()()() THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1957 



The total mean square voltage error, cr, is equal to the sum of the mean | 

 square quantizing errors introduced at each level, so that, 



c^ = Z^. = i¥E^(ei)(Ae)/ (4a) 



i } 



= AE (Ae)/[P(e,)(A<.),l (4b) 



4 



which may be rewritten as 



<r ^ -hL (Ae)/py (4c) 



3 



since the discrete probability appropriate to the / step is given by 



p, = f ' P{e) de ^ [P{ej)(Ae),] (5) 



Hence, 



a ^ ^[{AeYUy = (Ae)Vl2 (6) 



Thus, the total mean square error voltage is equal to one-twelfth the 

 average of the square of the input \-oltage step size when the steps are 

 sufficient!}^ small (and therefore numerous) to justify the approximations 

 employed in the deduction of (6). In applying (6), it is important to note 

 that (4) implies that this is a weighted average over the steps traversed 

 (or "excited") by the signal. 



In the special case of uniform step size, substitution of (Ae)j = (Ae) = 

 const, reduces (6) to the simple form 



(To = [o-]Ae=const = (Ae)Vl2 (7) 



Equations (6) and (7) provide the basis for the qualitative interpreta- 

 tion of quantizing error power which has already been discussed in con- 

 nection with Fig. 1. 



The deduction of (6) from (4a) is implicit in the work of Panter and 

 Dite. The absence of an explicit formulation of (6) therein* results from 

 the direct application of the equivalent of (4b) to a specific problem in- 

 volving a particular algebraic expression for {Ae)j . 



A prior, equivalent derivation of (7), based on a graphical represen- 

 tation of (e — Cj) as a sawtooth error function for uniform quantization 

 has been given by Bennett.' Although this derivation bypassed (6), 

 Bennett has also analj^zed compressed signals by means of an expression 



* The present notation has been chosen to resemble that of Reference 5 in order 

 to facilitate direct comparison by the reader. 



