Introduction
THE RECENT DEVELOPMENT Of various methods of modulation such
as PCM and PPM which
exchange bandwidth for signaltonoise ratio has intensified
the interest in a general theory of communication. A basis
for such a theory is contained in the important papers of
Nyquist[1] and Hartley[2]
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 the information.
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. The significant 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 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 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.
The logarithmic measure is more convenient for various reasons:
1. It is practically more useful. Parameters of engineering
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
measure 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 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 be called 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 flipflop
circuit, can store one bit of information. N such devices
can store N bits, since the total number of possible
states is 2^{N} and
log_{2}2^{N}
= N. If the base 10 is used the units may be called decimal
digits. Since
log_{2}M
= log_{10}M / log_{10}2

= 3.32 log_{10}M,

a decimal digit is about 3 1/3 bits.
A digit wheel on a desk computing machine has tenstable positions
and therefore has a storage capacity of one decimal digit.
In analytical work where integration and differentiation are
involved the base e is sometimes useful. The resulting
units of information will be called natural units. Change
from the base a to base b merely requires multiplication
by log_{b} a.
By a communication system we will mean a system of the type
indicated schematically in Fig. 1. It consists of essentially
five parts:
1. An information source which produces a message
or sequence of messages to be communicated to the receiving
terminal. The message may be of various types: (a) A sequence
of letters as in a telegraph or teletype system; (b) A single
function of time f(t) as in radio or telephony; (c)
A function of time and other variables as in black and white
television – here the message may be thought of as a
function f(x, y, t) of two space coordinates and time,
the light intensity at point (x, y) and time t on a
pickup tube plate; (d) Two or more functions of time, say
f(t), g(t), h(t) – this is the case in "three
dimensional" sound transmission or if the system is intended
to service several individual channels in multiplex; (e) Several
functions of several variables – in color television
the message consists of three functions f(x, y, t), g(x,
y, t), h(x, y, t) defined in a threedimensional 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 operation 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 properly to construct the signal.
Vocoder systems, television and frequency 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. During transmission, or
at one of the terminals, the signal may be perturbed by noise.
This is indicated schematically in Fig. 1 by the noise source
acting on the transmitted signal to produce the received signal.
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 intended.
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 discrete symbols. A typical case is telegraphy where the
message is a sequence of letters and the signal a sequence
of dots, dashes and spaces. A continuous system is one in
which the message and signal are both treated as continuous
functions, e.g., radio or television. A mixed system is one
in which both, discrete and continuous variables appear, e.g.,
PCM transmission of speech.
We first consider the discrete case. This case has applications
not only in communication theory, but also in the theory of
computing rnachines, the design of telephone exchanges and
other fields. In addition the discrete case forms a foundation
for the continuous and mixed cases which will be treated in
the second half of the paper.
Notes
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.
