COMPAUISO.N OF SIGNALLIXG ALPHABETS 505 



2. The low-pass Jiltcr 



The second channel is an ideal low-pass filter which attenuates com- 

 plelel}' all t're(iuencies above a cutoff frequency W cycles per second and 

 which passes frequencies below IT witliout attenuation. The channel is 

 supposed capable of handling only signals with average power P or less. 

 Before the signal emerges from the channel, the channel adds to it a 

 noise signal with average poAver N'. The noise is supposed to be white 

 Gaussian noise limited to the fi-equency band | J' 1 < W. The capacity 

 of this channel is 



C = TF]og(^l + ^^) (2) 



l)its per second. 



Shannon's theorems prove that encoding schemes exist for signalling 

 at rates near C vnih. arbitrarily small rates of errors without actually 

 giving a constructive method for performing the encoding. It is of some 

 interest to compare encoding systems which can easily be devised with 

 these ideal systems. In Part I of this paper some schemes for signalling 

 o\er th(^ binary channel will be compared with ideal systems. In Part 

 II the same will be done for the low-pass filter channel. 



Part I 



THE BINARY CHAXXEL 



1. Error-Correcting Alphabets 



Imagine the message source to produce messages which are sequences 

 of letters drawn from an alphabet containing K letters. We suppose that 

 the letters are equally likely and that the letters which the source pro- 

 duces at different times are independent of one another. (If the source 

 given is a finite state source which does not fit this simple description, 

 it can be converted into one which approximately does by a preliminary 

 encoding of the type described in Shannon's Theorem 9.) To transmit 

 the message over the binary channel we construct a new alphabet of 

 7v letters in which the letters are different sequences of binary digits of 

 some fixed length, say D digits. Then the new alphabet is used as an en- 

 coding of the old one suitable for transmission over the channel. For 

 example, if the source produced sequences of letters from an alphabet 

 of 3 letters, a typical encoding with D = b might convert the message 



