuate even when (Ae) is not very small, for a probability density 

 which (as is the case for speech'^) approaches zero together with its de- 

 rivatives at both ends of the (voltage) range under consideration .^^ 



Therefore, we are not presently concerned with the limitations imposed 

 by this approximation. 



4. Subjective Effects Beyond the Scope of the Present Analysis 



We shall have occasion to study graphs depicting the signal to quan- 

 tizing error power ratio as a function of signal power. Although these 

 curves, and the ecjuations they represent, wall always be of interest for the 

 case where even the weakest signal greatly exceeds the corresponding 

 error power, there exists the possibility of rash extrapolation to the 

 region where this inequality is reversed. Unfortunately, such extra- 

 polation may have little or no meaning. * This is particularly clear when 

 one considers that signals incapable of exciting at least the first quan- 

 tizing step, in the absence of companding, will be absolutely incapable 



* This is implicit in the deduction of Equation (6). 



