INSTANTANEOUS COMPANDING OF QUANTIZED SIGNALS 657 



Whereas the first two techniques result in an increase of bandwidth and 

 system complexity, the third requires only a modest increase in instru- 

 mentation without any increase in bandwidth.* This investigation is 

 therefore dcA'oted to the study of nonuniform step size as a means of 

 reducing quantizing impairment. 



C. Physical Implications of Nonuniform Quantization 

 I . Quantizing Error as a Function of Step Size 



Quantizing impairment may profitably be expressed in terms of the 

 total mean square error voltage since the ratio of the mean square signal 

 voltage to this quantity is equal to the signal-to-quantizing error power 

 ratio. 



In evaluating the mean square error voltage, we begin by considering 

 a complex signal, such as speech at constant volume, whose pulse sam- 

 ples yield an amplitude distribution corresponding to the appropriate, 

 probability density. These pulse samples may be expected to fall within, 

 or "excite", all the steps assigned to the signal's peak-to-peak ^•oltage 

 range. It will be assumed that, for quality telephon}^, the steps will be 

 sufficient^ small, and therefore numerous, to justify the assumption 

 that the probability density is effectively constant within each step, 

 although it maj^ be expected to vary from step to step. Thus the con- 

 tinuous cur^'e representing the probability density as a function of in- 

 stantaneous amplitude is to be replaced by a suitable histogram. 



If the midst ep voltage is assigned to all amplitudes falling in a par- 

 ticular quantizing interval, the absolute value of the error in any pulse 

 sample will be limited to values between zero and half the size of the 

 step in question; when combined with the assumed approximation of 

 a uniform probability density within each step, this choice minimizes 

 the mean square error introduced at each level. ^ Summation of the latter 

 quantity over all levels then yields the result that the total mean square 

 (luantizing error voltage is equal to one-twelfth the weighted a^■erage of 

 the square of the size of the voltage steps traversed (i.e., excited) by the 

 input signal. The direct consideration of the physical meaning of this 

 result (which, as (6) below, will constitute the basis of all subsequent 

 calculations) will now be shown to provide a simple ciualitative descrip- 

 tion of the implications of nonuniform quantization. 



* We refer to bandwidth in the transmission medium as determined by the pulse 

 repetition rate, which, in the time division multiplex applications envisioned 

 herein, is given bj' the product: (samjjling rate) X (number of digits or pulses per 

 sample) X (number of channels). 



