SPECTRA OF QUANTIZED SIGNALS 



453 



These results may be applied to the staircase transducer. The output 

 may be resolved into the input signal plus the error. The sampling fre- 

 quency is assumed to exceed its minimum required value of twice the top 

 signal frequency. The component of the output that is equal to the origi- 

 nal signal can therefore be separated at the receiver by a filter passing the 

 original signal band. A similar statement cannot be made for the error 

 component, for it has been found to extend over a vastly greater range than 

 the original signal. To calculate the total distortion received in the signal 

 band, we can multiply the distortion spectrum by the switching function and 

 sum up all sideband contributions to the original signal band. Each har- 



20 





35 



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$ 50 



^ 2 4 6 8 10 20 40 60 80 100 200 400 600 1000 



SAMPLING frequency/signal BAND WIDTH 



Fig. 5 — Total distortion in signal band from quantizing and sampling a random noise 

 wave. Full load on the quantizer is 12 db above the r.m.s. value of input. 



monic of the switching function makes such contributions by beating with 

 a band of the error spectrum above and below the frequency of the har- 

 monic. These contributions add as power when the sampling frequency is 

 independent of the individual frequencies contained in the signal. The 

 total error power accepted by the signal band filter decreases as the sampHng 

 frequency is increased because each harmonic of the sampling frequency is 

 thereby pushed upward into a less dense portion of the error spectrum. In 

 the limit as the sampling frequency is made indefinitely large, we return to 

 the non-sampled case, that of the staircase transducer only. 



Figure 5 shows the calculated curves of distortion in the signal band 

 plotted as a function of ratio of sampling frequency to signal band width. 

 The curves have downward slopes approaching asymptotes corresponding 

 to the area from zero to unity under the corresponding curves of Fig. 4. 



