G58 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1957 



2. Properties of the Mean Square Excited Step Sr2e 



Fig. 1(a) shows the range of input voltages, between the values +1 

 and —V divided into N equal quantizing steps (i.e., uniformly quan- 

 tized); Fig. 1(b) depicts the same range divided into A^ tapered steps, 

 corresponding to nonuniform quantization. 



Consider a complex signal, such as speech, whose distribution of in- 

 stantaneous amplitudes at constant vokmie results in the excitation 

 (symmetrically about the zero level) of the steps in the moderately large 

 interval A^ — .Y'. The quantizing error power will be shown to be pro- 

 portional to the (weighted) average of the square of the excited step size. 

 For uniform quantization, it is clear, from Fig. 1(a), that this average 

 is a constant, independent of the statistical properties of the signal. For 

 a nonuniformly quantized signal, [Fig. 1 (b)], the mean square excited step 

 size is reduced by the chvision of the identical interval X — X' into more 

 steps, most of which are smaller than those shown in Fig. 1(a). Apprecia- 

 tion of the full extent to which the quantizing error power ma}^ so be 

 reduced rec^uires the added recognition that the few larger c^uantizing 

 steps in the range X — X', corresponding to excitation by comparati'V'ely 

 rare speech peaks, are far less significant in their contribution to the 

 weighted average than the small steps in the ^dcinity of the origin, due 

 to the nature of the probability density of speech at constant volume.'^ 



It is also clear that weaker signals, corresponding to a contraction of 

 the interval X — X', enjoy the greatest potential tapering advantage 

 since their excitation may be confined to steps which are all appreciably 

 smaller than those in Fig. 1(a). However, if the interval X — X' were to 



(a) 



Fig. 1 — (a) Distribution of step.s of equal size corrosijonding to direct, uniform 

 quantization; (b) nonuniform quantization of this range into the same nmiiber of 

 steps. The function of the instantaneous compandor is to jirovide such nonuniform 

 quantization in the manner ilhistrated in Fig. 2. 



