714 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



provided E = 1 when F = 1 and provided — - = — ' at the transition point 



dV V t 



where the expandor voltages are designated Vt and Et. From (5) we write 



E, = ,<^-^>/^'. (6) 



Let "expansion ratio" be defined by the ratio of Em/Et to Vm/Vt where 

 Em and Vm are the maximum values of the expandor output and input 

 voltages. Recall that Em = Vm = lySet K equal to the expansion ratio and 

 write 



K = J'. (7) 



The expansion ratio may be represented as a function of Vt by replacing 

 Et in (7) by its value in (6), viz., 



K=Vte''- '''''''*, (8) 



Expressed in decibels, 



i^(in db) = 20 logio iVte^'-'''^"''). (9) 



When the value of K given by (9) refers to the compressor, it is called 

 "compression ratio." This follows from the identical (Fig. 3) compressor 

 and expandor characteristics after input and output designations are inter- 

 changed. 



The manner in which K and Et are related to Vt is given by (8) and (6) 

 respectively. These relationships are plotted in Fig. 6. Clearly, if any one 

 of the three parameters is fixed, the entire expandor characteristic is known. 



Signal-to-Noise Ratio 



Let 5i represent the noise susceptibility of the system during intervals 

 when the magnitude of the signal voltage is within the exponential range 

 of the expandor. By differentiating (5) with respect to V and using (4) 

 we get 



,, = 1 e'-"'"' (10) 



Vt 



which relates noise susceptibility to expandor input voltage, V, which in 

 turn equals the compressed signal voltage. To express ^2 as a function of the 

 normal signal voltage, apply (5) to (10), viz., 



^: = f . (11) 



Vi 



