A Mathematical Theory of Communication
Reprinted with corrections from The Bell System Technical Journal,
Vol. 27, pp. 379â€“423, 623â€“656, July, October, 1948.
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 intensiï¬ed the interest in a general theory of communication. A
basis for such a theory is contained in the important papers of Nyquist1 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 ï¬nal destination of the information.
The fundamental problem of communication is that of reproducing at one point either exactly or ap-
proximately 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. The signiï¬cant 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 ï¬nite then this number or any monotonic function of this number
can be regarded as a measure of the information produced 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 deï¬nition must be generalized considerably when we consider the inï¬‚uence 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.
The logarithmic 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 in-
tuitively 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 two identical
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 loga-
rithm 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 be called binary digits, or more brieï¬‚y bits, a word suggested by
J. W. Tukey. A device with two stable positions, such as a relay or a ï¬‚ip-ï¬‚op 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
3:32 log10 M;
1Nyquist, H., â€œCertain Factors Affecting Telegraph Speed,â€ Bell System Technical Journal, April 1924, p. 324; â€œCertain Topics in
Telegraph Transmission Theory,â€ A.I.E.E. Trans., v. 47, April 1928, p. 617.
2Hartley, R. V. L., â€œTransmission of Information,â€ Bell System Technical Journal, July 1928, p. 535.
log2 M = log10 M=log10 2