726 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



static discharges. The best type of signal for one channel may be very 

 poor for another. 



In the following sections it is assumed that the channel transmission 

 characteristic is flat in ampUtude and delay over a definite band and 

 zero outside. It is also assumed that the channel has a definite peak signal 

 power limitation, and that the noise is white gaussian noise. Such a 

 channel is no mere academic ideal. It is in fact quite closely approached 

 in practice by many circuits. Moreover, the conclusions based on these 

 assumptions can usually be modified or extended to other actual cases, 

 such as that of noise with non-uniform spectral distribution (as for 

 example the coaxial cable). 



If the bandwidth of the channel is W, we can (using single sideband 

 modulation, if necessary) transmit over it without distortion from fre- 

 quency limitation signals containing frequencies from to W (or —W 

 to W in the Fourier sense). Such a wave can assume no more than 2W 

 independent amphtudes per second. Any set of samples of the wave 



taken at regular intervals -^ serves to specify the wave completely. 



The wave may be thought of as a series of (sin x)/x pulses centered on 

 the samples and of proportional height, and indeed the wave may be 

 reconstructed from the samples in this fashion. This is the well-known 

 sampling theorem . Thus a message source of bandwidth W can supply 

 at most 2W independent symbols (samples) per second, and this same 

 number can be transmitted as overlapping, but independently dis- 

 tinguishable pulses by a circuit of bandwidth W. 



Since, as will appear later, channels which are to transmit signals 

 resulting from efficient statistical encoding must be relatively invailner- 

 able to noise, we shall assume that the pulses on the channel are quan- 

 tized. This allows regenerative repeatering to be used to eliminate the 

 accumulation of noise . If there are b quantizing levels, and if the levels 

 are sufficiently separated so that the probability of noise causing in- 

 correct readings is negligibly small, then the capacity of the channel in 

 bits/sec is^ 



C = 2W log2 h. (1) 



Such a circuit talks in an alphabet of h "letters" and uses a language 

 in which all combinations of these letters are allowed. There are no for- 

 bidden or impossible "words". The circuit has a vocabulary of h one- 

 letter words, h two-letter words, 6" n-letter words. The basic ineffi- 

 ciency in present day electrical communication is that we build circuits 

 with unrestricted vocabularies and then send signals over them which 



