640 BELL SYSTEM TECHNICAL JOURNAL 



thermal noise of power N when the average transmitter power is P is given by 



C = W log ^ . 



This means of course that by sufficiently involved encoding systems we 



P + N 

 can transmit binary digits at the rate W log2 — — — bits per second, with 



arbitrarily small frequency of errors. It is not possible to transmit at a 

 higher rate by any encoding system without a definite positive frequency of 

 errors. 



^ To Approximate this limiting rate of transmission the transmitted signals 

 must approximate, in statistical properties, a white noise. A system which 

 approaches the ideal rate may be described as follows: Let M = 2^ samples 

 of white noise be constructed each of duration T. These are assigned 

 binary numbers from to (M — 1). At the transmitter the message se- 

 quences are broken up into groups of ^ and for each group the corresponding 

 noise sample is transmitted as the signal. At the receiver the M samples are 

 known and the actual received signal (perturbed by noise) is compared with 

 each of them. The sample which has the least R .M .S . discrepancy from the 

 received signal is chosen as the transmitted signal and the corresponding 

 binary number reconstructed. This process amounts to choosing the most 

 probable {a posteriori) signal. The number M of noise samples used will 

 depend on the tolerable frequency e of errors, but for almost all selections of 

 samples we have 



^ . ^ . log M{e, T) ,,,, P+ N 

 Lmi Lun .^ ' ' = W log — i-— , 



SO that no matter how small e is chosen, we can, by taking T sufficiently 



P 4- N 

 large, transmit as near as we wish to TW log — ~ — binary digits in the 



time T. 



P A- N 

 Formulas similar to C = W log — ^^ — for the white noise case have 



been developed independently by several other writers, although with some- 

 what different interpretations. We may mention the work of N. Wiener, 

 W. G. TuUer, and H. Sullivan in this connection. 



In the case of an arbitrary perturbing noise (not necessarily white thermal 

 noise) it does not appear that the maximizing problem involved in deter- 



•This and other properties of the white noise case are discussed from the geometrical 

 point of view in "Communication in the Presence of Noise," loc. cit. 

 7 "Cybernetics," loc. cit. 

 •Sc. D. thesis, Department of Electrical Engineering, M.I.T., 1948. 



