# 1948Shannon-a mathematical thgeory of communication.pdf

A Mathematical Theory of Communication By C. E. SHANNON INTRODUCTION T HE recent development of various methods of modulation such as PCM and PPM which exchange bandwidth for signal-to-noise ratio has in- tensified the interest in a general theory of communication. A basis for such a theory is contained in the important papers of Nyquist’ and Hartley* 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 mea 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 in- formation 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 gen- eralized 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 importance * Nyquist, H., “Certain Factors Affecting Telegraph Speed,” BellSysletn Technical Jwr- nol, April 1924, p. 324; “Certain Topics in Telegraph Transmission Theory,” A. I. E. E. Ttans., v. 47, April 1928, p. 617. 1 Hartley, R. V. L., “Transmission of Information,” Bell System Teclanid Journal, July 1928, p. 535. Vol. 27, PP. 379.423, 623.656, July, October, 1948 Copyright 1948 by AMERICAN TBLEPIIONE AND TELEGRAPII Co. Printed in U. S. A. Reissued December. 1957 5 6 C. E. Shannon 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 com- parison 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 bils, 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. iV such devices can store N bits, since the total number of possible states is 2N and log,2N = N. If the base 10 is used the units may be called decimal digits. Since log2 M = log10 M/log102 = 3.32 log,, M, a decimal digit is about 3f bits. A digit wheel on a desk computing machine has ten stable 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 logb 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 iitforntalion source which produces a message or sequence of mes- sages to be communicated to the receiving terminal. The message may be of various types: e.g. (a) A sequence of letters as in a telegraph or teletype system; (b) A single function of time f(l) 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, 1) of two space coordi- nates 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(l), g(l), h(l)-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 A Mathematical Theory of Communication several variables-in color television the message consists of three functions f(x, y, I), g(r, y, I, It (f) Various combinations also occur, for example in television with an associated audio channel. 2. A lransmitter 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 IN~OoRu~k4~10N TRANSMITTER RECEIVERS DESTINATION t - SIGNAL RECEIVED SIGNAL c MESSAGE Ib MESSAGE El NOISE SOURCE Fig. l-Schematic diagram of a general communication system. properly to construct the signal. Vocoder systems, television, and fre- quency modulation are oiher 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 de certain sequences only may be allowed. These will be possible signals for the channel. Thus in telegraphy suppose the symbols are: (1) A dot, consisting of line closure for a unit of time and then line open for a unit of time; (2) A dash, consisting of three time units of closure and one unit open; (3) A letter space consisting of, say, three units of line open; (4) A word space of six units of line open. We might place the restriction on allowable sequences that no spaces follow each other (for if two letter spaces are adjacent, it is identical with a word space). The question we now consider is how one can measure the capacity of such a channel to transmit information. In the teletype case where all symbols are of the same duration, and any sequence of the 32 symbols is allowed the answer is easy. Each symbol represents five bits of information. If the system transmits n symbols per second it is natural to say that the channel has a capacity of 5n bits per second. This does not mean that the teletype channel will always be trans- mitting information at this rate-this is the maximum possible rate and whether or not the actual rate reaches this maximum depends on the source of information which feeds the channel, as will appear later. A Mathematical Theory of Communication In the more general case with different lengths of symbols and constraints on the allowed sequences, we make the following delinition: Definition: The capacity C of a discrete channel is given by where N(T) is the number of allowed signals of duration 7’. It is easily seen that in the teletype case this reduces to the previous result. It can be shown that the limit in question will exist as a finite num- ber in most cases of interest. Suppose all sequences of the symbols Sr , - . . , S, are allowed and these symbols have durations 11, . . . , t, . What is the channel capacity? If N(1) represents the number of sequences of d’uration 1 we have N(t) = N(1 - 11) + N(1 - 12) + + . . + N(1 - 1,) The total number is equal to the sum of the numbers of sequences ending in Sl,SZ, *-* , S, and these are N(1 - 1r), N(1 - is), . . . , N(1 - I~), respec- tively. According to a well known result in finite differences, N(1) is then asymptotic for large I to Xi where X0 is the largest real solution of the characteristic equation: XL’ + xf2 + . . . + X’” = 1 and therefore c = log x0 In case there are restrictions on allowed sequences we may still’often ob- tain a difference equation of this type and find C from the characteristic equation. In the telegraphy case mentioned above N(1) = N(1 - 2) + NO - 4) + N(1 - 5) + N(1 - 7) + N(1 - 8) + N(1 - 10) as we see by counting sequences of symbols according to the last or next to the last symbol occurring. Hence C is - log ~0 where ~0 is the positive root of 1 = c;” -I- l.f4 -I- PK 4 P7 + PUB + P’O* Solving this we find C = 0.539. A very general type of restriction which may be placed on allowed se- quences is the following: We imagine a number of possible states al , a2 , * * . , a,. For each state only certain symbols from the set ’ be the duration of the sul symbol which is allowable in state i and leads to state j. Then the channel capacity C is equal to log W where W is the largest real root of the determinant equation: where 6ij = 1 if i = J’ and is zero otherwise. DASH DOT DASH WORD SPACE Fig. 2-Graphical representation of the constraints on telegraph symbols. For example, in the telegraph case (Fig. 2) the determinant is: -1 (1Y-2 + w-“ (P 4-i Iv-“) (w-” + 1r4 - 1) = O On expansion this leads to the equation given above for this case. 2. TIIE DISCRETE SOURCE OF INFORMATION We have seen that under very general conditions the logarithm of the number of possible signals in a discrctc channel increases linearly with time. The capacity to transmit information can be specified by giving this rate of increase, the number of bits per second required to specify the particular signal used. We now consider the information source. How is an information source to be described mathematically, and how much information in bits per sec- ond is produced in a given source? The main point at issue is the effect of statistical knowledge about the source in reducing the required capacity A Mathematical Theory of Communication 11 of the channel, by the use of proper encoding of the information. In tcleg- raphy, for example, the messages to be transmitted consist of sequences of letters. These sequences, however, are not completely random. In general, they form sentences and have the statistical structure of, say, Eng- lish. The letter E occurs more frequently than Q, the sequence TIS more frequently than XI’, etc. The existence of this structure allows one to make a saving in time (or channel capacity) by properly encoding the mes- sage sequences into signal sequences. This is already done to a limited ex- tent in telegraphy by using the shortest channel symbol, a dot, for the most common English letter E; while the infrequent letters, Q, X, 2, arc rcpre- sented by longer sequences of dots and dashes. This idea is carried still further in certain commercial codes where common words and phrases arc represented by four- or five-letter code groups with a considerable saving in average time. The standardized greeting and anniversary telegrams now in use extend this to the point of encoding a sentence or two into a relatively short sequence of numbers. We can think of a discrete source as generating the message, symbol by symbol. It will choose successive symbols according to certain probabilities depending, in general, on preceding choices as well as the particular symbols in question. A physical system, or a mathematical model of a system which produces such a sequence of symbols governed by a set of probabilities is known as a stochastic process.3 We may consider a discrete source, there- fore, to bc represented by a stochastic process. Conversely, any stochastic process which produces a discrete sequence of symbols chosen from a finite set may be considered a discrete source. This will include such cases as: 1. Natural written languages such as English, German, Chinese. 2. Continuous information sources that have been rendered discrete by some quantizing process. For example, the quantized speech from a PCM transmitter, or a quantized television signal. 3. Mathematical cases where we merely define abstractly a stochastic process which generates a sequence of symbols. The following are ex- amples of this last type of source. (A) Suppose we have five letters A, B, C, D, E which are chosen each with probability .2, successive choices being independent. This would lead to a sequence of which the following is a typical example. BDCBCECCCADCBDDAAECEEA ABBDAEECACEEBAEECBCEAD This was constructed with the use of a table of random numbers.4 a See, for example, S. Chandrasekhar, “Stachastic Problems in Physics and Astronomy,” Rcuicws o Modern Plrysics, v. 15, No. 1, January 1943, p. 1. 4 Ken d all and Smith, “Tables of Random Sampling Numbers,” Cambridge, 1939. 12 C. E. Shannon (B) Using the same five letters let the probabilities be .4, .l, .2, .2, .l respectively, with successive choices independent. A typical message from this source is then: AAACDCBDCEAADADACEDA EADCABEDADDCECAAAAAD (C) A more complicated structure is obtained if successive symbols are not chosen independently but their probabilities depend on preced- ing letters. In the simplest case of this type a choice depends only on the preceding letter and not on ones before {hat. The statistical structure can then be described by a set of transition probabilities pi(j), the probability that letter i is followed by letter j. The in- dices i and j range over all the possible symbols. A second cquiv- alent way of specifying the structure is to give the “digram” prob- abilities p(i, j), i.e., the relative frequency of the digram i j. The letter frequencies p(i), (the probability of letter i), the transition probabilities pi(j) and the digram probabilities p(i, j) are related by the following formulas. PC4 = 7 PCi, j) = 7 P(j) 4 = T P(jPi(4 P(i9.d = PCi)PiW 7 Pi(i) = 7 PC4 = z P(i,j) = 1. As a specific example suppose there arc three letters A, B, C with the prob- ability tables: ;B$+O B++ i B 2~ 27 0 c+ Z. 5’1 , SC , . . . , S, . In addi- tion there is a set of trans