378 BELL SYSTEM TECHNICAL JOURNAL
S. A. SCHEI,KUNOFF, .B.A., M.A. in Mathematics, The State College of Washington, 1923; Ph.D. in Mathematics, Columbia University, 1928. The Bell System Technical Journal Engineering Department, West~rn Electric Company, 1923-25; Bell Tele phone Laboratories, 1925-26. Department of Mathematics, State College rol. XXVII July, HNS No.3 of . \Vashington, 1926-29. Bell Telephone Laboratories, 1929-. Dr. Schelkunoff has been engaged in mathematical research, especially in the field of electromagnetic theory. A Mathematical Theory of Communication
,V. H. WISE, B.S., Montana State College, 1921; M.A., University of By C. E. SHANNON Oregon, 1923; Ph.D., California Institute of Technology, 1926. American Telephone and Telegraph Company, Department of Development and INTRODUCTION Research, 1926-34; Bell Telephone Laboratories, 1934-. Dr. Wise has been HE recent development of various methods of modulation such as rCM 1 engaged in various theoretical investigations relating to transmission theory. T and PPM which exchange bandwidth for signal-to-noise ratio has in tensitied the interest in a general theory of communication. A basis for l such a theory is contained in the important papers of Nyquist 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 1 the savings possible due to the statistical structure of the original message l and due to the nature of the tinal 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 I entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected iro111 a set of possible messages. The system must be designed to operate for each possible selection, not just the one which will actually i 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 1! 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 inl1uence 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
1 Nyquist, H., "Certain Factors Affecting Telegraph Speed," Bell System Terlmical J,ll/r Ilul, April 1924, p. 324; "Certain Topics in Telegraph Transmission Theory," A. I. E. E. T,ans., v. 47, April 1928, p. 617. 2 Hartley, R. V. L., "Transmission of Information," Bell System Tec/mical JIJunllll, July 1928, p. 535. 379
------.-.:.;...... ;..-~==-.,.-'~--.------.- 3&.l BELL SYSTEJI TECHNICAL JOL'RNAL Jf A THEM. I TiCAL THJ such as time, bandwidth, number of relays, etc., tend to vary linearly with several variables--in color television the message consists of three functions the logarithm of the number of possibilities. For example, adding one relay ! j(.l:; y, t), g(x, y, I), hex, y, t) defined in a three-dimensional continuum to a group doubles the number of possible states of the relays. It adds 1 I we may also think of these three functions as components of a vector tield to the base 2 logarithm of this number. Doubling the time roughly squares ..... ~fined in the region-similarly, several black and white televi$ion sources the number of possible messages, or doubles the logarithm, etc. ,,{ould produce "messages" consisting of a llumber of functions of three 2. It is nearer to our intuitive feeling as to the proper measure. This is variables; (f) Various combinations also occur, for example itl television closely related to (1) since we intuitively measure entities by linear com- ! with an associated audio channel. parison with common standards. One feels, for example, that two punched 2. A trallsmitler which operates on the message in some way to produce a cards should have twice the capacity of one for information storage, and two signal suitable for tra.nsmission over the channel. In telephony this opera identical channels twice the capacity of one for transmitting information. tion consists merely of changing sound pressure into a proportional electrical 3. It is mathematically more suitable. Many of the limiting operations current. In telegraphy we have an encoding operation which prouuces a are simple in terms of the logarithm but would require clumsy restatement in seqpence of dots, dashes and spaces on the channel corresponding to the terms of the number of possibilities. messiige. In a multiplex PCM system the different speech functions must The choice of a logarithmic base corresponds to the choice of a unit for be sampled, compressed. quantized and encoded, and 1Inally interleaved measuring information. If the base 2 is used the resulting units may he called binary digits, or more briedy 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. III such devices can store N bits, since the total number of possible states is 2.v and log22N = N. If the base 10 is MESSAGE used the units may be called decimal digits. Since ME~~AGE log2 M = 10glO M /loglo2 = 3.32 loglo }[, NOISE ,;.'" a decimal digit is about 3! bits. A digit wheel on a desk computing machine SOURCE has ten stable positions and therefore has a storage capacity of one decimal Fig. l--Schematic diagram of a gem'ral communication system. digit. In analytical work where integration and differentiation are involved llroperly to construct the signal. Voeoder systems, television, and fre the base e is sometimes useful. The resulting units of information will he quency modulation are other examples of complex operations applied to the called natural units. Change from the base a. to base b merely requires message to obtain the signal. multiplication by 10gb a. 3. The (hannel is merely the medium used to transmit the signal from By a communication system we will mean a system of the type indicated transmitter to receiver. It may be a pair of wires, a coaxial cable, a baml of schematically in Fig. 1. It consists of essentially five parts: radio frequencies, a beam of light, etc. 1. An illformation source which produces a message or sequence of mes 4. The receiver ordinarily performs the inverse operation of that done by sages to be communicated to the receiving temlinaL The message may he the transmitter, reconstructing the message from the signal. of various types: e.g. (a) A sequence of letters as in a telegraph or teletype S. The destination is the person (or thing) for whom the message is in system; (b) A single function of time j(t) as in radio or telephony; (c) A tended. function of time and other variables as in black and white television--here We wish tu consider certain general problems involving cummunicatiun the message may be thought of as a function j(x, y, t) of two space coordi systems. To do this it is first necessary to represent the various clements nates and time, the light intensity at point (x, y) and time t on a pickup tube involved as mathematical entities, suitably idealized from their physical plate; (d) Two or more functions of time, say jet), g(t), h(t)-this is the counterparts. \:Ve may roughly classify communication systems into three case in "three dimensional" sound transmission or if the system is intended main categories: discrete, continuous and mixt(l. By a discrete system we to sen'ice several individual channels in multiplex; (e) Several fUllctions of will mean one in which both the message and the signal are a sequence of 382 BELL SYSTEM TECIl.:VICAL JOURNAL /IIATHEJI.1TfCAL THEORI- OF COMMUNiCATiON 383 discrete symbols. A typical case is telegraphy where the message is a In the more general case with different lengths of symbols and constraints sequence of letters and the signal a sequence of dots, dashes and spaces. on the allowed sequences, we make the following definition: A continuous system is one in which the message and signal are both treated Definition: The capacity C of a discrete channel is given by as continuous functions, e.g. radio or television. A mixed system is one in which both discrete and continuous variables appear, e.g., PCM transmis C = Lim l_og-i'\l:_(J~) sion of speech. T ->co \Ve first consider the discrete case. This case has applications not only where N (T) is the number of allowed signals of duration T. in communication theory, but also in the theory of computing machines, It is easily seen that in the teletype case this reduces to the previous the design of telephone exchanges and other fields. In addition the discrete result. It can be showll that the limit in question will exist as a finite num case forms a foundation for the continuous and mixed cases which will be ber in most cases of interest. Suppose all sequences of the symbols S\ , ... , treated in the second half of the paper. Sn are allowed and these symbols have durations 11 , ... , in. What is the channel capacity? If N(t) represents the number of sequences of duration PART I: DISCRETE NOISELESS SYSTEMS t we have 1. THE DISCRETE NOISELESS CHANNEL N(t) = N(t - 11) + N(t - 12) + ... + N(t - In) Teletype and telegraphy are two simple examples of a discrete channel The total number is equal to the sum of the numbers of sequences ending in for transnllttmg mtormation. Generally, a discrete channel will mean a 51, S2 , ... ,Sn and these are NO - II), N(t - 12), ••• , N(t - In), respec system whereby a sequence of choices from a finite set of elementary sym tively. According to a well known result in finite differences, N(t) is then bols Sl ... S n can be transmitted from one point to another. Each of the asymptotic for large t to X~ where Xo is the largest real solution of the J symbols S; is assumed to have a certain duration in time Ii seconds (not characteristic equation: ~ " necessarily the same for different S i , for example the dots and dashes in X-tl + r t2 + ... + x-t,. = 1 ~ telegraphy). It is not required that all possible sequences of the Si be cap ~ able of transmission on the system; certain sequences only may be allowed. and therefore ~ These will be possible signals for the channel. Thus in telegraphy suppose C = log Ko the symbols are: (1) A dot, consisting of line closure for a unit of time and In case there are restrictions on allowed sequences we may still often ob then line open for a unit of time; (2) A dash, consisting of three time units tain a difference equation of this type and find C from the characteristic of closure and one unit open; (3) A letter space consisting of, say, three units equation. In the telegraphy case mentioned above of line open; (4) A word space of si.x units of line open. We might place the restriction on allO\\'able sequences that no spaces follow each other (for N(t) = N(t - 2) + N(t - 4) + N(I - 5) + N(t - 7) + N(I - 8) if two letter spaces are adjacent, it is identical with a word space). The + N(t - 10) question we now consider is how one can measure the capacity of such a channel to transmit information. as we see by counting sequences of symbols according to the last or next to In the teletype case where all symbols are of the same duration, and any the last symbol occurring. Hence C is - log p.o where JAo is the positive sequence of the 32 symbols is allowed the answer is easy. Each symbol root of 1 = fJ.2 + p.4 + fJ.5 + fJ.7 + fJ.8 + p.lO. Solving this we find C = 0.539. represents five bits of information. If the system transmits n symbols , A very general type of restriction which may be placed on allowed se per second it is natural to say that the channel has a capacity of Sit bits per ~!. quences is the following: We imagine a number of possible states al ,a2 , ... , second. This does not mean that the teletype channel will always be trans am. For each state only certain symbols from the set Sl , ... , Sn can be mitting information at this rate--this is the maximum possible rate and transmitted (different subsets for the different states). When one of these whether or not the actual rate reaches this maximum depends on the source has been transmitted, the state changes to a new state depending both on of information which feeds the channel, as will appear later. the old state and the particular symbol transmitted. The telegraph case is a simple example of this. There are twu states depending on whether or not ..... 38-1- BELL SrSl'EM TECHNICAL JUURNAL JJA TllEMA TlctJ. TiIA'ORJ" OF ('OM.lJ{'NICAnON 385 a space was the last symbul transmitted. Ii so then only a dot or a dash of the channel, by the use of proper encuding of the information. In teleg can be sent next and the state always changes. If not, any symbol can be raphy, for example, the messages to be transmitted consist of sequences transmitted and the state changes if a space is sent, otherwise it remains of letters. These sequences, however, are not completely random. In the same. The conditions can be indicated in a linear graph as shown in general, they form sentences and have the statistical structure of, say, Eng Fig. 2. The junction points correspond to the states and the lines indicate lish. The letter E occurs mure frequently than Q, the sequence TH more the symbols possible in a state and the resulting state. In Appendix I it is frequently than XP, etc. The existence of this structure allows one to shown that if the conditions on allowed sequences can be described in this make a saving in time (or channel capacity) by properly encoding the mes form C will exist and can be calculated in accordance with the following sage sequences into signal sequences. This is already done to a limited ex result: tent in telegraphy by using the shortest channel symbol, a dot, for the most Theorem 1: Let bi;) be the duration of the sth symbol which is allowable in common English letter E; while the infrequent letters, Q, X, Z are repre state i and leads to state J. Then the channel capacity C is equal to log sented by longer sequences of dots and dashes. This idea is carried still 11' where H' is the largest real root of the determinant equation: further in certain conunercial codes where common words and phrases are represented by four- or five-letter code groups with a considerable saving in L.....J'"' lV-b,)(,,) - Ojj I o. I average time. The standardized greeting and anniversary telegrams now to where Oil 1 if i = j and is zero otherwise. in use extend this the point of encoding a sentence or two into a relatively short se4uence of numbers. DASH We can think of a discrete source as generating the message, symbol by symbol. It will choose successive symbols according to certain probabilities DOT 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 DASH produces such a sequence of symbols governed by a set of probabilities is 3 WORD SPACE known as a stochastic process. We may consider a discrete source, there Fig. 2--Graphical representation of the constraints on telegraph symbols. fore, to be represented by a stochastic pro::ess. Conversely, any stochastic process which produces a discrete sequence of symbols chosen from a finite For example, in the telegraph case (Fig. 2) the determinant is: set may be rO:lsidered a discrete source. Tbis will include such cases as: 2 1. Natural written languages such as English, tierman, Chinese. - 1 Or- + 1l,-4) I 2. Continuous information sources that have been rendered discrete by some I = 0 (Jr---;; + Jr'-- G) (TV-:! + W-4 - 1) I quantlzlIlg process. For example, the quantized speech from a peM On expansion this leads to the equation given above for this case. transmitter, or a quantized t60levision signal. 3. Mathematical cases where we merely define abstractly a stuchastic 2. TIlE DISCRETE SOURCE OF INFORMATIOK process which generates a sequence of symbols. The following are ex We have seen that under very general conditions the logarithm of the amples of this last type of source. number of possible signals in a discrete channel increases linearly with time. (A) Suppose we have five letters A, B, C, D, E which are chosen each The capacity to transmit information can be specified by giving this rate of with probability .2, successive choices being independent. This increase, the number of hits per second required to specify the particular would lead to a sequence of which the folluwing is a typical example. signal used. RDCBCECCCADCBDDAAECEEA We now consider the information source. How is an information source ABBDAEECACEEBAEECBCEAD to be described mathematically, and how much information in bits per sec This was constructed with the use of a table of random numbers.' ~ ond i~ produced in a given source? The main point at issue is the effect of 3 See, for example, S. Chandrasekhar, "Stochastic Problem~ in Physics and Astronomy," statistical knowledge about the sourCe in reducing the required capacity Ret'ie-,,'s 4 JJ odern PIi_vsics, v. 15, No.1, January 19-B. p. 1. I Kendall and Smith. "Tables of Random Samplin~ Numbers," ('alllbrid~('. 1939. 386 BELL SYSTEM TECHNICA]. 10URNAL JUTHEMATICAL THEORr OF COJfMUN1CAT10N 387 (B) Using the same tive letters let the probabilities be .4, .1, .2, .2,.1 abilities piJ(k) would be required. Continuing in this way one ob respectively, with successive choices independent. A typical tains successively more complicated stochastic processes. In the message from this source is then: general ll-graIll case a set of 1l-gram probabilities PUI , i 2 , ••• , iT.) • ;,,-1 AAACDCBDCEAADADACEDA or of transition probabilities Pi l iz ..... (i,.) is required to EADCABEDADDCECAAAAAD specify the statistical structure. (C) A more complicated structure is obtained if successive symbols are (D) Stochastic processes can also be defined which produce a text con not chosen independently but their probabilities depend on preced sisting of a sequence of "wonls." Suppose there are rive letters ing letters. In the simplest case of this type a choice depends only A, B, C, D, E and 16 "words" in the language with associated on the preceding letter and not on ones before that. The statistical probabilities: structure can then be described by a set of transition probabilities .10 A ..16 BEBE .11 CABED .04 DEB Pi(j), the probability that letter i is followed by letter j. The in .04ADEB .04 BED .05 CEED .15 DEED dices i and j range over all the possible symbols. A second equiv .05 ADEE .02 BEED .08 DAB .01 EAB alent way of specifying the structure is to give the "digram" prob .01 BADD .05 CA .04 DAD .05 EE abilities p(i, j), i.e., the relative frequency of the digram i j. The letter irequencies pCi), (the probability of letter i), the transition Suppose successive "words" are chosen independently and are probabilities Pi(j) and the digram probabilities P(i, j) are related by separated by a space. A typical message might be: the following formulas. DAB EE A BEBE DEED DEB ADEE ADEE EE DEB BEBE BEBE BEBE ADEE BED DEED DEED CEED ADEE A DEED p(i) = L p(i, j) = L p(j, i) L p(j)Pi(i) DEED BEBE CABED HEBE BED DAR DEED ADEB j i If all the words are of [mite length this process is equivalent to one p(i, j) = P(i)Pi(j) of the preceding type, but tIlt' description may be simpler in terms of the word structure and probabilities. \Ve may also generalize L p.;(j) = L p(i) = L p(i, j)' = l. ii i.i here and introduce transition probabilities between words, etc. These artificial languages are useful in constructing simple problems and As a specitic example suppose there are three letters A, B, C with the prob examples to illustrate various possibilities. We can also approximate to a ability tables: natural language by means of a series of simple artiflcial languages. The zero-order approximation is obtained by choosing all letters with the same Pi(j) I .I p(i) p(i, j) j I probability and independently. The first-order approximation is obtained A B C A B C ____ 0 __ ~---.------by choosing successive letters independently but each letter having the J_ \I 4 1 5 A 0 5" 5 A --;.-f A 0 Yi, T"K same probability that it does in the natural language. Thus, in the first order approximation to English, E is chosen with probability .12 (its fre 1 1 16 ~ B 0 B B -2-78 0 2 2 -2-"- 27 quency in nomlal English) and W with probability .02, but there is no in I 2 1 1 4 ] C i 5 To Clf C :i'f Y~-5- r35 fluence between adjacent letters and no tendency to form the preferred digrams such as TH, ED, etc. In the second-order approximation, digram A typical message from this source is the following: structure is introduced. After a letter is chosen, the next one is chosen in ABBABABABABABABBBABBBBBAB accordance with the frequencies with which the various letters follow the ABABARABBHACACABBABBBBABB first one. This requires a table of digram frequencies Pi(j). In the third ABACBBBABA order approximation, trigram structure is introduced. Each letter is chosen The next increase in complexity would involve trigram frequencies with probabilities which depend on the preceding two letters. but no more. The choice of a letter would depend on the preceding two letters but not on the message before that point. A set of tri • Letter, digram and trigram irequencies are given in "Secret and Urgent" Ly FlctclllT Pratt, Blue Ribbon Books 1939. Word frequencies are taLulated in "Relative Frequency gram frequencies pU, j, k) or equivalently a set of transition prob- of English Speech Sounds," G. Dewey, Harvard University Press, 1923. 31)8 l;[ BELL SY:·,TE.. TECli"Vll'AL .TOFRXAL J-[ATH.E:'vfATfCA.!. THEORl" OF COMMUNICATION 3o~ 3. THE SERIES OF ApPl{OXLU.'l.TIONS TO ENGUSH words can easily he placed in sentences without unusual or strained con To give a visual idea of how this series of J>rocesses approaches a language, structions. The particular sequence of ten words "attack on an English typical sequences in the approximations to English have been constructed writer that the character of this" is 110t at all unreasonable. It appears and are given below. In all cases we have assumed a 27-symbol "alphabet," then that a sufficiently complex stochastic process will give a satisfactory the 26 letters and a space. representation of a discrete source. 1. Zero-order approximatioll (symbols iJl(lependem and equi-probable). The tirst two samples were constructed by the use of a book of random XF()ML RXKHRJFFJUJ ZLPWCFWKCYJ numbers in conjunction with (for example 2) a table of letter frequencies. FFJEYVK( \~SGXYD ()PAAMKBZAACIRZLHJQJJ This method might have been continued for (3), (-1-), and (5), since digram, 2. First-order apl>roximation (symbols independent but with frequencies trigram, and word frequency tables are available, but a simpler equivalent of English text). method was used. To construct (3) for example, one opens a book at ran neRO HLI RCWR :NMl.ELWIS EU LL NBNESEHYA TH EEL dom and selects a letter at random on the page. This letter is recorded. ALHENHTTPA OOBTTVA NAH BRL The book is then opened to another page and one reads until this letter is 3. Second-order approximation (digrani structure as in English). enrountered. The succeeding letter is then recorded. Turning to another ON IE ANTSOLJTI~YS ARE T INCTORE ST BE S DEAMY page this second letter is searched for and the succeeding letter recorded, ACHIN D ILONASIVE TUCOO\\'E AT TEASONARE FFSO etc. A similar pn>cess was used for (-1-), (5), and (6). It would be interest TIZI\" ANDY TOBE SEACE CTISBE ing if further approximations could be cOllstructed, but the labor involved -1-'. Third-order approximation (trigram structure as in English). becomes enormous at the next stage. IX NO 1ST LAT WHEY CRATICT FROURE BIRS GROCID PUNDENOME OF DEMONSTFRES OF THE REPTAGIK IS 4. (;RAPIHCAL REPRESENTATWI\ m' A MARKOPF PR()CES~ REGOACTIONA OF eRE Stochastic processes of the type described above are known mathe 5. First-Order Word Approximation. Rather than continue with tetra matically as discrete Markoff processes and have been extensively studied in gram, ... , Il-gram structure it is easier and better to jump at this the literature.6 The general case can be described as follows: There exist a point to word units. Here words are chosen independently but with finite number of possible "states" of a system; SI , S2 , ... , Sn. In addi their appropriate frequencies. tion there is a set of transition probabilities; Pi(j) the probability that if the REPRESENTIN(~ A~D SPEEDILY IS A1\ (~OOD APT OR system is in state S i it will next go to state Sj. To make this Markoff COME CAN DIFFERENT NATURAL HERE HE THE A IX process into an information source we need only assume that a letter is pro CAME THE TO OF TO EXPERT GRAY COME TO FUR duced for each transition from one state to another. The states will corre NISHES THE LI~E MESSAGE HAD BE THESE. spond to the ':residue of influence" from preceding letters. 6. Second-Order Word Approximation. The word transition probabil- The situation can be represented graphically as shown in Figs. 3, 4 and 5. ities are correct but no further structure is included. The "states" are the junction points in the graph and the probabilities and THE HEAD AND I~ FRONTAL ATTACK ON AN ENGLISH letters produced for a transition are given beside the corresponding line. WRITER THAT THE CHARACTER OF THIS POINT IS Figure 3 is for the example B in Section 2, while Fig. -1- corresponds to the THEREFORE AXOTHER METHOD FOR THE LETTERS example C. In Fig. 3 there is only one state since successive letters are THAT THE TIME OF WHO E\'ER TOLD THE PROBLEM independent. In Fig. -1- there are as many states as letters. If a trigram FOR AX UNEXPECTED example were constructed there would be at most Jl2 states corresponuing The re~elllblance to ordinary English text increases quite noticeably at to the possible pairs of letters preceding the OIle being chosen. Figure 5 each of the above steps. Note that these samples have reasonably good is a graph for the case of word structure in example D. Here S corresponds structure out to about twice the range that is taken into account in their to the ':space" symbol. constructioll. Thus in (3) the statistical process insures reasonable text o For a detailed treatment see M. Frl'chet. ,. Methods des iunctions arhitraircs. Tlworie for two-letter sequence, but four-letter sequences from the sample can des enenements en chaine oans It' cas o'un nimllJre fini o'etats possilJlcs" Paris, Gauthi('r usually be iitted into good sentences. In (6) sequences of four or more Yillars. 1938. 390 BELL SYSTEM TECHNICAL JOURNAL MATIIEMATICAL THEOl\Y OF COMMUN1CIT10N 391 5. ERGODIC AND MIXED SOURCES ess is the same in statistical properties. Thus the letter frequencies, As we have indicated above a discrete source for our purposes can be con digram frequencies, etc., obtained from particular sequences will, as the sidered to be represented by a Markoff process. Among the possible discrete lengths of the sequences increase, approach defmite limits independent of Markoff processes there is a group with special properties of significance in the particular sequence. Actually this is not true of every sequence but the set for which it is false has probability zero. Roughly the ergodic property !4 A.lB means statistical homogeneity. All the examples of artiticial language:-. given above arc ergodic. This property is related to the structure of the corresponding graph. If the graph E .2 7 .1 C has the following two properties the corresponding process will be ergodic: 1. The graph does not consist of two isolated parts A and B such that it is ¥o .2 impossible to go from junction points in part A to junction points in Fig. 3-A graph correbponding to the source in example B. part B along lines of the graph in the direction of arrows and also im possible to go from junctiolls in part B to junctions in part A. 2. A closed series of lines in the graph with all arrO\vs on the lines pointing in the same orientation will be called a '·circuit." The "length" of a circuit is the number of lines in it. Thus in Fig. 5 the series BERES is a circuit of length 5. The second pruperty required is that the c greatest common divisor uf the lengths of all circuits in the graph be Fig. 4-A graph corresponding to the source in example C. one. n the first condition is satisfied but the secoml one violated by having the greatest common divisor equal to d > 1, the sequences have a certain type of periodic structure. The various sequences fall into d different classes which are statistically the same apart from a shift of the origin (i.e., which letter ill the sequence is called letter 1). By a shift of from 0 up to d - 1 any sequence can be made statistically equivalclIt to any other. A simple example with d = 2 is the following: There are three possihle letters a, b, c. Letter a is followed with either b or c with probabilities 1 and 5 respec tively. Either b or c is always followed by letter <1. Thus a typical sequence IS abacacacabacababacac This type of situation is not of much importance for our work. If the first condition is violated the graph may be separated into a set of subgraphs each of which satisfies the first condition. We will assume that I the second condition is also satisfied for each suhgraph. We have in this case what may be called a "mixed" source made up of a number of pure Fig. 5-A graph corrt:sponding to the source in example D. components. The components correspond to the various suhgraphs. t H L 1 , L2 , L.3 , ••• are the component sources we may write communication theory. This special class consists of the "ergodic" proc ~ esses and we shall call the corresponding sources ergodic sources. Although L = PILI + PtL 2 + PaLa + ... a rigorous definition of an ergodic process is somewhat involved, the general idea is simple. In an ergodic process every sequence produced by the proc- where Pi is the probability of the component source Li . 7 These are restatements in terms of the graph of conditions given in Freehet. ...... _--"-"-"" 392 BELL SYSTE.M TECHNICAL JOURI'lAL JJATHEMAT/CAJ. T1IFDkl" OF COMMUN1CATION- 393 Physically the situation represented is this: There are several different function of /l. With equally likely events there is more choice j or un sources 1.1 , L2 , L,l , ... which are each of homogeneous statistical structure certainty, when there are more possible events. (i.e., they are ergodic). vVe do not know a priori which is to be used, but 3. If a choice be broken down into two successive choices, the original once the sequence starts in a given pure component L i it continues indefi H should be the weighted sum of the individual values of H. The nitely according to the statistical structure of that component. meaning of this is illustrated in Fig. o. At the left we have three As an example one may take two of the processes defined ahove and possibilities PI = L h = ~, p:; =~. On the right \ve first choose be assume PI = .2 and P2 = .8. A sequence from the mixed source tween two possibilities each with probability ~, and if the second occurs make another choice ""ith probahilities ~, ~. The tinal results have L = .2 1.1 + .8 L2 the same probabilities as hefore. We require, in this special case, would be obtained by choosing first 1.1 or 1.2 with probabilities .2 and .S that and after this choice generating a sequence from whichever was chosen. H(~, L ~) = H(L!) + ~H(~, A) ,.,.- Except when the contrary is stated we shall assume a source to be ergodic. This assumption enables one to identify averages along a sequence with The coetticient .~ is because this second choice only occurs half the time. ,( averages over the ensemble of possible sequences (the probability of a dis crepancy being zero). For example the relative frequency of the letter A /2 1/2 in ,a particular infinite sequence will be, with probability une, equal to its relative frequency in the ensemble of sequences. 13 If P; is the probability of state i and Pi(j) the transition probability to state j, then for the process to be stationary it is dear that the Pi must 1/6 satisfy equilihrium conditions: ...... 39-1 BELL SYSTKM TECHl{]CAL JOUR... VAL MATHEMATlCAL THEURI" OF COMMUNICATION 395 i PI, "', p". If X is a chance variable we will write H(x) for its entropy; while thus x is not an argument of a function but a label for a number, to differen tiate it from H(y) say, the entropy of the chance variable y. H(x) = - L p(i, j) log L: p(~' j) i.i i, The entropy in the case of two possibilities with probabilities p and q = 1 - p, namely I H(y) = - L: p(i, j) log L: p(i, j). 1 '.j H = - (p log P + q log q) It is easily shown that is plotted in Fig. 7 as a function of p. The quantity H has a number of interesting properties which further sub H(x, y) ~ H(x) + H(y) I stantiate it as a reasonable measure of choice or information. with equality only if the events are independent (i.e., p(i, j) = pU) p(j»). The uncertainty of a joint event is less than or equal to the sum of the individual uncertainties. 1.0 4. Any change toward equalization of the probabilities PI , Pz, ... , pn .9 increases II. Thus if PI < h and we increase PI, decreasing pz an equal amount so that PI and h are more nearly equal, then H increases. More .8 generally, if we perform any "averaging" operation on the Pi of the form .7 p~ = L aij Pi H .6 j BITS .5 where L aij = L au = 1, and all aij :2: 0, then H increases (except in the • i .41 special case where this transformation amounts to no more than a permuta tion of the pj with H of course remaining the same) . .3 ." ""If 5. Suppose there are two chance events x and y as in 3, not necessarily .2 independent. For any particular value i that :r can assume there is a con ditional probability pJj) that y has the value j. This is given by , . pC i, j) 00 PiU) = L-jc'i, j) . .1 .2 .3 A .5 .6 .7 .8 .9 1.0 P i Fig. i-Entropy in the case of two possibilities with probabilities p and (1 _ p). We define the conditiollal entropy of y, H z(y) as the average of the cntropy of y for each value of x, weighted according to the probability of getting 1. 11 = 0 if and only if all the but one are zero, this one having the Pi that .particular x. That is value unity. Thus only when we are certain of the outcome does H vanish. Othenvise II is positive. FIx(y) = - L: p(i, j) log pJj). i.i 2. For a given 1Z, H is a maximum and equal to log Jl when all the Pi are This quantity measures how uncertain we are of y on the average when we equal ( i.e., This is also intuitively the most uncertain situation. ~). know x. Substituting the value of p,(j) we obtain 3. Suppose there are two events, :r and y, in question with m possibilities Ilx(Y) = - 2.: p(i, j) log p(i, j) + L p(i, j) log L p(i, j) for the first and n for the second. Let p(i, j) be the probability of the joint i j ij j occurrence of i for the first and j for the second. The entropy of the joint ii event is = H(x, y) - H(x) H(x, y) = - L pCi, j) log pCi, j) or "1' H(x, y) = H(x) + H .(y) ~. ''"-_.-. '" ... _-.,. .--- 396 BELL SYSTEM TECH.V1CAL JOFR,VAL JfATH£:1IAIHAL IHAORF OF COMMUNiCATION 39i The uncertainty (or entropy) of the joint event x, y is the uncertainty of x log P =-'= ;V L Pi log Pi plus the uncertainty of y when x is known. i 6. From 3 and 5 we have log P == -NIl H(x) + HCy) ~ H(.r, y) = H(x) + H x(Y) 11 -± It'll 1/P Hence .'Ii H(y) ~ H x(Y) II is thus approximately the logarithm of the reciprocal probability of a The uncertainty of y is never increased by knowledge of x. It will be de typical long sequence divided by the number of symbols in the sequence. creased unless x and yare independent events, in which case it is not changed. The same result holds for any source. Stated more precisely we have (see Append ix III): ... 7. THE ENTROPY ()F .\:\ Il\:FORMATIOl\ SOURCE Theorem 3: Given any E > 0 and 0 > 0, we can find an No such that the se ) Consider a discrete source of the ti.nite state type considered above. quences of any length N ~ N u fall into two classes: 1. A set whose total probability is less than E. ~ For each possible state i there will be a set of probabilities Pi(j) of pro 'oj ducing the various possible symbols j. Thus there is an entropy Hi for 2. The remainder, all of whose members have probabilities satisfying tl1(' .,;J each state. The entropy of the source will be defined as the average of inequality ~ H' = mH Lim l~~g n(q) 11 N~oo 5/ - where In is the average number of symbols produced per second. H or H' measures the amount of information generated by the source per symbol when tj does 110t equal 0 or 1. or per second. If the logarithmic base is 2, they will represent bits per We may interpret log 1l(1/) as the number of bits required to specify the symbol or per second. sequence when we consider only the most probable sequences with a total If successive symbols are independent then H is simply - ~ Pi log P,· probability q. Then 1?if,7v'!....(ql is the number of bits per symbol for the where Pi is the probability of symbol i. Suppose in this case we consider a long message of N symbols. It will contain with high probability about speCllIcation. The theorem says that for large iV this will be independent of PIN occurrences of the first symbol, hN occurrences of the second, etc. q and equal to H. The rate of growth of the logarithm of the number of Hence the probabili ty of. this particular message will he roughly reasonably probable sequences is given by H, regardless of our interpreta tion of ';reasonahly probable." ()ue to tllese results, which are proved in P = til.\' tr.v· .. P;.",\ appendix III, it is possible for most purposes to treat the long sequences a:;. or 1lX llN though there were just 2 of them, each with a probability 2-- • 39/) BELL SYSTEM TECHNICAL JOURNAL MATHEMATICAL TlIJ.~URY OF COMMUNICATiON 39tJ The next two theorems show that Hand H' can be determined by limit ~ancy of ordinary English, not considering statistical structure over greater ing operations directly from the statistics of the message sequences, without. Ilistances than about eight letters is roughly 50vicJ. This means that when reference to the states and transition probabilities between states. we write English half of what we write is determined by the structure of the Theorem 5: Let p(B.) be the probability of a sequence Bi of symbols from language and half is chosen freely. The fIgure 50% was found by several the source. Let independent methods which all gave results in this neighborhood. One is by calculation of the entropy of the approximations to English. A second &.v = - ~ ~ p(B;) log pCB;) method is to delete a certain fraction of the letters from a sample of English text and then let someone attempt to resture them. If they can be re where the sum is over all sequences B. containing N symbols. Then· G N stored when 50% are deleted the redundancy must be greater than 50%. is a monotonic decreasing function of Nand .\ third method depends on certain known results in cryptography. ITwo extremes of redundancy in English prose are represented by Basic Lim G... = 11. N-+ and Lim PN H. 8. REPRESENTATION OF THE ENCODING AND DECODING OPERATIONS N-+ better approximation. In fact F N is the entropy of the Nth order approxi transducer is a sequence of input symbols and its output a sequence of out mation to the source of the type discussed above. If there are no statistical put symbols. The transducer may have an internal memory so that its iniluences extending over more than N symbols, that is if the conditional output depends not only on the present input symbol but also on the past probability of the next symbol knowing the preceding (N - 1) is not history. We assume that the internal memory is finite, i.e. there exists changed by a knowledge of any before that, then FIr = H. Ff'/ of course is a finite number m of possible states of the transducer and that its output is the conditional entropy of the next symbol when the (N - 1) preceding a function of the present state and the present input symbol. The next ones are known, while G ".. is the entropy per symbol of blocks of N symbols. state will be a second function of these two quantities. Thus a transducer The ratio of the entropy of a source to the maximum value it could have can be described by two functions: while still restricted to the same symbols will be called its Tela/hie e1ltropy. Yn = j(:Xn , a,,) This is tlle maximum compression possible when we encode into the same alphabet. One minus the relative entropy is the redundancy. The redun- an+l g(xn , an) ...... J BELL SY,'iTEM TECHNICAL JOURNAL J/ATHEJIATICAL TllEORr OF COM1~IUNIC'AT10:Y -lOl where: Xn is the n tl• input symbol, By proper assigllllent of the transition probabilities the entropy of sym a" is the state of the transducer when the ,/" input symbol is introduced, bols on a channel can be maximized at the channel capacity. Y1l is the output symbol (or sequence of output symbols) produced when x" is introduced if the state is Ct". 9. THE FUNDAMENTAL THEOREM FOR A NOISELESS CHANNEL If the output symbols of one transducer can be identified with the input We will now justify our interJ)retation oi H as the rate of generating symbols of a second, they can be connected in tanJem and the result is also information by proving that 11 determines the channel capacity required a transducer. If there exists a second transJucer which operates on the out with most efficient coding. put of the 1irst and recovers the original input, the first transducer will be Theorem 9: Let a source have entroJ)Y H (bits per symbol) and a channel called non-singular and the second will be called its inverse. have a capacity C (bits per second). Then it is possible to encode the output Theorem 7: The output of a finite state transducer driven by a finite state . h' . C; ot t e source 111 such a way as to transmIt at. the average rate-- - f symbols statistical source is a finite state statistical source, with entropy (per unit II time) less than or equal to that of the input. If the transducer is non per second over the channel where f is arbitrarily small. It is not possihle singular they are equal. C to transmit at an average rate greater than -- . Let a represent the state of the source, which produces a sequence of II symbols Xi ; and let (3 be the state of the transducer, which produces, in its C The converse part of the theorem, that Ii cannot be exceeded, may be ~ output, blocks of symbols Yi. The combined system can be represented i by the <'product state space" of pairs (a, p). Two points in the space, proved by noting that the entropy uf the channel input per second is equal i (aI, (31) and (a2 #2), are connected by a line if al can produce an x which to that of the source, since the transmitter must be non-singular, and also ~ changes 131 to ;:h , and this line is given the probability of that x in this case. this entropy cannot exceed the channel capacity. Hence H' ~ C and the The lille is labeled with the block of Yi symbols produced by the transducer. number of symbols per second = H'/11 ~ C/H. The entropy of the output can be calculated as the weighted sum over the The first part of the theurem will be proved in two different ways. The states. If we sum first on (3 each resulting term is less than or equal to the first method is to consider the set of all sequences of.v symbols produced by corresponding term for a, hence the entropy is not increased. If the trans the source. For N large we can divide these into two groups, one containing H ducer is non-singular let its output be connected to the inverse transducer. less than i/{-+-~I.\ members and the second containing less than 2 l\' members 1 If H~ . 11; and H~ are the output entropies of the source, the first and (where R is the logarithm of the number of different symbols) and having a second transducers respectively, then ll~ ~ H; ~ H; = H; and therefore total probability less than J.I.. As N increases 'rJ and J.I. approach zero. The H~ = H~. number of signals of duration T in the channel is greater than iC-IJ) T with Suppose we have a system of constraints on possible sequences of the type 8 small when T is large. If we choose which can be represented by a linear graph as in Fig. 2. If probabilities p~}) were assigned to the various lines connecting state i to state j this would T = (~+ x) N become a source. There is one particular assignment which maximizes the resulting entropy (sec Appendix IV). then there will be a sutlicient number of sequences of channel symbols for Theorem 8: Let the system of constraints considered as a channel have a the high probability group when Nand T are sufficiently large (however capacity C. If we assign small X) and also some additional ones. The high probability group is coded in an arbitrary one to one way into this set. The remaining sequences (.j B j c-t(~) Pij = -- 'I Bi are represented by larger sequences, starting and ending with one of the I sequences not used for the high prohability group. This special sequence where f,~;) is the duration of the sth symbol leading from state i to state j acts as a start and stop signal for a different cocle. In between a sufficient and the R i satisfy time is allowed to give enough different sequences for all the low probability H,. = L B) c-t~j) messages. This will require I -'.j ~ then H is maximized and equal to C. 1\ = (~+ ~) LV .,l 402 BELL SYSTEAI TECHNICAL JOZ,'RNAL MATHEMATICA/~ THEORl" OF COlvIAIUI'{lCATION 403 where rp is small. The mean rate of trallsmission in message symbols per second will then be greater than - r.p, 10).." p" ::; H' < N 'i.ps log ps 1 [. (1 - 0) NT + 0 NTJ-l = [(1 - 0) (11)c + A + 0 ('RC + rp )J-l As N increases -'i:,ps log ps approaches H, the entropy of the source and H' approaches H. We see from this that the inefficiency in coding, when only a finite delay of As .:.V increases 0, A and rp approach zero and the rate approaches ~ . N symbols is used, need not be greater than ~~ plus the dit1erence between Another method of performing this coding and proving the theorem can be described as follows: Arrange the messages of length N in order of decreas the true entropy H and the entropy GN calculated for sequences of length N. ing probability.,-1 and suppose their probabilities are PI ~ h ~ Pa ... ~ Pn . The per cent excess time needed over the ideal is therefore less than Let p. = I: Pi ; that is p. is the cumulative probability up to, but not GN 1 1 Ij+HN-l. .lll..:iadillg, p.,. We first encode into a binary system. The binary code for message s is obtained by expanding PM as a binary number. The expansion This method of encoding is substantially the same as one found inde is carried out to m. places, where ms is the integer satisfying: pendently by R. M. Fano.\! His method is to arrange the messages of length N in order of decreasing probability. Divide this series into two groups of 1 1 log2 - ~ ms < 1 + log2 - as nearly equal proLability as possible. If the message is in the first group P8 ps its 1irst binary digit will be 0, othenvise 1. The groups are similarly divided Thus the mes::oages of high probability are represented by short codes and into subsets of nearly equal probability and the particular subset determines those of low probability by long codes. From these inequalities we have the second binary digit. This process is continued until each subset contains only one message. It is easily seen that apart from minor differences (gen 1 1 - ... --.,-.----~--. -to ... BELL SrSTEJl TECHNICIL JOURNAL JrATHEMATlCAL 1'HEOR1' OF t:OMMl'N1CATlO,V ·lOS matching of probabilities to corresponding lengths of sequences. With a .A 0 good code the logarithm of the reciprocal probability of a long message B 10 must Le proportional to the duration of the corresponding signal, in fart C llU -1 D 111 , i Il~)!/ _C I The average number of binary digits used in encoding a sequence of N sym bols will be must be small for all but a small fraction of the long messages. I If a suurce can produce only one particular message its entropy is zero, N(~ X 1 + 1 X 2 + i X 3) = IN and no channel is required. For example, a computing machine set up to It is easily seen that the Linary digits 0, 1 have probabilities ~, l so the 11 fur calculate the successive digits of 7r produces a detlnite sequence with no the coded sequences is one bit per symbol. Since, on the average, we have ~ chance element. No challiel is required tu "transmit" this to another binary symbols per original letter, the entropies on a time basis are the point. One could construct a second machine to compute the same sequence same. The maximum possible entropy for the original set is log ..j. = 2, at the point. However, this may Le impractical. In such a case we can occurring when A, B, C, D have probabilities t, i, L {. Hence the relative choose to ignore some or all of the statistical knowledge we have of the entropy is~. We can translate the binary sequences into the original set of source. We might consider the digits of 7r to be a random sequence in that symbols on a two-to-one basis by the following table: we construct a system capable of sending any sequence of digits. In a similar way we may choose to use some of our statistlcal knowledge of Eng 00 A' lish in constructing a code, but not all of it. In such a case we consider the 01 H' source with the maximum entropy subject to the statistical conditions we 10 C' wish to retain. The entropy of this source determines the channel capacity 11 D' which is necessary and sufficient. In the 7r example the only information This double process then encodes the original message into the same symbols retained is that all the digits are chosen from the set 0, 1, ... , 9. In the but with an average compression ratio ~ . case of English one might wish to use the statistical saving possible due to letter frequencies, but nothing else. The maximum entropy source is then As a second example consider a source which produces a sequence oj .1 's the first approximation to English and its entropy determines the required and E's ,,,ith probability p for A and q for R. If P < < q we have channel capacity. H = -log pJl (I _ p) J p 11. EXAMPLES -p log P (1- p)(l~-J')IP As a simple example of some of these results consider a source which e produces a sequence of letters chosen from among A, B, C, D with prob - p log- abilities ~" L ~, 1, successive symbols being chosen independently. We p have In such a case one can construct a fairly good coding of the message on a 0, 1 channel by sending a special sequence, say 0000, for the infrequent H = - 0 log! + t log 1 + ~ log 1) :;ymbol A and then a sequence indicating the number of B's following it. t bits per symbol. This could be indicated by the binary representation with all numbers con taining the special sequence deleted. All numbers up to 16 are represented Thus we can approximate a coding system to encude messages from this as usual; 16 is represented by the next binary number after 16 which does source into binary digits with an average of t binary digit per symbol. not contain four zeros, namely 17 = 10001, etc. In this case we can actually achieve the limiting value by the following code It can be shown that as p ~ 0 the coding approaches ideal provided the (obtained by the method of the second proof of Theorem 9): length of the special sequence is properly adjusted. 406 BELL SYSTEM TECHNiCAL JOFRNAL J[ATHEMATICAL THEORY OF C01'l,fJ.l[UlollCATION 4U7 PART II: THE DISCRETE CHANNEL WITH NOISE All of these entropies can be measured on a per-second or a per-~ymbol basis. 11. REPRESENTATION Or' A NOISY DISCRETE CHANNEL W c now consider the case where the signal is perturbed by noise during 12. EQUIVOCATION AND CHANNEL CAPACITY transmission or at one or the other of the terminals. This means that the If the channel is noisy it is not in general possible to reconstruct the urig received signal is not necessarily the same as that sent out by the trans inal message or the transmitted signal with certaiJlty by any operation on the mitter. Two cases may be distinguished. If a particular transmitted signal received signal E. There are, however, ways of transmitting the information always produces the same received sil::,nal, i.e. the received signal is a definite which are optimal in cumbating noise. This is the problem which we now function of the transmitted signal, thcn the effect may be called distortion. consider. If this function has an inverse--no two transmitted signals producing the Suppose there are two possible sym bois 0 and 1, and we are transmitting same received signal--distortion may be corrected, at least in principle, by at a rate of 1000 symbols per second with probabilities po = PI = ~. Thus merely performing the inverse functional operation on the received signal. our source is producing information at the rate of 1000 bits per second. Dur The case of interest here is that in which the signal does not always undergo ing transmission the noise introduces errors so that, on the average, 1 in 100 the same change in transmission. In this case ,ve may assume the received is received incorrectly (a 0 as 1, or 1 as 0). What is the rate of transmission signal E to be a function of the transmitted signal S and a second variable, of information? Certainly less. than 1000 bits per second since about 1% the noise N. of the received symbols are incorrect. Our first impulse might be to say the E = f(5, N) rate is 990 bits per second, merely subtracting the expected number of errors. The noise is considered to be a chance variable just as the message was This is not satisfactory since it fails to take into account the recipient's above. In general it may be represented by a suitable stochastic process. lack of knowledge of where the errors occur. We may carry it to an extreme The most general type of noisy discrete channel we shall consider is a general case and suppose the noise so great that the received symbols are entirely ization of the finite state noise free channel described previously. We independent of the transmitted symbols. The probability of receiving 1 is assume a finite number of states and a set of probabilities ~ whatever was transmitted and similarly for O. Then about half of the received syn1bols are correct due to chance alone, and we would be giving Pa,Jj3, j). the system credit for transmitting 500 bits per second while actually no This is the probability, if the channel is in state a and symbol i is trans information is being transmitted at all. Equally "good" transmission mitted, that symbol j will be received and the channel left in state p. Thus would be obtained by dispensing with the channel entirely a.nd flipping a a and (3 range over the possible states, i over the possible transmitted signals coin at the receiving point. and j over the possible received signals. In the case where successive sym Evidently the proper correction to apply to the amount of information bols are independently perturbed by the noise there is only OIle state, and transmitted is thc amount of this information which is missing in the re the channel is described by the set of transition probabilities Pi(j), the prob ceived signal, or alternatively the uncertainty when we have rcceived a ability of transmitted symbol i being received as j. signal of what was actually sent. From our previous discussion oi entropy If a noisy channel is fed by a source there are two statistical processes at as a measure of uncertainty it seems reasonable to use the conditional work: the source and the noise. Thus there are a number of entropies that entropy of the message, knowing the received signal, as a measure of this can be calculated. First there is the entropy H(x) of the source or of the missing information. This is indeed the proper definition, as we shall see input to the channel (these will be equal if the transmitter is non-singular). later. Following this idea the rate of actual transmission, R, would be oh The entropy of the output of the channel, i.e. the received signal, will be tained by subtracting from the rate of production (i.e., the entropy of the denoted by H(y). In the noiseless case H(y) = H(x). The joint entropy of source) the average rate of conditional entropy. input and output will be H(xy). Finally there are two conditional entro pies HAy) and H/x) , the entropy of the output when the input is known R = l1(x) - Hy(x) and conversely. Among these quantities we have the relations The conditional entropy Hy(x) will, for convenience, he called the equi H(x, y) = H(x) + H .,(y) = H(y) + lly(x) vocation. It mea.sures the average ambiguity of the received signal. - ...... "'l: .... \t... '~. ______._, .... .J.08 BELL SL),TEM TECHNICAL JOURNAL JfATHEJIAT1C.!L THEORr OF COJIMUNICATlON 409 In the example considered above, if a 0 is received the a postericri prob If we identify x as the output of the source, y as the received signal and z ability that a 0 was transmitted is .99, and that a 1 was transmitted is as the signal sent over the correction channel, then the right-hand side is the .01. These figures are reversed if a 1 is received. Hence equivocation less the rate of transmission over the correction channel. If Hy(x) = - r.99 log .99 + 0.01 log 0.01] the capacity of this channel is less than the equivocation the right-hand side will be greater than zero and HI.z(x) ~ O. But this is the uncertainty of = J)S} bits/symbol what was sent, knO\ving both the received signal and the correction signal. or 81 hits per second. We may say that the system is transmitting at a rate If this is greater than zero the frequency of errors cannot be arbitrarily 1000 - S1 = <)19 bits per second. In the extreme case where a 0 is equally small. likely to be received as a 0 or 1 and similarly for 1, the a posteriori proba Example: bilities are 1, -} and Suppose the errors occur at random in a sequence of hinary digits: proba ... Hy(x) - rt log 1 + ~ log ~] bility p that a digit is wrong and q = 1 - P that it is right. These errors can be corrected if their position is known. Thus the correction channel 1 bit per symbol 5 need only send information as to these positions. This amounts to trans- ~5... or 1000 bits per second. The rate of transmission is then 0 as it should ."j be. CORRECTION DATA '.,. - - -- ~: The following theorem gives a direct intuitive interpretation of the ;; I ... equivocation and also serves to justify it as the unique appropriate measure . .lJ We consider a communication system and an observer (or auxiliary device) ~ :.., who can see both what is sent and what is recovered (with erro:s "'04 OB5E RVER ~ due to noise). This observer notes the errors in the recovered message and transmits data to the receiving point over a "correction channel" to enable ~ )01 the receiver to correct the errors. The situation is indicated schematically .... in Fig. 8. :; -M~ :, Theorem 10: If the correction channel has a capacity equal to Hy(x) it is r ~M- possible to so encode the correction data as to send it over this channel 50URCE l'RAN5MITTER RECEIVE DEVICE ~ and correct all but an arbitrarily small fraction E of the errors. This is not ".. possible if the channel capacity is less than Hy(x). Fig. 8-Schematic diagram of a correction system . Roughly then, Hy(x) is the amount of additional information that must be mittillg from a source which prorluces binary digits with probability p for supplied per second at the receiving point to correct the received message. 1 (correct) and q for 0 (incorrect). This requires a channel of capacity To prove the 11rst part, consider long sequences of received message M' and corresponding original message M. There will be logarithmically - [p log P + q log q] THt,(x) of the llf's which could reasonably have produced each M'. Thus which is the equivocation of the original system. we have TlIlI (x) binary digits to send each T seconds. This can be done The rate of transmission R Call be written in two other forms due to the with E frequency of errors on a channel of capacity H.v(.r). identities noted above. We have The second part can be proved by noting, first, that for any discrete chance variables x, y, z R = H(x) - Hy(x) H y(.1:, ;;) 2:: Hy(:x) = H(y) - H:r;(y) The left-hand side can be expanded to give = H(x) + H(y) - lJ(x, y). HII(;;) + Hyz(x) ~ HJr) The first delining expression has already been interpreted as the amuunt of lll1z(x) ~ Hy(x) - Hy(s) ~ 1J II (.r) lI(-:-) information sent less the uncertainty of what was sent. The serond meas- , ..... ".,.. -.'""... _, ..... _------410 BELL SYSTEM TECHNICAL JOURNAL JfATHEJlATlLII, THfWRr OF COMMUNICATION 411 ures the amount received less the part of this which is due to noise. The These results are the main ju~ti1icati()n for the definition of C and will third is the sum of the two amounts less the joint entropy and therefore in a now be proved. sense is the number of bits per second common to the two. Thus all three expressions have a certaill intuitive significance. Theorem 11. Let a discrete channel have the capacity C and a discrete source the entropy per second H. Ii H :s; C there exists a coding system The capacity C of a noisy channel should be the maximum possible rate such that the output of the source can be transmitted over the channel with of transmission, i.e., the rate when the source is properly matched to the an arbitrarily small frequency of errors ~ or an arbitrarily small equivocation). channel. We therefore define the channel capacity by If H > C it is possible to encode the source ~o that the equivocation is less C = Max (IJ(x) - Hy(x» than 11 - C + E where E is arbitrarily small. There is no method of encod ing which gives an equivocation less than H - C. where the maximum is with respect to all possible information sources used The method of proving the first part of this theorem is not by exhibiting ./ '" as input to the channel. If the channel is noiseless, Hy(x) = O. The defini a coding method having the desired properties, but by showing that such a tion is then equivalent to that already given for a noiseless channel since the code must exist in a certain group of codes. In fact we will average the maximum entropy for the channel is its capacity. frequency of errors over this group and show that this average can be made less than E. If the average of a set of numbers is less than t there must . ~.-- ...... , .~.,.,..--.------MATIJEMAT1CIL TlJEOkl' OF CUMMUlll1Cr1TlON 413 ·U2 BELL SYSTEM TECIIZ'.;JCAL JOURNAL The probability that nont' of the point:". in the fan is a message (apart from All the f'S and b's implied by the words "small" and "about" in these statements approach zero as we allow T to increase and So to approach the the actual originating message) is maximizing source. P = [1 - 2T (n H(r))pl'Hu (.c1 The situation is summarized in Fig. 10 where the input sequences are points on the left and output sequences poinb on the right. The fan of Xow R < H(x) Hy(.r) so R - H(x) -11~(x) - 7} with 7} positive. cross lines represents the range of possible causes for a typical output. Consequently Now suppose \ve have another source producing information at rate R 2" TH:,\J')-T~j2l'lly(X) p = 11 with R < c. In the period T this source will have 2TR high probability outputs. We wish to associate these' with a selection of the possible channe approaches (as T -'1 oc) 1 - TT~ E • Hence the probability of an error approaches zero alld the tirst part of the M • theorem is proved. • • The second part of the theorem is easily shown by noting that we could • • merely send C bits per second from the source, completely neglecting the • remainder of the information generated. At the receiver the neglected part 2Hh"T~ gives an equivocation H(x) - C and the part transmitted need only add f. HIGH PROBABILITY 2 (y)T H This limit can also be attained in many other ways, as will be shown when we MESSAGES • HIGH PROBABILITY consider the continuous case. RECEIVED SIGNALS -REASONABL'E CAUSES· The last statement of the theorem is a simple consequence of our definition • FOR EACH E • of C. Suppose we can encode a source with R = C + u in such a way as to • • obtain an equivocation Hy(x) = U - f with t positive. Then R = H(x) = C + (] and H(.\:) - H (x) = C f :~ lI + with f positive. This contradicts the definition of C as the maximum of ~2 • REASONABLE EFFECTS. l1(x) - Hy(x). FROM EACH M • Actually more has been proved than was stated in the theorem. If the Fig. lU-Schematic representation of the relations bet\\een. inputs and outputs in a average of a set of numbers is within t of their maximum, a fraction of at channel. most y~-can be more than y;below the maximum. Since f is arbitrarily small we can say that almost all the systems are arbitrarily close to the ideal. inputs in such a way as to get a small frequency of errors. \Ve will set up this association in all possible ways (using, however, only the high proba 14. DISCUSSION bility group of inputs as determined by the source So) and average the fre quency of errors for this large class of possible coding systems. This is the The demonstration of theorem 11, while not a pure existence proof, has same as calculating the frequency of errors for a random association of the some of the deficiencies of such proofs. An attempt to obtain a good messages and channel inputs of duration T. Suppose a particular output approximation to ideal coding hy following the method of the proof is gen Yl is observed. 'Vhat is the probability of more than one message in the set erally impractical. In fact, apart from some rather trivial cases and certain limiting situations, no explicit description of a series of approxima of possible causes of yt? There are 27'R messages distributed at random in tion to the ideal has been found. Probahly this is no accident but is related 21'1l(r) points. The probability oi a particular point being a message is thus to the difticulty of giving an explicit construction for a good approximation 2T (K-H(X» to a random sequence. J.-...._ ...... _~~ ..... 414 ilELL SYSTEJf TECHNICIL jO(!l<.i'v'AL J,LU'HEly! A 1'1 C.If. T1Jl~() N r OF <-"OM J/ if N rcA 1'1 ON 4lS ~ An approximation to the ideai would have the property that if the signal L 15. EXAMPLE 01<' A. ])lSCREn: CHANNEL AND ITS CAPAcrn.: i' is altered in a reasonable way by the noise, the original can still be recovered. ¥ A simple example of a discrete channel is indicated in Fig. 11. There ! [n other words the alteration will not in general bring it closer to another r~ reasonable signal than the original. This is accomplisherl at the cost of a are three possible symbols. The first is never affected by noise. The second i; and third each have probability p of coming through undisturbed, and (/ I., certain amount of redunrlancy in the corling. The rerlundancy must be f: introduced in the proper way to combat the particular noise structure of being changed into the other of the pair. We have (letting a = - [p log ~ ~ involved. However, any redundancy in the source will usually help if it is ~;. t utilized at the receiving point. In particular, if the source already has a ~. certain redundancy and no attempt is made to eliminate it in matching to the P t; channel, this redundancy will help combat noise. For example, in a noiseless ./ telegraph channel one could save about 50% in time by proper encoding of TRANSMITTEOZRECEIVEO SYMBOLS SYMBOLS the messages. This is not done and most of the redundnacy of English q i remains in the channel symbols. This has the adYantage, however, of I· allowing considerable noise in the channel. A sizable fraction of the letters P I, can be received incorrectly and still reconstructed by the context. In Fig. II-Example of a discrete channel. fact this is probably not a bad approximation to the ideal in many cases, since the statistical structure of English is rather involved and the reasonable p + q log q] and P and (J be the probabilities of using the first or serond English sequences are not too far (in the sense required for theorem) from a symbols) random selection. H(x) = -Plog P - 2(>logQ As in the noiseless case a delay is generally required to approach the ideal I encoding. It now has the additional function of allowing a large sample of Hy(x) = 2Qa noise to affect the signal before any judgment is made at the receiving point We wish to choose P and Q in such a way as to maximize H(x) - H y(x), as to the original message. Increasing the sample size always sharpens the subject to the constraint P + 2Q = 1. Hence we consider possible statistical assertions. The content of theorem 11 and its proof can be formulated in a somewhat U = - P log P - 2Q log Q - 2Qa + XC]> + 2Q) different way which exhibits the connection with the noiseless case more aU -1 - log P + x = 0 clearly. Consider the possible signals of duration T and suppose a subset aF of them is selected to be used. Let those in the subset all be used with equal J, probability, and suppose the receiver is constructed to select, as the original au ~! - 2 - 1 log Q - la + 2X = o. signal, the most probable cause from the subset, when a perturbed signal aQ i is received. We define N(T, q) to be the maximum number of signals we Eliminating X i.;' can choose for the subset such that the probability of an incorrect inter ~. pretation is less thaH or equal to 1/.. log P = log Q + u f: = 1'1 1) 1· log NCT, q). I ... h I I . P ()e" = Vfj rleurt:m -: .un --- -r--- = ( , w lere C IS t e C lamle rapaCIty, pro- ~ 7 ~.QO 1 vided that q does not eyual () or 1. ? = JJ iJ +2 V~!i+2 In other words. no matter how we set our limib of reliability, we call distinguish reliably in time T enough messages to correspond to about CT The challnel capacity is then bits, ,,,,hen T is sufficiently large. Theorem 12 can be compared with the loc, {J 2 definition of the capacity of a noiseless channel given in section 1. c + t"> -;i -116 BELL .')'l'STEM TECHNICAL .JOURNAL JfATHEMATICAL THRORl' OF COMMUNICATION 417 :\ ote how this checks the obvious values in the cases p = 1 and P = !. If each input symbol has the same set of probabilities on the lines emerging In the first, ~ = 1 and C = log 3, which is correct since the channel isthen irom it, and the same is true of each output symbol, the capacity can be noiseless with three possible symbols. If P = ~, ~ = 2 and C = log 2. easily calculated. Examples are shown in Fig. 12. In such a case ll:.(y) Here the second and third symbols cannot be distinguished at all and act is independent of the distribution of probabilities on the input symbols, and together like one symbol. The first symbol is used with probability P = is given by -~ Pi log Pi where the Pi are the values of the transition proba .~ and the second and third together with probability!. This may be bilities from any input symbol. The channel capacity is distributed in any desired way and still achieve the maximum capacity. Max lH(y) flAY)! For intermediate values of P the channel capacity will lie between log 2 and log 3. The distinction between the second and third symbols conveys = MaxH(Y)+~PilogPi' some information but not as much as in the noiseless case. The first symbol The maximum of H(y) is clearly log m where m is the number oi output is used somewhat more frequently than the other two because of its freedom from noise. 16. THE CHAl\1\EL CAPACITY 11\ CEkTAIN SPECIAL CASES Ii the noise atIects successive channel symbols independently it can be described by a set of transition probabilities Pij. This is the probability, if symbol i is sent, that j will be received. The maximum channel rate is then given by the maximum of L PiPij log L PiP;j - L FiPii log Pij i. iii.i 1/2 where we vary the Pi subject to ~Pi 1. This leads by the method of Lagrange to the equations, abe Fig. 12-Exarnples of di:,crete channels with the same transition probahilities for each input and for each output. .l!L~_ = iJ. L psi log L Pi Pii s 1,2, .... i i symbols, since it is possible to make them all equally probable by making the input symbols equaUy probable. The channel capacity is therefore Multiplying by P s and summing on s shows that iJ. = -C. Let the inverse uf P"i (if it exists) be h'l so that L hs/Psj = Dtj. Then: C = log m + 2; Pi log Pi. In Fig. 12a it would be L:: h"t psi log psi - log L:: Pi Pit -c L:: h.t • s.i i C = log 4 - log 2 = log 2 . Hence: This could be achieved by using only the 1st and 3d symbols. In Fig. 12b L:: Pi P,:/ = exp lC L:: h"t + L:: h.t psi log Psi} i -" 8, i C = log 4 - 1 log 3 - Slog b or, = Jog 4 - log" - t log 2 Pi = L:: hi! exp [C L:: It. t + L:: hst psi log Psi]. = log 2;. t • s.j t This is the system of equations for determining the maximizing values of In Fig. 12c we have Pi, with C to be determined so that ~ Pi = 1. \Vhen this is done C will be C = log 3 - ~. log 2 - t log 3 - k log 6 the channel capacity, and the Pi the proper probabilities for the channel 3 symbols to achieve this capacity. log 2~-jY 6~ . ----- JIAf1J&MATIC.t l. THEON.!' OF COMJll'NICATION 4-19 .t.l~ BELL ."tTSTEJI TECHNICAL JUURNAI, where , ... are the length of the symbols which may be chot>en Suppose the symbols fall into several groups such that the noise never bL , b~j b7i in state i and lead to state j. These are linear difference equations and the causes a symbol in (Jne group to be mistaken for a symbol in another group. behavior as L ~ Xi must be of the type rIX Let the capacity for the nth group be C" when we use only the symbols in this group. Then it is easily shov~'n that, for best use of the entire set, N j = AjU"/' t he total probability P" of all symbols in the nth group should be e Substituting in the difference equation 2 " n P i2c r< /I) II'L L: Ai WL-bj-:) f' -i.oS ~ Within a group the probability is distributed just as it would be if these ~ were the only symbols being used. The channel capacity is or ~ - '"' 4 jr--b (.~ ) C = log ~2cl' • ,i 1} L....J 4. i ~ , I , I w 'is AN EXAMPLE OF E}'FICIENT CODING 17. '"' ,'"' • bl' i ) i ~ ~~ H - ij - Oij .L = O. 1 The following example, although somewhat unrealistic, is a case in which s ~- exact matching to a noisy channel is possible. There are two channel For this to be possible the determinant symbols, 0 and 1, and the noise affects them in blocks of seven symbols. A , - • • '"' (S ) I block of seven is either transmitted without error, or exactly one symbol of DOV) = \ au I = : ~ IJ'b;i - Oij ,; I the seven is incorrect. These eight possibilities are equally likely. We have C = Max [H(y) - H.r(Y)] must vanish and this determines 11', which is, of course, the largest real root ~ of D = O. ~ = +17 -~logl] + The quantity C is then given by = 4 bits/symbol. ;I An efficient code, allowing complete correction of errors and transmitting at C = Lim ~o_~_ ~~..l~!f~~ log W - ].->00 L ~fic the rate C, is the following (found by a method due to R. Hamming): '~ Let a block of seven symbols be Xl, X 2, ••• X j . Of these ,\'3, Xli, X6 and and we also note that the same growth properties result if we require that all ~ X7 are message symbols and chosen arbitrarily by the source. The other blocks start in the same (arbitrarily chosen) state. lc;-'!o ",'If three are redundant and calculated as follows: ~.~ , APPENDIX 2 X.I is chosen to make a = X 4 + Xli + X6 + X7 even v- .. ...'1. 2 " {:J = X2 + Xa + Xs + X7 •• DERiVATlON OF H = -~ Pi log Pi lor ,; f~' ..'\.1 l' = Xl + X3 + Xli + X, 1 1" -- - L et H ---, 1--, ... ,-1) = A ()11. 1<-, rom con( alOn (3) we can decompose When a block of seven is received a, {3 and l' are calculated and if even called ( n It It zero, if odd called one. The binary number a j3 l' then gives the subscript a choice from slll equally likely possihilities into a series of m choices each of the Xi that is incorrect (if 0 there was no error). from s equally likely possibilities and obtain APPENDIX 1 A(sm) = In A(s) THE GROWTH OF THE NUMBER Q}' BLOCKS OE SYMBOLS WITH A Similarly FINITE STATE CONDITION A (til) = 1/ .-t (I) Let N i(L) be the number of blocks of symbols of length L ending in state i. Then we have We can choose 11 arbitrarily large alltl rind an In to satisfy -"AI,) = L X;\L - b~"/) s'" ::; t < /111+1) i. ------.' 420 BELL SYSTEM TECHNICAL JOURNAL MATHEMATICAL THEONI' OF COMMUNICATION 421 Thus, taking logarithms and dividing by Il log s, work is traversed in a long sequence of length 1"1 is about proportiunal to the In log t In 1 probability of being at i and then choosing this path, P ipijN. If N is large --< --- <-- + -- or 'm _ log! I < E n - log s - n 17 0 t: 1 i1, log s I enough the probability of percentage error ± in this is less than so that for all but a set of small probability the actual numbers lie within the limits where E is arbitrarily small. Now from the monotonic property of A (It) (P iPij ± b)'y ,- A (s"') ~ A (t) ~ A (S"'+I) Hence nearly all sequences have a probability p given by P = IIp~~il"i.t:b)N m A(s) ~ nA(t) ~ (m + 1) .ti(s) Hence, dividiug by nA(j), and ~o~p is limited by ' f!t _ A (t) I '?! < ~0 < ,!t + ~ or ~~g P = ~(p,: Pij ± 0) log Pij 11 - A(s) - It 1l / 11 A(s) < E N or A (t) _ log t j < 2E A(t) = -:-K log t I log p, ;\ I,A(s) log s - : - N- - ~Pi Pij log Pij < '1· where K must be positive to satisfy (2). This proves theorem 3. N ow suppose we have a choice from 11 possibilities with commeasurable prob- Theoreill -1 follows immediately frum this on calculating upper and lower abilities where the Iti are integers. We can break down a choice bounds for Il(q) based on the possihle range uf values of p in Theorem 3. Pi = !tn,-~ In the mixed (nut ergodic) case if from ~Jlj possibilities into a choice from n possibilities with probabilities Pi . .. P" and then, if the ith was chosen, a choice from 1l; with equal prob L = ~ Pi Li abilities. Using conditiun 3 again, we equate the total choice from 2;llj and the entropies of .the compollents are H1 ~ H2 ~ ... Hn we have the as computed by two methods 2:: '1 L' log n(q) ( ) . i . .. 1fleorem: nn -- -- = .p q IS a ( ecreaslllg step tunctlOn, 1'; log 2:.ni = R(PI , ... , pn) + K~ Pi log Iti N->oo N Hence ~-l s ",(q) = 11 s in the interval 1: ai < q < 1: ai . II = K [~ Pi log ~ lIi - !t Pi log Id 1 1 To prove theorems 5 and 6 lirst note that FN is munotonic decreasing be L - " 1 It; L - ",,' I -.l\. - Pi og ~- = -.l\. ~ Pi og Pi . cause increasing N adds a subscript to a conditional entropy. A simple 2; n i substitution for PH, (Sj) in the definition of FN shows that If the Pi are incommeasuraule, they may be approximated by rationals ar.d FN = N G - (N - 1) GN - 1 the same expression must hold by our continuity assumption. Thus the N expression holds in general. The choice of coefficient K is a matter of con and summing this for all N gives GN = ~ ~ F N • Hence GN ~ F ... and GN venience and amounts to the choice of a unit of measure. monotonic decreasing. Also they must approach the same limit. By using APPENDLX 3 theorem 3 we see that Lim GN = H. l'{-too THEO.KEMS OK ERGODIC SOURCES APPENDIX -1 If it is possible to go from any state with P > 0 to any other along a path MAXTMlZING THE RATE :FOR A SVSTEiI.l OF l \INST1U1NT~ of probability P > 0, the system is ergodic and the strong law of large num bers can be applied. Thus the number of times a given path Pij in the net- Suppose we have a set of constraints on sequences of symbols that is of the finite state type and can be represented therefore by a linear graph . .~~-~--~------423 422 BELL STlSTE1vI TECHNICAL JOURlllAL lvlATHElvlA.TICAL fHEORl' OF CO.lvIMUN1CATION Let C~i) be the lengths of the various ::;ymbols that can occur in passing from Pj " 1!.-,: C-J'i j state i to state j. What distribution of probabilities Pi for the different .. Bi 13i states and p~i) for choosing symbol s in state i and going to state j maximizes the rate of generating information under these con::;traints? The constraints So that if Ai satisfy "" C-(i l define a discrete channel and the maximum rate must be less than or equal ""'''Y i . "Yi to the capacity C of this channel, since if all blocks of large length were Pi = Ri"Yi equally likely, this rate would result, and if possible this would be best. We will show that this rate can be achieved by proper choice of the p. and p~i). Both of the sets of equations for B i amI "Y; can be satisfied since C is ::;uch that The rate in question is \ C-{ i i - Oij I = 0 - ';;PiP~·;> log t~'/ N In this case the rate is ~p~p("'> .e~8)-- ~1 ..., (d I.} 'I .i~ ~1\ P'j log C-(~j pl-) j'(8) i: L e t f ij L.-"" 1; j • E'• VI< l entIf' y or a maXImum Pi](.) k exp {ij • 'I'h e COI1- '2;Pi Pij (ij straints on maximizatiun are '2;P i 1, L p,:j 1, ~ P,(pij - OiJ = o. i Hi Hence we maximize -;:;p,: Pij log 13i c ip~p~ fij (' = A P, '2;rJjP h ) _~_~iPiiJ!!'tk Pii + L + 'J:./J-,pij + i (p'J - ii "LP i Pij iii i . but ae ~l1PJl + log Pii~.±}i~i!~j + A + /J-i + TJi p, = o. ~Pi P'i(log Bi - log Hi) = L Pi log Bi - -;:;Pi log Hi = 0 apij - -~M2 j Solving for Pii Hence the rate is C and as this could never be exceeded this is the maximum, justifying the assumed solution. pij = .·LBjD-fii. (1'0 be colllillwd) Since L Pij 1, Ail = L BjD-f;j BjIJ--tij P'ii 2:- B,-D~r..: . The correct value of J) is the capacity C and the B are solutions of It j t Hi = ~ BjC-I;i ~ for then t B· i Pij = __J C- .j I Bi "Lp. ~t C-iij = P 'Bi J I~ or L.. ..lc.< __•