MATHEMATICAL THEORY OF COMMUNICATION 645 



and the channel capacity is greater than 



ire* N 



S 

 We now wish to show that, for small — (peak signal power over average 



white noise power), the channel capacity is approximately 

 C = PFlog(l+|). 



/ s\ s 



More precisely C/W log ( 1 + - ) -^ 1 as — — » 0. Since the average signal 



5 

 power P is less than or equal to the peak S, it follows that for all — 



C<W'log(l+ j)<fFlog(l+|). 



Therefore, if we can find an ensemble of functions such that they correspond 



to a rate nearly TF log f 1 + - J and are limited to band W and peak S the 



result will be proved. Consider the ensemble of functions of the following 

 type. A series of / samples have the same value, either + \/S or — \/S, 

 then the next / samples have the same value, etc. The value for a series 

 is chosen at random, probability | for + \/S and | for — \/5 If this 

 ensemble be passed through a filter with triangular gain characteristic (unit 

 gain at D.C.), the output is peak limited to =h5. Furthermore the average 

 power is nearly S and can be made to approach this by taking / sufl5ciently 

 large. The entropy of the sum of this and the thermal noise can be found 

 by applying the theorem on the sum of a noise and a small signal. This 

 theorem will apply if 



S 

 is sufficiently small. This can be insured by taking — small enough (after 



/ is chosen). The entropy power will be 5 + A^ to as close an approximation 

 as desired, and hence the rate of transmission as near as we wish to 



PFlog 



e4-o 



