462 BELL SYSTEM TECHNICAL JOURNAL 



square value of the signal itself. We also point out that the inversion for- 

 mula for the Fourier integral enables us to express yp^ in terms of Wf , thus: 



yj/r = \ Wf cos IwTfdf (2.2) 



It also may be shown that the ratio ^t/^o cannot have values outside the 

 interval from —1 to +1. 



The correlation theorem furnishes a powerful analytical tool for the 

 solution of modulation problems because the calculation of the average xf/r 

 is often a straightforward process, while direct calculation of Wf may be a 

 very devious one. Once \{/r has been obtained, Eq. (2.1) brings the highly 

 developed theory of Fourier integrals to bear on the computation of w/ . 



We shall give the derivation of Wf for quantizing noise making use of the 

 correlation function. In the analysis we shall apply a number of other 

 needed theorems with appropriate references given for proof. 



Our first problem is that of calculating the spectrum of the output of the 

 staircase transducer, Fig. 1, when the spectrum of the input signal is given. 

 Let Wf represent the power spectrum of the input signal and \f/r the auto- 

 correlation function. The two quantities are related by (2.1) and it is 

 sufficient to express our results in terms of either one. If the instantaneous 

 value of the input signal is represented by Ei, and that of the output by 

 £2, the staircase function may be defined mathematically by: 



(2.3) 



E2 = mEq , Eq K El < Eq , 



w = 0, =tl, ±2, ••• 

 The error is the difference between Ei and £2 and may be written as 



e(l) = El - £2 = £1 - mEo , ^Jl^ E, < E, < ^JUlZLI r, (2.4) 



The error characteristic is plotted in Fig. 3. 



One approach depends on a knowledge of the probability density function 

 p{V\j V2) of the variables Vi = Ei at time / and V2 = £2 at time / + r. 

 The definition of this function is th3itp{Vi,V2) dVidV^ istheprobabiUty that 

 Vi and Vi lie in a rectangle of dimensions dVi and ^1^2 centered on the 

 point V\, V2 of the KiF2-plane. The function p{Vi,V2) has been calculated 

 for certain types of signals and in theory could be computed for any signal 

 by standard methods. If it is assumed known, we may determine the 



