452 BELL SYSTEM TECHNICAL JOURNAL 



the spectrum becomes flatter over a wider range, but with d smaller maxi- 

 mum density. The area under each curve represents the total mean power 

 in the corresponding error wave and is found to agree quite accurately 

 with the approximate result of Eq. (1.1). The distortion power faUing in 

 the signal band is represented by the area included under the curve from 

 zero to unit abscissa. 



Quantizing the magnitude only is not a technically attractive method of 

 transmission because of the wide frequency band required to preserve the 

 discrete values of the quanta. Thus in a 128-step system, a full load sinus- 

 oidal signal passes through 64 different steps each quarter cycle and hence 

 would require transmitting 256 successively different magnitudes during 

 each period of the signal frequency. We therefore consider the second prob- 

 lem — that of sampling the quantized magnitudes. 



The theory of periodic sampling of signals is a limiting case of com- 

 mutator modulation theory as previously shown by the author.^ We may 

 think of a periodically closed switch in series with the line and source as 

 producing a multiplication of the signal by a switching function. The 

 switching function has a finite value during the time of switch closure and is 

 zero at other times. It may be expanded in a Fourier series containing a 

 term of zero frequency, the repetition frequency of switch closure, and all 

 harmonics of the latter. Multiplication of the signal by the Fourier series 

 representing the constant component of the switching function gives a term 

 proportional to the signal itself. Multiplication of the signal by the funda- 

 mental component of the switching function gives upper and lower side- 

 bands on the repetition frequency. Likewise multiplication by the har- 

 monics gives sidebands on each harmonic. The signal is separable from the 

 sidebands on a frequency basis if the signal band does not overlap the lower 

 sideband on the repetition frequency. This leads to the condition for no 

 distortion in time division: the highest signal frequency must be less than 

 one-half the repetition frequency. 



To apply the above theory to instantaneous sampling we let the duration 

 of switch closure in one period approach zero. We then approach the con- 

 dition of one signal value in each period, so that the repetition frequency 

 now becomes the sampling frequency. Clearly the sampling frequency 

 must slightly exceed twice the highest signal frequency. We also note that 

 as the contact time tends toward zero, the switching function approaches a 

 periodically repeated impulse. The important terms of the Fourier series 

 representing the switching function accordingly become a set of harmonics 

 of equal amplitude with a constant component equal to half the amplitude 

 of the typical harmonic. On multiplication of this series by the signal, we 

 get a set of sidebands of equal amplitude including the one corresponding to 

 the original signal itself, the sideband on zero frequency. 



