INSTANTANEOUS COMPANDING OF QUANTIZED SIGNALS 665 



of transmission. Under these circumstances, companding may actually 

 res wscz7ate a signal; the mathematical description of resuscitation (as 

 anything short of infinite improvement) is clearl}^ beyond the scope 

 of the present analysis. 



At the other extreme, it is probable that there exists a limit of error 

 power suppression beyond which listeners will fail to recognize an}^ fur- 

 ther improvement. Our anal3^sis will not be useful in describing this 

 region of subjective saturation. Furthermore, it is possible that the sub- 

 jective improvement afforded a listener by adding to the number of 

 quantizing steps, or companding, may depend on the initial and final 

 states, even before subjective saturation is reached. For example, it is 

 entirely possible that the change from 5 to 6 digits per code group may 

 provide a degree of improvement which appears different to the listener 

 from that corresponding to the increase from 6 to 7 digits, although the 

 present mathematical treatment does not recognize such a distinction. 



II. EVALUATION OF MEAN SQUARE QUANTIZATION ERROR (a) 



A.* Generalization of the Analysis of Panter and Dite 



The mean square error voltage, a-> , associated with the quantization 

 of voltages assigned to the / voltage interval, ey , is adopted as the sig- 

 nificant measure of the error introduced by cjuantization. If e> is to repre- 

 sent any voltage, e, in the range 



0, - [. - i^'] . . . [. + <^; 



= Rj (1) 



then 



aj = I \e - ejfPie) de (2) 



where (e — Cj) is the voltage error imparted to the sample amplitude by 

 quantization and P{e) is the probability density of the signal. The loca- 

 tion of ej at the center of the voltage range assigned to this level mini- 

 mizes Gj since we shall assume an effectively constant value of P{e) 

 within the confines of a single step. 



If the value of P(e) is approximated by the constant value P{ej) 

 appropriate to Cj in (2), it follows that 



aj = (Ae)/P(e,)/12 (3) 



* 



This passage contains mathematical details which may be omitted, in a first 

 reading, without loss of continuity. 



