MATHEMATICAL THEORY OF COMMUNICATION 643 



subject to the constraint that all the functions /(/) in the ensemble be less 



than or equal to \/S, say, for all /. A constraint of this type does not work 



out as well mathematically as the average power limitation. The most we 



S 

 have obtained for this case is a lower bound valid for all — , an "asymptotic" 



/ . S\ . S 



upper band ( valid for large — 1 and an asymptotic value of C for - small. 



Theorem 20: The channel capacity C for a band W perturbed by white 

 thermal noise of power N is bounded by 



s 



where S is the peak allowed transmitter power. For sufficiently large — 



- S + N 

 C <W\og'^^ (l + e) 



where e is arbitrarily small. As - ^ (and provided the band W starts 

 at 0) 



C-^T^log(l + |). 



S 

 We wish to maximize the entropy of the received signal. If - is large 



this will occur very nearly when' we maximize the entropy of the trans- 

 mitted ensemble. 



The asymptotic upper bound is obtained by relaxing the conditions on 

 the ensemble. Let us suppose that the power is limited to S not at every 

 instant of time, but only at the sample points. The maximum entropy of 

 the transmitted ensemble under these weakened conditions is certainly 

 greater than or equal to that under the original conditions. This altered 

 problem can be solved easily. The maximum entropy occurs if the different 

 samples are independent and have a distribution function which is constant 

 from — \/S to + V^- The entropy can be calculated as 



W log 4S. 

 The received signal will then have an entropy less than 

 W\og (45+ 27reA^)(l + e) 



