468 BELL SYSTEM TECHNICAL JOURNAL 



The curves of Fig. (4) were obtained in this way. The relation between k 

 and the number of digits N is based on the assumption of the rms value of 

 signal reaching one-fourth the instantaneous overload voltage of the 

 quantizer. Since zero signal voltage is in the middle of the quantizing 

 range 2^Eq , the overload signal measured from zero is 2^" Eq. The mean 

 square signal input is \{/o . Therefore 



l^'-'Eo = 4v^o (2.31) 



or from (2.9) 



k = 1/4^-' . (2.32) 



We thus have obtained the spectrum of the quantizing errors without 

 sampling. To apply our results to the sampling case we sum up all con- 

 tributions from each harmonic of the sampling rate beating with the noise 

 spectrum from quantizing only. The resulting power spectrum is given by 



00 



^/ = fi/ + Z (fin/,-/ + fin/,+/), <f <fs/2. (2.33) 



If y is the ratio of sampling frequency to signal band width and ^o(y) is the 

 ratio of quantizing power received in the signal band to the applied signal 

 power, 



Aoiy) = fio(l) + Z Mny + D + ^o(ny - 1)]. (2.34) 



n = l 



This is the equation used in calculating the curves of Fig. (5). 



APPENDIX I 



Relation Between Mean Squares of Signal and Its Samples 



We have already shown that there is a unique relationship between a 

 signal occupying the band of all frequencies less than fc , and the sampled 

 values of the signal taken at a rate/« = 2fc . If we are given the signal wave, 

 we can obviously determine the samples; and if we are given the samples, 

 we can determine the signal wave since it is the response of an ideal low-pass 

 filter of cutoff frequency fc to unit impulses multiplied by the samples. If 

 we apply samples of a signal containing components of frequency greater 

 than fc , the output of the filter is a new signal with frequencies confined to 

 the band from zero to fc and yielding the same sampled values as the original 

 wideband signal. 



We now consider the problem of determining the mean square value of the 

 samples of an arbitrary function /(O- Let the samples be taken at / = 

 wr, « = 0, ± 1, db 2, • • , where T = 1/2/. = \/f, . 



