SPECTRA OF QUANTIZED SIGNALS 



449 



an approximate value for the mean square error, which will later be shown to 

 be sufficiently accurate in most cases of practical importance. This ap- 

 proximation consists of calculating the mean square value of a straight Une 

 going from minus half a step to plus half a step with arbitrary slope. If 



Fig. 2 — A quantized signal wave and the corresponding error wave. 



lU 1 



< 



A 



\-\^^\\^\W 



Er,X 



9_7_5 31 lol 1 ^ I 1 



~2 2 2 "2 "2 2 2 2 2 2 



INPUT VOLTAGE, E| 



Fig. 3 — Characteristic of the errors in quantizing. 



xE, 



£o is the voltage corresponding to one step, and ^ is the slope, the equation 

 of the typical Une is: 





(1.0) 



where e is the error voltage and t is the time referred to the midpoint as 

 origin. Then the mean square error is 



■-i.L 



or one twelfth the square of the step size 



^0/2. 1^ 



(1.1) 



