INSTANTANEOUS COMPANDING OF QUANTIZED SIGNALS 673 



It is convenient to define the quantization error voltage ratio, 



^ _ RMS Error Voltage _ . ,-5.1 . . 



RMS Input Signal Voltage " ^''^^'^ ' 



which takes the form 



D = log (1 + m)[1 + {C/nf + 2AC/4/y/^N (17) 



when we define the quantities 



_ T— , ,- _ Average Absolute Input Signal Voltage , . 



^-|e|/Ve2 RMS Input Signal Voltage ^ ^ 



and 



Compressor Overload Voltage 

 RMS Input Signal Voltage 



„ ,^ , /= compressor uverioaa voiiage /.„>, 



C = V/Ve' = T^^/ra T..„„. q;„,..i Ar^u.^^ ^1^^ 



The simple linear proportionality of Ae to (F ± ixe) results from the 

 properties of the logarithmic function in differentiation. Other, seemingly 

 more simple compression equations, when differentiated, yield much 

 more complicated and unwieldy expressions for Ae. The value of this 

 simplicity is evident in the absence, from (14), of moments of e higher 

 than the second. 



If we set A = 0, (17) reduces to one deduced by Panter and Dite"; 

 their analysis erroneously associated A with e = rather than with 

 Ye~\, as a result of their tacit assumption that (12a) and (12b) are identi- 

 cal. They also imposed the restriction of considering only that class of 

 input signals having peak values coincident with the compandor over- 

 load voltage, by defining V as the peak value of the signal in specifying C. 

 The definition of C in terms of the independent properties of both signal 

 (e') and compandor (V) is then converted into one based solely on the 

 properties of the signal. This interpretation leads to conclusions quite 

 different from those to be presented here. 



B. Logarithmic Companding in the Presence of "DC Bias" 



It has heretofore been assumed that the input signal is symmetrically 

 disposed about the zero voltage level since it may be expected that e = 

 for speech. Although this is a standard assumption, subsequent discus- 

 sion will disclose that it is probable, in actual practice, for the average 

 value of the input signal to be introduced at a point other than the ori- 

 gin of the compression characteristic. In terms of Fig. 1, the signal is 



