INSTANTANEOUS COMPANDING OF QUANTIZED SIGNALS 677 



speech in time division multiplex systems which employ binary number 

 encoding. 



A. Quantitative Description of Conventional Operation (eo = 0) 



1 . Number of Quantizing Steps (N) 



The number of steps, N, is related to the choice of code. For a binary 

 code, the relation takes the form, N = 2", where n is the number of 

 binarj^ digits per code group. 



In the present discussion it will usually be convenient to regard n and 

 A'' as fixed in order to permit comparison of various companding charac- 

 teristics under the same coding conditions. Since both D and Do are in- 

 verseh^ proportional to N, the quotient (D/Do)~, which is equal to the 

 ratio of the quantizing error power in the presence of companding to 

 that in the absence of companding, is independent of N. Consec^uently, 

 as will be evident in the discussion of (37), the relative diminution of 

 quantizing error (in db) afforded by companding is also independent of 

 N* 



However, the value of N mil determine the signal to quantizing error 

 power ratio to which the companding improvement is to be added. Thus 

 the number of digits per code group required for a particular application 

 will ultimately be determined by the value of A^ which, in combination 

 with suitable companding, will suffice to produce an acceptably low value 

 of quantizing error power in relation to signal power. 



2. Compandor Overload Voltage (V) 



The compandor overload voltage, V, will be determined by the full 

 load power objectives for the proposed system. Specifically, T^ will be 

 equal to the amplitude of the sinsuoidal voltage corresponding to "full 

 modulation." 



3. Relative Signal Power (C) 



The quantity C = V/ie^Y will, for a given value of T', be deter- 

 mined by the rms signal voltage (e^)'. 



The range of C values appropriate to a given system will therefore 

 reflect the distribution of volumes to be encountered. In fact, the signal 



* It must of course be understood that this independence requires a value of 

 N sufficientl}' large to justify the approximations involved in the deduction of 

 (27) and (33). 



