650 BELL SYSTEM TECHNICAL JOURNAL 



represent the situation as a discrete case. This will not change the evalua- 

 tion function by more than an arbitrarily small amount (when the cells are 

 very small) because of the continuity assumed for p{x, y). Suppose that 

 Pi{x, y) is the particular system which minimizes the rate and gives Ri . We 

 choose from the high probability y's a set at random containing 



2(Ri + €)r 



members where € -^ as T -^ oo . With large T each chosen point will be 

 connected by a high probability line (as in Fig. 10) to a set of x's. A calcu- 

 lation similar to that used in proving Theorem 11 shows that with large T 

 almost all x's are covered by the fans from the chosen y points for almost 

 all choices of the y's. The communication system to be used operates as 

 follows: The selected points are assigned binary numbers. When a message 

 X is originated it will (with probability approaching 1 as T — > oo ) lie within 

 one at least of the fans. The corresponding binary number is transmitted 

 (or one of them chosen arbitrarily if there are several) over the channel by 

 suitable coding means to give a small probability of error. Since Ri < C 

 this is possible. At the receiving point the corresponding y is reconstructed 

 and used as the recovered message. 



The evaluation vi for this system can be made arbitrarily close to vi by 

 taking T sufficiently large. This is due to the fact that for each long sample 

 of message x{t) and recovered message y{t) the evaluation approaches vi 

 (with probability 1). 



It is interesting to note that, in this system, the noise in the recovered 

 message is actually produced by a kind of general quantizing at the trans- 

 mitter and is not produced by the noise in the channel. It is more or less 

 analogous to the quantizing noise in P. CM. 



28. The Calculation of Rates 



The definition of the rate is similar in many respects to the definition of 

 channel capacity. In the former 



/? = Max//p(.,,)log^^<fx<^, 



with P{x) and Vi = 11 P(x, y)p(x, y) dx dy fixed. In the latter 



C = Min l\ P(x, y) log ^^ , dx dy 

 Pix) JJ P{x)P{y) 



with Px{y) fixed and possibly one or more other constraints (e.g., an average 

 power limitation) of the form K = ff P(x, y) \{x, y) dx dy. 



