4m 



BELL SVSTBM TECHNICAL JOURNAL 



Flatness of the distortion spectrum with frequency within the signal band 

 is demonstrated by Fig. 10. Two kinds of input were used here — a flat 

 band of thermal noise and a set of 16 sine waves with frequencies distributed 

 throughout the band. Results in the two cases were practically the same. 

 The theoretical levels of distortion power for the band widths of the measur- 

 ing filters (95 cps) are shown by the horizontal lines. 



In the experimental results given here use has been made of laboratory 

 studies by Messrs. A. E. Johanson, W. A. Klute, and L. A. Meacham. 



S -25 



m 



o 



z -30 



-35 



-40 



^ -50 



z 



w -55 



UJ 



§-60 



2 



0.2 0.4 0.6 0.8 (.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 



MIDBAND FREQUENCY IN KILOCYCLES PER SECOND 



3.0 3.2 



Fig. 10 — Spectral density of distortion in signal band from quantizing and sampling. 

 The quantizing steps were equal and the quantizer was fully loaded by a random noise 

 or 16- tone input signal with mean power = —2.5 dbm. 



2. Theoretical Analysis 



The correlation theorem discovered by N. Wiener" may be stated as 

 follows: Let xf/r represent the average value of the product I{t)I{t + r), 

 where /(/) is the value of a variable such as current or voltage at time ^, 

 and /(/ 4- t) is the value at a time r seconds later. Mathematically: 



yPr = /(/)/(/ -h r) 



= Lim 



r Jo 



/(/)/(/ -h t) dt 



(2.0) 



From analogy with statistical theory, ypr is called the correlation of /(/) 

 with itself, or the autocorrelation function of the signal. Since we shall not 

 deal here with the correlation of two signals, we shall shorten our terms and 

 call ^r simply the correlation of /(/). Let Wf df represent the mean power in 

 the output of an ideal bandpass filter of width df centered at/. We assume 

 that the ideal filter is designed to work between resistances of one ohm each 

 and that the input signal /(/) is delivered to the filter from a source with 

 internal resistance of one ohm. (The use of unit resistances does not re- 

 strict the generality of the results, since equivalent transmission performance 



