644 BELL SYSTEM TECHNICAL JOURNAL 



S 

 with € -^ as — — > 00 and the channel capacity is obtained by subtracting 



the entropy of the white noise, W log l-weN 



-S + N 

 W log (45 + 2TreN){\ + e) - TF log {lireN) = W log ""^ ^ (1 + e). 



This is the desired upper bound to the channel capacity. 



To obtain a lower bound consider the same ensemble of functions. Let 



these functions be passed through an ideal filter with a triangular transfer 



characteristic. The gain is to be unity at frequency and decline linearly 



down to gain at frequency W. We first show that the output functions 



of the filter have a peak power limitation 5 at all times (not just the sample 



^. , , sin ItWI . . , , ^, 



pomts). First we note that a pulse gomg mto the filter produces 



1 sin irWt 



2 {irWiy 



in the output. This function is never negative. The input function (in 

 the general case) can be thought of as the sum of a series of shifted functions 



sin lirWt 

 ^ lirWt 



where a, the amplitude of the sample, is not greater than v^- Hence the 

 output is the sum of shifted functions of the non-negative form above with 

 the same coefficients. These functions being non-negative, the greatest 

 positive value for any / is obtained when all the coefficients a have their 

 maximum positive values, i.e. -s/S. In this case the input function was a 

 constant of amplitude -s/S and since the filter has unit gain for D.C., the 

 output is the same. Hence the output ensemble has a peak power S. 



The entropy of the output ensemble can be calculated from that of the 

 input ensemble by using the theorem dealing with such a situation. The 

 output entropy is equal to the input entropy plus the geometrical mean 

 gain of the filter; 



flogG^<f/=flog(i^J<i/=-2W' 

 Hence the output entropy is 



W log 45 - 21F = W log ^ 



