# A mathematical theory of communication

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From The Mathematical Theory of Communication, c 1949 by the Board of Trustees of the University of Illinois. Used with the permission of University of Illinois Press.

A Mathematical Theory of Communication
By C. E. SHANNON

HE recent development of various methods of modulation such as PCM and PPM which exchange bandwidth for signal-to-noise ratio has intensiﬁed the interest in a generaltheory of communication. A basis for such a theory is contained in the important papers of Nyquist 1 and Hartley2 on this subject. In the present paper we will extend the theory to include a number of new factors, in particular the effect of noise in the channel, and the savings possible due to the statistical structure of the original message and due to the nature of the ﬁnal destination of theinformation. The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. Thesigniﬁcant aspect is that the actual message is one selected from a set of possible messages. The system must be designed to operate for each possible selection, not just the one which will actually be chosen since this is unknown at the time of design. If the number of messages in the set is ﬁnite then this number or any monotonic function of this number can be regarded as a measure of the informationproduced when one message is chosen from the set, all choices being equally likely. As was pointed out by Hartley the most natural choice is the logarithmic function. Although this deﬁnition must be generalized considerably when we consider the inﬂuence of the statistics of the message and when we have a continuous range of messages, we will in all cases use an essentially logarithmic measure. Thelogarithmic measure is more convenient for various reasons: 1. It is practically more useful. Parameters of engineering importance such as time, bandwidth, number of relays, etc., tend to vary linearly with the logarithm of the number of possibilities. For example, adding one relay to a group doubles the number of possible states of the relays. It adds 1 to the base 2 logarithm of this number.Doubling the time roughly squares the number of possible messages, or doubles the logarithm, etc. 2. It is nearer to our intuitive feeling as to the proper measure. This is closely related to (1) since we intuitively measures entities by linear comparison with common standards. One feels, for example, that two punched cards should have twice the capacity of one for information storage, and twoidentical channels twice the capacity of one for transmitting information. 3. It is mathematically more suitable. Many of the limiting operations are simple in terms of the logarithm but would require clumsy restatement in terms of the number of possibilities. The choice of a logarithmic base corresponds to the choice of a unit for measuring information. If the base 2 is used the resulting units may becalled binary digits, or more brieﬂy bits, a word suggested by J. W. Tukey. A device with two stable positions, such as a relay or a ﬂip-ﬂop circuit, can store one bit of information. N such devices can store N bits, since the total number of possible states is 2N and log2 2N = N. If the base 10 is used the units may be called decimal digits. Since log2 M = log10 M log10 2 = 3 32 log10 M

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1 Nyquist, H., “Certain Factors Affecting Telegraph Speed,” Bell System Technical Journal, April 1924, p. 324; “Certain Topics in Telegraph Transmission Theory,” A.I.E.E. Trans., v. 47, April 1928, p. 617. 2 Hartley, R. V. L., “Transmission of Information,” Bell System Technical Journal, July 1928, p. 535.

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