450 BELL SYSTEM TECHNICAL JOURNAL 



Not all the distortion falls within the signal band. The distortion may 

 be considered to result from a modulation process consisting of the appli- 

 cation of the component frequencies of the original signal to the non-linear 

 staircase characteristic. Kigh order modulation products may have fre- 

 quencies quite remote from those in the original signal and these can be ex- 

 cluded by a filter passing only the signal band. It becomes of importance, 

 therefore, to calculate the spectrum of the error wave. This we shall do in 

 the next section for a generalized signal using the method of correlation, 

 which is based on the fact that the power spectrum of a wave is the Fourier 

 cosine transform of the correlation function. The result is then applied to a 

 particular kind of signal, namely one having energy uniformly distributed 

 throughout a definite frequency band and with the phases of the components 

 randomly distributed. This is a particularly convenient type of signal 

 because it in effect averages over a large number of possible discrete fre- 

 quency components within the band. Single or double-frequency signal 

 waves are awkward for analytical purposes because of the ragged nature of 

 the spectra produced. The amplitudes of particular harmonics or cross- 

 products of discrete frequency components are found to oscillate violently 

 with magnitude of input. The use of a large number of input components 

 smooths out the irregularities. 



The type of spectra obtained is shown in Fig. 4. Anticipating binary 

 coding, we have shown results in terms of the number of binary digits used. 

 The number of different magnitudes available are 16, 32, 64, 128, and 256 

 for TV = 4, 5, 6, 7 and 8 digits, respectively. Here a word of explanation is 

 needed with respect to the placing of the scale of quantized voltages. A 

 signal with a continuous distribution of components along the frequency 

 scale is theoretically capable of assuming indefinitely great values of instan 

 taneous voltage at infrequent instants of time. An actual quantizer (stair- 

 case transducer) has a finite overload value which must not be exceeded and 

 hence can have only a finite number of steps. This difficulty is resolved 

 here by the experimentally observed fact that thermal noise, which has the 

 type of spectrum we have assumed for our signal, has never been observed 

 to exceed appreciably a voltage four times its root-mean-square value. 

 Hence we have placed the root-mean-square value of the input signal at 

 one-fourth the overload input to the staircase. This fixes the relation be- 

 tween step size and the total number of steps. In the actual calculation 

 the number of steps is taken as infinite; the effect of the assumed additional 

 steps beyond 2^ is negligible because of the rarity of excursion into this range. 



The curves of Fig. 4 are drawn for the case in which the signal band starts 

 at zero frequency. The original signal band width is represented by one 

 unit on the horizontal scale. The relatively wide spread of the distortion 

 spectrum is clearly shown. As the number of digits (or steps) is increased 



