INSTANTANEOUS COMPANDING OF QUANTIZED SIGNALS 699 



where P(e) has been assumed to be an even function. The function, 

 v(e), which will minimize (^4-1), subject to the usual l)oundary condi- 

 tions at e = and c = V, may be obtained by solving the Euler differ- 

 ential e(iuation of the vaiiational problem. For (^4-1), this takes the 

 form 



(dv/de) = KP"^ (A-2) 



where the constant K is given by 



K = V / f P'" de (A-3) 



Hence the minimum quantizing error is given by 



f P"'de /sN' (A-4) = 



o'min — ^ 



2. Representation of Speech by an Exponential Distribution of Amplitudes 



We shall assume, as in (25), that the distribution of amplitudes in 

 speech at constant volume may be represented by 



P(e) = G exp (-Xe) for e ^ (A-5) 



where P{-e) = P(e), G = X/2, and X' = 2/?. With this choice of P(e), 

 the solution of (A-2) f is 



(v/V) = l-exp[(-V2C/3)(./F)] 

 1 - exp(-V2C/3) 



Thus, for any given relative volume (i.e., for each value of C = V /{e^f'^), 

 (A-6) specifies the compression characteristic required to minimize the 

 quantizing error power. 



We are therefore led to study the properties of the family of charac- 

 teristics of the form 



iv/V) = 1 - exp(-me/F) ^^^, ^ c ^ V (A -7) 

 1 — exp ( — m) 



* An alternate derivation of (A-2) and (A-4), has been given by Panter and 

 Dite,' who also acknowledge a prior and dif'fereni deduction by P. R. Aigrain. 

 Upon reading a preliniinary version of the present manuscript, B. McMillan called 

 my attention toS. P. Lloyd's related, l)ut luipublished work, which proved to con- 

 tain still another derivation. I am grateful to Dr. Lloj'd for access to this material. 



t In the vocabulary of analytical dj'namics, the direct integrability of the 

 Euler equation may be ascribed to the existence of an "ignorable" or "cyclic" 

 coordinated^ 



