MATHEMATICAL THEORY OF COMMUNICATION 



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several variables — in color television the message consists of three functions 

 /C^'j y^ t), g{^', y, 0, K^, y, defined in a three-dimensional continuum — 

 we may also think of these three functions as components of a vector field 

 defined in the region — similarly, several black and white television sources 

 would produce ''messages" consisting of a number of functions of three 

 variables; (f) Various combinations also occur, for example in television 

 with an associated audio channel. 



2. A transmitter which operates on the message in some way to produce a 

 signal suitable for transmission over the channel. In telephony this opera- 

 tion consists merely of changing sound pressure into a proportional electrical 

 current. In telegraphy we have an encoding operation which produces a 

 sequence of dots, dashes and spaces on the channel corresponding to the 

 message. In a multiplex PCM system the different speech functions must 

 be sampled, compressed, quantized and encoded, and finally interleaved 



INFORMATION 



SOURCE TRANSMITTER 



RECEIVER 



SIGNAL 



D 



MESSAGE 



RECEIVED 

 SIGNAL 



DESTINATION 



MESSAGE 



NOISE 

 SOURCE 



Fig. 1 — Schematic diagram of a general communication system. 



properly to construct the signal. Vocoder systems, television, and fre- 

 quency modulation are other examples of complex operations applied to the 

 message to obtain the signal. 



3. The channel is merely the medium used to transmit the signal from 

 transmitter to receiver. It may be a pair of wires, a coaxial cable, a band of 

 radio frequencies, a beam of light, etc. 



4. The receiver ordinarily performs the inverse operation of that done by 

 the transmitter, reconstructing the message from the signal. 



5. The destination is the person (or thing) for whom the message is in- 

 tended. 



We wish to consider certain general problems involving communication 

 systems. To do this it is first necessary to represent the various elements 

 involved as mathematical entities, suitably idealized from their physical 

 counterparts. We may roughly classify communication systems into three 

 main categories: discrete, continuous and mixed. By a discrete system we 

 will mean one in which both the message and the signal are a sequence of 



