Reprinted with corrections from The Bell System Technical Journal, Vol. 27, pp. 379­423, 623­656, July, October, 1948. A Mathematical Theory of Communication By C. E. SHANNON INTRODUCTION The recent development of various methods of modulation such as PCM and PPM which exchange bandwidth for signal-to-noise ratio has intensified the interest in a general theory of communication. A T basis for such a theory is contained in the important papers of Nyquist1 and Hartley2 on this subject. In thepresent paper we will extend the theory to include a number of new factors, in particular the effect of noisein the channel, and the savings possible due to the statistical structure of the original message and due to thenature of the final destination of the information. The fundamental problem of communication is that of reproducing at one point either exactly or ap- proximately a message selected at another point. Frequently the messages have meaning; that is they referto or are correlated according to some system with certain physical or conceptual entities. These semanticaspects of communication are irrelevant to the engineering problem. The significant aspect is that the actualmessage is one selected from a setof possible messages. The system must be designed to operate for eachpossible selection, not just the one which will actually be chosen since this is unknown at the time of design. If the number of messages in the set is finite then this number or any monotonic function of this number can be regarded as a measure of the information produced when one message is chosen from the set, allchoices being equally likely. As was pointed out by Hartley the most natural choice is the logarithmicfunction. Although this definition must be generalized considerably when we consider the influence of thestatistics of the message and when we have a continuous range of messages, we will in all cases use anessentially logarithmic measure. The logarithmic measure is more convenient for various reasons: 1. It is practically more useful. Parameters of engineering importance such as time, bandwidth, number of relays, etc., tend to vary linearly with the logarithm of the number of possibilities. For example,adding one relay to a group doubles the number of possible states of the relays. It adds 1 to the base 2logarithm of this number. Doubling the time roughly squares the number of possible messages, ordoubles the logarithm, etc. 2. It is nearer to our intuitive feeling as to the proper measure. This is closely related to (1) since we in- tuitively measures entities by linear comparison with common standards. One feels, for example, thattwo punched cards should have twice the capacity of one for information storage, and two identicalchannels twice the capacity of one for transmitting information. 3. It is mathematically more suitable. Many of the limiting operations are simple in terms of the loga- rithm but would require clumsy restatement in terms of the number of possibilities. The choice of a logarithmic base corresponds to the choice of a unit for measuring information. If the base 2 is used the resulting units may be called binary digits, or more briefly bits,a word suggested byJ. W. Tukey. A device with two stable positions, such as a relay or a flip-flop circuit, can store one bit ofinformation. Nsuch devices can store Nbits, since the total number of possible states is 2Nand log2 2N N. = If the base 10 is used the units may be called decimal digits. Since log2 M log log = 10 M= 10 2 3 32 log = : 10 M; 1Nyquist, H., "Certain Factors Affecting Telegraph Speed," Bell System Technical Journal,April 1924, p. 324; "Certain Topics in Telegraph Transmission Theory," A.I.E.E. Trans.,v. 47, April 1928, p. 617. 2Hartley, R. V. L., "Transmission of Information," Bell System Technical Journal,July 1928, p. 535. 1 =============================================================================== INFORMATION SOURCE TRANSMITTER RECEIVER DESTINATION SIGNAL RECEIVED SIGNAL MESSAGE MESSAGE NOISE SOURCE Fig. 1 -- Schematic diagram of a general communication system. a decimal digit is about 3 1 bits. A digit wheel on a desk computing machine has ten stable positions and 3 therefore has a storage capacity of one decimal digit. In analytical work where integration and differentiationare involved the base eis sometimes useful. The resulting units of information will be called natural units.Change from the base ato base bmerely 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 information sourcewhich produces a message or sequence of messages to be communicated to the receiving terminal. The message may be of various types: (a) A sequence of letters as in a telegraphof teletype system; (b) A single function of time f tas 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 afunction f x y tof two space coordinates and time, the light intensity at point x yand time ton a ; ; ; pickup tube plate; (d) Two or more functions of time, say f t, g t, h t-- this is the case in "three- dimensional" sound transmission or if the system is intended to service several individual channels inmultiplex; (e) Several functions of several variables -- in color television the message consists of threefunctions f x y t, g x y t, h x y tdefined in a three- dimensional continuum -- we may also think ; ; ; ; ; ; of these three functions as components of a vector field defined in the region -- similarly, severalblack and white television sources would produce "messages" consisting of a number of functionsof three variables; (f) Various combinations also occur, for example in television with an associatedaudio channel. 2. A transmitterwhich operates on the message in some way to produce a signal suitable for trans- mission over the channel. In telephony this operation consists merely of changing sound pressureinto a proportional electrical current. In telegraphy we have an encoding operation which producesa sequence of dots, dashes and spaces on the channel corresponding to the message. In a multiplexPCM system the different speech functions must be sampled, compressed, quantized and encoded,and finally interleaved properly to construct the signal. Vocoder systems, television and frequencymodulation are other examples of complex operations applied to the message to obtain the signal. 3. The channelis 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 receiverordinarily performs the inverse operation of that done by the transmitter, reconstructing the message from the signal. 5. The destinationis the person (or thing) for whom the message is intended. We wish to consider certain general problems involving communication systems. To do this it is first necessary to represent the various elements involved as mathematical entities, suitably idealized from their 2 =============================================================================== physical counterparts. We may roughly classify communication systems into three main categories: discrete,continuous and mixed. By a discrete system we will mean one in which both the message and the signalare a sequence of discrete symbols. A typical case is telegraphy where the message is a sequence of lettersand the signal a sequence of dots, dashes and spaces. A continuous system is one in which the message andsignal are both treated as continuous functions, e.g., radio or television. A mixed system is one in whichboth discrete and continuous variables appear, e.g., PCM transmission of speech. We first consider the discrete case. This case has applications not only in communication theory, but also in the theory of computing machines, the design of telephone exchanges and other fields. In additionthe discrete case forms a foundation for the continuous and mixed cases which will be treated in the secondhalf of the paper. PART I: DISCRETE NOISELESS SYSTEMS 1. THE DISCRETE NOISELESS CHANNEL Teletype and telegraphy are two simple examples of a discrete channel for transmitting information. Gen-erally, a discrete channel will mean a system whereby a sequence of choices from a finite set of elementarysymbols S1 Sncan be transmitted from one point to another. Each of the symbols Siis assumed to have ; : : : ; a certain duration in time tiseconds (not necessarily the same for different Si, for example the dots anddashes in telegraphy). It is not required that all possible sequences of the Sibe capable of transmission onthe system; certain sequences only may be allowed. These will be possible signals for the channel. Thusin telegraphy suppose the symbols are: (1) A dot, consisting of line closure for a unit of time and then lineopen for a unit of time; (2) A dash, consisting of three time units of closure and one unit open; (3) A letterspace consisting of, say, three units of line open; (4) A word space of six units of line open. We might placethe 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 ofsuch 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 nsymbols per second it is natural to say that the channel has a capacity of 5nbits per second. This does notmean that the teletype channel will always be transmitting information at this rate -- this is the maximumpossible rate and whether or not the actual rate reaches this maximum depends on the source of informationwhich feeds the channel, as will appear later. In the more general case with different lengths of symbols and constraints on the allowed sequences, we make the following definition:Definition: The capacity Cof a discrete channel is given by log N T C Lim = T T ! where N Tis the number of allowed signals of duration T. 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 number in most cases of interest. Suppose all sequences of the symbolsS1 Snare allowed and these symbols have durations t1 tn. What is the channel capacity? If N t ; : : : ; ; : : : ; represents the number of sequences of duration twe have N t N t t1 N t t2 N t tn = , + , + + , : The total number is equal to the sum of the numbers of sequences ending in S1 S2 Snand these are ; ; : : : ; N t t1 N t t2 N t tn, respectively. According to a well-known result in finite differences, N t , ; , ; : : : ; , is then asymptotic for large tto Xtwhere X 0 0 is the largest real solution of the characteristic equation: X t t tn , 1 X, 2 X, 1 + + + = 3 =============================================================================== and therefore C log X0 = : In case there are restrictions on allowed sequences we may still often obtain a difference equation of this type and find Cfrom the characteristic equation. In the telegraphy case mentioned above N t N t 2 N t 4 N t 5 N t 7 N t 8 N t 10 = , + , + , + , + , + , as we see by counting sequences of symbols according to the last or next to the last symbol occurring.Hence Cis log 2 4 5 7 8 10 0 where 0 is the positive root of 1 . Solving this we find , = + + + + + C 0 539. = : A very general type of restriction which may be placed on allowed sequences is the following: We imagine a number of possible states a1 a2 am. For each state only certain symbols from the set S1 Sn ; ; : : : ; ; : : : ; can be transmitted (different subsets for the different states). When one of these has been transmitted thestate changes to a new state depending both on the old state and the particular symbol transmitted. Thetelegraph case is a simple example of this. There are two states depending on whether or not a space wasthe last symbol transmitted. If so, then only a dot or a dash can be sent next and the state always changes.If not, any symbol can be transmitted and the state changes if a space is sent, otherwise it remains the same.The conditions can be indicated in a linear graph as shown in Fig. 2. The junction points correspond to the DASH DOT DOT LETTER SPACE DASH WORD SPACE Fig. 2 - - Graphical representation of the constraints on telegraph symbols. states and the lines indicate the symbols possible in a state and the resulting state. In Appendix 1 it is shownthat if the conditions on allowed sequences can be described in this form Cwill exist and can be calculatedin accordance with the following result: s Theorem 1:Let b be the duration of the sth symbol which is allowable in state iand leads to state j. i j Then the channel capacity Cis equal to logWwhere Wis the largest real root of the determinant equation: s W b , i j i j 0 , = s where i j 1 if i jand is zero otherwise. = = For example, in the telegraph case (Fig. 2) the determinant is: 1 W2 4 , W, , + 0 W3 6 2 4 = : , W, W, W, 1 + + , On expansion this leads to the equation given above for this case. 2. THE DISCRETE SOURCE OF INFORMATION We have seen that under very general conditions the logarithm of the number of possible signals in a discretechannel increases linearly with time. The capacity to transmit information can be specified by giving thisrate 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 second is produced in a given source? The main point at issue is theeffect of statistical knowledge about the source in reducing the required capacity of the channel, by the use 4 =============================================================================== of proper encoding of the information. In telegraphy, for example, the messages to be transmitted consist ofsequences of letters. These sequences, however, are not completely random. In general, they form sentencesand have the statistical structure of, say, English. The letter E occurs more frequently than Q, the sequenceTH more frequently than XP, etc. The existence of this structure allows one to make a saving in time (orchannel capacity) by properly encoding the message sequences into signal sequences. This is already doneto a limited extent in telegraphy by using the shortest channel symbol, a dot, for the most common Englishletter E; while the infrequent letters, Q, X, Z are represented by longer sequences of dots and dashes. Thisidea is carried still further in certain commercial codes where common words and phrases are representedby four- or five-letter code groups with a considerable saving in average time. The standardized greetingand anniversary telegrams now in use extend this to the point of encoding a sentence or two into a relativelyshort sequence of numbers. We can think of a discrete source as generating the message, symbol by symbol. It will choose succes- sive symbols according to certain probabilities depending, in general, on preceding choices as well as theparticular symbols in question. A physical system, or a mathematical model of a system which producessuch a sequence of symbols governed by a set of probabilities, is known as a stochastic process.3 We mayconsider a discrete source, therefore, to be represented by a stochastic process. Conversely, any stochasticprocess which produces a discrete sequence of symbols chosen from a finite set may be considered a discretesource. 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 se- quence of symbols. The following are examples 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 typicalexample. B D C B C E C C C A D C B D D A A E C E E AA B B D A E E C A C E E B A E E C B C E A D. This was constructed with the use of a table of random numbers.4 (B) Using the same five letters let the probabilities be .4, .1, .2, .2, .1, respectively, with successive choices independent. A typical message from this source is then: A A A C D C B D C E A A D A D A C E D AE A D C A B E D A D D C E C A A A A A D. (C) A more complicated structure is obtained if successive symbols are not chosen independently but their probabilities depend on preceding letters. In the simplest case of this type a choicedepends only on the preceding letter and not on ones before that. The statistical structure canthen be described by a set of transition probabilities pi j, the probability that letter iis followed by letter j. The indices iand jrange over all the possible symbols. A second equivalent way ofspecifying the structure is to give the "digram" probabilities 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 3See, for example, S. Chandrasekhar, "Stochastic Problems in Physics and Astronomy," Reviews of Modern Physics, v. 15, No. 1, January 1943, p. 1. 4Kendall and Smith, Tables of Random Sampling Numbers,Cambridge, 1939. 5 =============================================================================== pi jand the digram probabilities p i jare related by the following formulas: ; p i p i jp j ip j pj i = ; = ; = j j j p i j p i pi j ; = pi jp ip i j1 = = ; = : j i i j ; As a specific example suppose there are three letters A, B, C with the probability tables: pi j j i p i p i j j ; A B C A B C A 0 4 1 A 9 A 0 4 1 5 5 27 15 15 i B 1 1 0 B 16 i B 8 8 0 2 2 27 27 27 C 1 2 1 C 2 C 1 4 1 2 5 10 27 27 135 135 A typical message from this source is the following: A B B A B A B A B A B A B A B B B A B B B B B A B A B A B A B A B B B A C A C A BB A B B B B A B B A B A C B B B A B A. The next increase in complexity would involve trigram frequencies but no more. The choice ofa letter would depend on the preceding two letters but not on the message before that point. Aset of trigram frequencies p i j kor equivalently a set of transition probabilities pi j kwould ; ; be required. Continuing in this way one obtains successively more complicated stochastic pro-cesses. In the general n-gram case a set of n-gram probabilities p i1 i2 inor of transition ; ; : : : ; probabilities pi i is required to specify the statistical structure. 1 i i n ; 2; :::; n1 , (D) Stochastic processes can also be defined which produce a text consisting of a sequence of "words." Suppose there are five letters A, B, C, D, E and 16 "words" in the language withassociated probabilities: .10 A .16 BEBE .11 CABED .04 DEB .04 ADEB .04 BED .05 CEED .15 DEED .05 ADEE .02 BEED .08 DAB .01 EAB .01 BADD .05 CA .04 DAD .05 EE Suppose successive "words" are chosen independently and are separated by a space. A typicalmessage might be: DAB EE A BEBE DEED DEB ADEE ADEE EE DEB BEBE BEBE BEBE ADEE BED DEEDDEED CEED ADEE A DEED DEED BEBE CABED BEBE BED DAB DEED ADEB. If all the words are of finite length this process is equivalent to one of the preceding type, butthe description may be simpler in terms of the word structure and probabilities. We may alsogeneralize here and introduce transition probabilities between words, etc. These artificial languages are useful in constructing simple problems and examples to illustrate vari- ous possibilities. We can also approximate to a natural language by means of a series of simple artificiallanguages. The zero- order approximation is obtained by choosing all letters with the same probability andindependently. The first-order approximation is obtained by choosing successive letters independently buteach letter having the same probability that it has in the natural language.5 Thus, in the first-order ap- proximation to English, E is chosen with probability .12 (its frequency in normal English) and W withprobability .02, but there is no influence between adjacent letters and no tendency to form the preferred 5Letter, digram and trigram frequencies are given in Secret and Urgentby Fletcher Pratt, Blue Ribbon Books, 1939. Word frequen- cies are tabulated in Relative Frequency of English Speech Sounds,G. Dewey, Harvard University Press, 1923. 6 =============================================================================== digrams such as TH, ED, etc. In the second-order approximation, digram structure is introduced. After aletter is chosen, the next one is chosen in accordance with the frequencies with which the various lettersfollow the first one. This requires a table of digram frequencies pi j. In the third-order approximation, trigram structure is introduced. Each letter is chosen with probabilities which depend on the preceding twoletters. 3. THE SERIES OF APPROXIMATIONS TO ENGLISH To give a visual idea of how this series of processes approaches a language, typical sequences in the approx-imations to English have been constructed and are given below. In all cases we have assumed a 27- symbol"alphabet," the 26 letters and a space. 1. Zero-order approximation (symbols independent and equiprobable). XFOML RXKHRJFFJUJ ZLPWCFWKCYJ FFJEYVKCQSGHYD QPAAMKBZAACIBZL-HJQD. 2. First-order approximation (symbols independent but with frequencies of English text). OCRO HLI RGWR NMIELWIS EU LL NBNESEBYA TH EEI ALHENHTTPA OOBTTVANAH BRL. 3. Second-order approximation (digram structure as in English). ON IE ANTSOUTINYS ARE T INCTORE ST BE S DEAMY ACHIN D ILONASIVE TU-COOWE AT TEASONARE FUSO TIZIN ANDY TOBE SEACE CTISBE. 4. Third-order approximation (trigram structure as in English). IN NO IST LAT WHEY CRATICT FROURE BIRS GROCID PONDENOME OF DEMONS-TURES OF THE REPTAGIN IS REGOACTIONA OF CRE. 5. First-order word approximation. Rather than continue with tetragram, , n-gram structure it is easier : : : and better to jump at this point to word units. Here words are chosen independently but with theirappropriate frequencies. REPRESENTING AND SPEEDILY IS AN GOOD APT OR COME CAN DIFFERENT NAT-URAL HERE HE THE A IN CAME THE TO OF TO EXPERT GRAY COME TO FURNISHESTHE LINE MESSAGE HAD BE THESE. 6. Second-order word approximation. The word transition probabilities are correct but no further struc- ture is included. THE HEAD AND IN FRONTAL ATTACK ON AN ENGLISH WRITER THAT THE CHAR- ACTER OF THIS POINT IS THEREFORE ANOTHER METHOD FOR THE LETTERS THATTHE TIME OF WHO EVER TOLD THE PROBLEM FOR AN UNEXPECTED. The resemblance to ordinary English text increases quite noticeably at each of the above steps. Note that these samples have reasonably good structure out to about twice the range that is taken into account in theirconstruction. Thus in (3) the statistical process insures reasonable text for two-letter sequences, but four-letter sequences from the sample can usually be fitted into good sentences. In (6) sequences of four or morewords can easily be placed in sentences without unusual or strained constructions. The particular sequenceof ten words "attack on an English writer that the character of this" is not at all unreasonable. It appears thenthat a sufficiently complex stochastic process will give a satisfactory representation of a discrete source. The first two samples were constructed by the use of a book of random numbers in conjunction with (for example 2) a table of letter frequencies. This method might have been continued for (3), (4) and (5),since digram, trigram and word frequency tables are available, but a simpler equivalent method was used. 7 =============================================================================== To construct (3) for example, one opens a book at random and selects a letter at random on the page. Thisletter is recorded. The book is then opened to another page and one reads until this letter is encountered.The succeeding letter is then recorded. Turning to another page this second letter is searched for and thesucceeding letter recorded, etc. A similar process was used for (4), (5) and (6). It would be interesting iffurther approximations could be constructed, but the labor involved becomes enormous at the next stage. 4. GRAPHICAL REPRESENTATION OF A MARKOFF PROCESS Stochastic processes of the type described above are known mathematically as discrete Markoff processesand have been extensively studied in the literature.6 The general case can be described as follows: Thereexist a finite number of possible "states" of a system; S1 S2 Sn. In addition there is a set of transition ; ; : : : ; probabilities; pi jthe probability that if the system is in state Siit will next go to state S j. To make this Markoff process into an information source we need only assume that a letter is produced for each transitionfrom one state to another. The states will correspond to the "residue of influence" from preceding letters. The situation can be represented graphically as shown in Figs. 3, 4 and 5. The "states" are the junction A .1 .4 B E .2 .1 C D .2 Fig. 3 -- A graph corresponding to the source in example B. points in the graph and the probabilities and letters produced for a transition are given beside the correspond-ing line. Figure 3 is for the example B in Section 2, while Fig. 4 corresponds to the example C. In Fig. 3 B C A A .8 .2 .5 .5 B C B .4 .5 .1 Fig. 4 -- A graph corresponding to the source in example C. there is only one state since successive letters are independent. In Fig. 4 there are as many states as letters.If a trigram example were constructed there would be at most n2 states corresponding to the possible pairsof letters preceding the one being chosen. Figure 5 is a graph for the case of word structure in example D.Here S corresponds to the "space" symbol. 5. ERGODIC AND MIXED SOURCES As we have indicated above a discrete source for our purposes can be considered to be represented by aMarkoff process. Among the possible discrete Markoff processes there is a group with special propertiesof significance in communication theory. This special class consists of the "ergodic" processes and weshall call the corresponding sources ergodic sources. Although a rigorous definition of an ergodic process issomewhat involved, the general idea is simple. In an ergodic process every sequence produced by the process 6For a detailed treatment see M. FrŽechet, MŽethode des fonctions arbitraires. ThŽeorie des ŽevŽenements en cha^ine dans le cas d'un nombre fini d'Žetats possibles. Paris, Gauthier- Villars, 1938. 8 =============================================================================== is the same in statistical properties. Thus the letter frequencies, digram frequencies, etc., obtained fromparticular sequences, will, as the lengths of the sequences increase, approach definite limits independentof the particular sequence. Actually this is not true of every sequence but the set for which it is false hasprobability zero. Roughly the ergodic property means statistical homogeneity. All the examples of artificial languages given above are ergodic. This property is related to the structure of the corresponding graph. If the graph has the following two properties7 the corresponding process willbe ergodic: 1. The graph does not consist of two isolated parts A and B such that it is impossible to go from junction points in part A to junction points in part B along lines of the graph in the direction of arrows and alsoimpossible to go from junctions in part B to junctions in part A. 2. A closed series of lines in the graph with all arrows 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 series BEBESis a circuit of length 5. The second property required is that the greatest common divisor of the lengthsof all circuits in the graph be one. D E B E S A B E E D A B D E S B D E C A E E B B D E A D B E E A S Fig. 5 -- A graph corresponding to the source in example D. If the first condition is satisfied but the second 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 ddifferent classes which are statistically the same apart from a shift of the origin (i.e., which letter in the sequence iscalled letter 1). By a shift of from 0 up to d 1 any sequence can be made statistically equivalent to any , other. A simple example with d 2 is the following: There are three possible letters a b c. Letter ais = ; ; followed with either bor cwith probabilities 1 and 2 respectively. Either bor cis always followed by letter 3 3 a. Thus a typical sequence is a b a c a c a c a b a c a b a b a c a c: 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 the second condition is also satisfied for each subgraph. We have inthis case what may be called a "mixed" source made up of a number of pure components. The componentscorrespond to the various subgraphs. If L1, L2, L3 are the component sources we may write ; : : : L p1L1 p2L2 p3L3 = + + + 7These are restatements in terms of the graph of conditions given in FrŽechet. 9 =============================================================================== where piis the probability of the component source Li. Physically the situation represented is this: There are several different sources L1, L2, L3 which are ; : : : each of homogeneous statistical structure (i.e., they are ergodic). We do not know a prioriwhich is to beused, but once the sequence starts in a given pure component Li, it continues indefinitely according to thestatistical structure of that component. As an example one may take two of the processes defined above and assume p1 2 and p2 8. A = : = : sequence from the mixed source L 2L1 8L2 = : + : would be obtained by choosing first L1 or L2 with probabilities .2 and .8 and after this choice generating asequence from whichever was chosen. 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 averages over the ensemble of possible sequences (the probabilityof a discrepancy being zero). For example the relative frequency of the letter A in a particular infinitesequence will be, with probability one, equal to its relative frequency in the ensemble of sequences. If Piis the probability of state iand pi jthe transition probability to state j, then for the process to be stationary it is clear that the Pimust satisfy equilibrium conditions: Pj Pipi j = : i In the ergodic case it can be shown that with any starting conditions the probabilities Pj Nof being in state jafter Nsymbols, approach the equilibrium values as N . ! 6. CHOICE, UNCERTAINTY AND ENTROPY We have represented a discrete information source as a Markoff process. Can we define a quantity whichwill measure, in some sense, how much information is "produced" by such a process, or better, at what rateinformation is produced? Suppose we have a set of possible events whose probabilities of occurrence are p1 p2 pn. These ; ; : : : ; probabilities are known but that is all we know concerning which event will occur. Can we find a measureof how much "choice" is involved in the selection of the event or of how uncertain we are of the outcome? If there is such a measure, say H p1 p2 pn, it is reasonable to require of it the following properties: ; ; : : : ; 1. Hshould be continuous in the pi. 2. If all the p 1 iare equal, pi , then Hshould be a monotonic increasing function of n. With equally = n likely events there is more choice, or uncertainty, when there are more possible events. 3. If a choice be broken down into two successive choices, the original Hshould be the weighted sum of the individual values of H. The meaning of this is illustrated in Fig. 6. At the left we have three 1 2 1 2 1 2 1 3 2 3 1 3 1 2 1 6 1 3 1 6 Fig. 6 -- Decomposition of a choice from three possibilities. possibilities p 1 1 1 1 , p2 , p3 . On the right we first choose between two possibilities each with = 2 = 3 = 6 probability 1 , and if the second occurs make another choice with probabilities 2 , 1 . The final results 2 3 3 have the same probabilities as before. We require, in this special case, that H1 1 1 H1 1 1 H2 1 2 ; 3 ; 6 = 2 ; 2 + 2 3 ; 3 : The coefficient 1 is because this second choice only occurs half the time. 2 10 =============================================================================== In Appendix 2, the following result is established: Theorem 2:The only Hsatisfying the three above assumptions is of the form: n H K pilog pi = , i1 = where Kis a positive constant. This theorem, and the assumptions required for its proof, are in no way necessary for the present theory. It is given chiefly to lend a certain plausibility to some of our later definitions. The real justification of thesedefinitions, however, will reside in their implications. Quantities of the form H pilog pi(the constant Kmerely amounts to a choice of a unit of measure) = , play a central role in information theory as measures of information, choice and uncertainty. The form of Hwill be recognized as that of entropy as defined in certain formulations of statistical mechanics8 where piisthe probability of a system being in cell iof its phase space. His then, for example, the Hin Boltzmann'sfamous Htheorem. We shall call H pilog pithe entropy of the set of probabilities p1 pn. If xis a = , ; : : : ; chance variable we will write H xfor its entropy; thus xis not an argument of a function but a label for a number, to differentiate it from H ysay, the entropy of the chance variable y. The entropy in the case of two possibilities with probabilities pand q 1 p, namely = , H plog p qlogq = , + is plotted in Fig. 7 as a function of p. 1.0 .9 .8 .7 H BITS .6 .5 .4 .3 .2 .1 0 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 p Fig. 7 -- Entropy in the case of two possibilities with probabilities pand 1 p. , The quantity Hhas a number of interesting properties which further substantiate it as a reasonable measure of choice or information. 1. H 0 if and only if all the pibut one are zero, this one having the value unity. Thus only when we = are certain of the outcome does Hvanish. Otherwise His positive. 2. For a given n, His a maximum and equal to log nwhen all the piare equal (i.e., 1 ). This is also n intuitively the most uncertain situation. 8See, for example, R. C. Tolman, Principles of Statistical Mechanics,Oxford, Clarendon, 1938. 11 =============================================================================== 3. Suppose there are two events, xand y, in question with mpossibilities for the first and nfor the second. Let p i jbe the probability of the joint occurrence of ifor the first and jfor the second. The entropy of the ; joint event is H x y p i jlog p i j ; = , ; ; i j ; while H x p i jlogp i j = , ; ; i j j ; H y p i jlogp i j = , ; ; : i j i ; It is easily shown that H x y H x H y ; + with equality only if the events are independent (i.e., p i j p i p j). The uncertainty of a joint event is ; = less than or equal to the sum of the individual uncertainties. 4. Any change toward equalization of the probabilities p1 p2 pnincreases H. Thus if p1 p2 and ; ; : : : ; we increase p1, decreasing p2 an equal amount so that p1 and p2 are more nearly equal, then Hincreases.More generally, if we perform any "averaging" operation on the piof the form p0 i ai j p j = j where i ai j 1, and all ai j 0, then Hincreases (except in the special case where this transfor- = j ai j= mation amounts to no more than a permutation of the p jwith Hof course remaining the same). 5. Suppose there are two chance events xand yas in 3, not necessarily independent. For any particular value ithat xcan assume there is a conditional probability pi jthat yhas the value j. This is given by p i j ; pi j = : j p i j ; We define the conditional entropyof y, Hx yas the average of the entropy of yfor each value of x, weighted according to the probability of getting that particular x. That is Hx y p i jlogpi j = , ; : i j ; This quantity measures how uncertain we are of yon the average when we know x. Substituting the value of pi jwe obtain Hx y p i jlog p i jp i jlogp i j = , ; ; + ; ; i j i j j ; ; H x y H x = ; , or H x y H x Hx y ; = + : The uncertainty (or entropy) of the joint event x yis the uncertainty of xplus the uncertainty of ywhen xis ; known. 6. From 3 and 5 we have H x H y H x y H x Hx y + ; = + : Hence H y Hx y : The uncertainty of yis never increased by knowledge of x. It will be decreased unless xand yare independentevents, in which case it is not changed. 12 =============================================================================== 7. THE ENTROPY OF AN INFORMATION SOURCE Consider a discrete source of the finite state type considered above. For each possible state ithere will be aset of probabilities pi jof producing the various possible symbols j. Thus there is an entropy Hifor each state. The entropy of the source will be defined as the average of these Hiweighted in accordance with theprobability of occurrence of the states in question: H PiHi = iPipi jlogpi j = , : i j ; This is the entropy of the source per symbol of text. If the Markoff process is proceeding at a definite timerate there is also an entropy per second H0 fiHi = i where fiis the average frequency (occurrences per second) of state i. Clearly H0 mH = where mis the average number of symbols produced per second. Hor H0 measures the amount of informa-tion generated by the source per symbol or per second. If the logarithmic base is 2, they will represent bitsper symbol or per second. If successive symbols are independent then His simply pilog piwhere piis the probability of sym- , bol i. Suppose in this case we consider a long message of Nsymbols. It will contain with high probabilityabout p1Noccurrences of the first symbol, p2Noccurrences of the second, etc. Hence the probability of thisparticular message will be roughly p pp1N pp2N ppnN = 1 2 n or log p: N pilog pi = i log p: NH = , log 1 p H = : = : N His thus approximately the logarithm of the reciprocal probability of a typical long sequence divided by thenumber of symbols in the sequence. The same result holds for any source. Stated more precisely we have(see Appendix 3): Theorem 3:Given any 0 and 0, we can find an N0 such that the sequences of any length N N0 fall into two classes: 1. A set whose total probability is less than . 2. The remainder, all of whose members have probabilities satisfying the inequality log p1 , H , : N log p1 , In other words we are almost certain to have very close to Hwhen Nis large. N A closely related result deals with the number of sequences of various probabilities. Consider again the sequences of length Nand let them be arranged in order of decreasing probability. We define n qto be the number we must take from this set starting with the most probable one in order to accumulate a totalprobability qfor those taken. 13 =============================================================================== Theorem 4: log n q Lim H = N N ! when qdoes not equal 0 or 1. We may interpret log n qas the number of bits required to specify the sequence when we consider only log n q the most probable sequences with a total probability q. Then is the number of bits per symbol for N the specification. The theorem says that for large Nthis will be independent of qand equal to H. The rateof growth of the logarithm of the number of reasonably probable sequences is given by H, regardless of ourinterpretation of "reasonably probable." Due to these results, which are proved in Appendix 3, it is possiblefor most purposes to treat the long sequences as though there were just 2HNof them, each with a probability2 HN , . The next two theorems show that Hand H0 can be determined by limiting operations directly from the statistics of the message sequences, without reference to the states and transition probabilities betweenstates. Theorem 5: Let p Bibe the probability of a sequence Biof symbols from the source. Let 1 GN p Bilogp Bi = , N i where the sum is over all sequences Bicontaining Nsymbols. Then GNis a monotonic decreasing functionof Nand Lim GN H = : N ! Theorem 6:Let p Bi S jbe the probability of sequence Bifollowed by symbol S jand pB S j ; i = p Bi S j p Bibe the conditional probability of S jafter Bi. Let ; = FN p Bi SjlogpB Sj = , ; i i j ; where the sum is over all blocks Biof N 1 symbols and over all symbols S j. Then FNis a monotonic , decreasing function of N, FN NGN N 1 GN1 = , , ; , 1 N GN Fn = ; N n1 = FN GN ; and LimN FN H. = ! These results are derived in Appendix 3. They show that a series of approximations to Hcan be obtained by considering only the statistical structure of the sequences extending over 1 2 Nsymbols. FNis the ; ; : : : ; better approximation. In fact FNis the entropy of the Nth order approximation to the source of the typediscussed above. If there are no statistical influences extending over more than Nsymbols, that is if theconditional probability of the next symbol knowing the preceding N 1 is not changed by a knowledge of , any before that, then FN H. FNof course is the conditional entropy of the next symbol when the N 1 = , preceding ones are known, while GNis the entropy per symbol of blocks of Nsymbols. The ratio of the entropy of a source to the maximum value it could have while still restricted to the same symbols will be called its relative entropy. This is the maximum compression possible when we encode intothe same alphabet. One minus the relative entropy is the redundancy. The redundancy of ordinary English,not considering statistical structure over greater distances than about eight letters, is roughly 50%. Thismeans that when we write English half of what we write is determined by the structure of the language andhalf is chosen freely. The figure 50% was found by several independent methods which all gave results in 14 =============================================================================== this neighborhood. One is by calculation of the entropy of the approximations to English. A second methodis to delete a certain fraction of the letters from a sample of English text and then let someone attempt torestore them. If they can be restored when 50% are deleted the redundancy must be greater than 50%. Athird method depends on certain known results in cryptography. Two extremes of redundancy in English prose are represented by Basic English and by James Joyce's book "Finnegans Wake". The Basic English vocabulary is limited to 850 words and the redundancy is veryhigh. This is reflected in the expansion that occurs when a passage is translated into Basic English. Joyceon the other hand enlarges the vocabulary and is alleged to achieve a compression of semantic content. The redundancy of a language is related to the existence of crossword puzzles. If the redundancy is zero any sequence of letters is a reasonable text in the language and any two-dimensional array of lettersforms a crossword puzzle. If the redundancy is too high the language imposes too many constraints for largecrossword puzzles to be possible. A more detailed analysis shows that if we assume the constraints imposedby the language are of a rather chaotic and random nature, large crossword puzzles are just possible whenthe redundancy is 50%. If the redundancy is 33%, three-dimensional crossword puzzles should be possible,etc. 8. REPRESENTATION OF THE ENCODING AND DECODING OPERATIONS We have yet to represent mathematically the operations performed by the transmitter and receiver in en-coding and decoding the information. Either of these will be called a discrete transducer. The input to thetransducer is a sequence of input symbols and its output a sequence of output symbols. The transducer mayhave an internal memory so that its output depends not only on the present input symbol but also on the pasthistory. We assume that the internal memory is finite, i.e., there exist a finite number mof possible states ofthe transducer and that its output is a function of the present state and the present input symbol. The nextstate will be a second function of these two quantities. Thus a transducer can be described by two functions: yn f xn n = ; n1 g xn n = ; + where xnis the nth input symbol, nis the state of the transducer when the nth input symbol is introduced, ynis the output symbol (or sequence of output symbols) produced when xnis introduced if the state is n. If the output symbols of one transducer can be identified with the input symbols of a second, they can be connected in tandem and the result is also a transducer. If there exists a second transducer which operateson the output of the first and recovers the original input, the first transducer will be called non-singular andthe second will be called its inverse. Theorem 7:The output of a finite state transducer driven by a finite state statistical source is a finite state statistical source, with entropy (per unit time) less than or equal to that of the input. If the transduceris non-singular they are equal. Let represent the state of the source, which produces a sequence of symbols xi; and let be the state of the transducer, which produces, in its output, blocks of symbols y j. The combined system can be representedby the "product state space" of pairs . Two points in the space 1 1 and 2 2 , are connected by ; ; ; a line if 1 can produce an xwhich changes 1 to 2, and this line is given the probability of that xin this case. The line is labeled with the block of y jsymbols produced by the transducer. The entropy of the outputcan be calculated as the weighted sum over the states. If we sum first on each resulting term is less than or equal to the corresponding term for , hence the entropy is not increased. If the transducer is non-singularlet its output be connected to the inverse transducer. If H0 , H0 and H0 are the output entropies of the source, 1 2 3 the first and second transducers respectively, then H0 H0 H0 H0 and therefore H0 H0 . 1 2 3 = 1 1 = 2 15 =============================================================================== Suppose we have a system of constraints on possible sequences of the type which can be represented by s a linear graph as in Fig. 2. If probabilities p were assigned to the various lines connecting state ito state j i j this would become a source. There is one particular assignment which maximizes the resulting entropy (seeAppendix 4). Theorem 8:Let the system of constraints considered as a channel have a capacity C logW. If we = assign s B j s p W,`ij i j= Bi s where is the duration of the sth symbol leading from state ito state jand the Bisatisfy `i j s B i j i BjW,` = s j ; then His maximized and equal to C. By proper assignment of the transition probabilities the entropy of symbols on a channel can be maxi- mized at the channel capacity. 9. THE FUNDAMENTAL THEOREM FOR A NOISELESS CHANNEL We will now justify our interpretation of Has the rate of generating information by proving that Hdeter-mines the channel capacity required with most efficient coding. Theorem 9:Let a source have entropy Hbits per symbol and a channel have a capacity Cbits per second . Then it is possible to encode the output of the source in such a way as to transmit at the average C rate symbols per second over the channel where is arbitrarily small. It is not possible to transmit at , H C an average rate greater than . H C The converse part of the theorem, that cannot be exceeded, may be proved by noting that the entropy H of the channel input per second is equal to that of the source, since the transmitter must be non-singular, andalso this entropy cannot exceed the channel capacity. Hence H0 Cand the number of symbols per second H0 H C H. = = = The first part of the theorem will be proved in two different ways. The first method is to consider the set of all sequences of Nsymbols produced by the source. For Nlarge we can divide these into two groups,one containing less than 2 H N + members and the second containing less than 2RNmembers (where Ris the logarithm of the number of different symbols) and having a total probability less than . As Nincreases and approach zero. The number of signals of duration Tin the channel is greater than 2 C T , with small when Tis large. if we choose H T N = + C then there will be a sufficient number of sequences of channel symbols for the high probability group whenNand Tare sufficiently large (however small ) and also some additional ones. The high probability group is coded in an arbitrary one- to-one way into this set. The remaining sequences are represented by largersequences, starting and ending with one of the sequences not used for the high probability group. Thisspecial sequence acts as a start and stop signal for a different code. In between a sufficient time is allowedto give enough different sequences for all the low probability messages. This will require R T1 N = + ' C where is small. The mean rate of transmission in message symbols per second will then be greater than ' 1 , T T 1 , 1 H R 1 1 , + = , + + + ' : N N C C 16 =============================================================================== C As Nincreases , and approach zero and the rate approaches . ' H Another method of performing this coding and thereby proving the theorem can be described as follows: Arrange the messages of length Nin order of decreasing probability and suppose their probabilities are p 1 1 p2 p3 pn. Let Ps s, pi; that is Psis the cumulative probability up to, but not including, ps. = 1 We first encode into a binary system. The binary code for message sis obtained by expanding Psas a binarynumber. The expansion is carried out to msplaces, where msis the integer satisfying: 1 1 log2 ms 1 log + : p 2 s ps Thus the messages of high probability are represented by short codes and those of low probability by longcodes. From these inequalities we have 1 1 ps 2ms 2ms1 : , The code for Pswill differ from all succeeding ones in one or more of its msplaces, since all the remainingPiare at least 1 larger and their binary expansions therefore differ in the first m 2ms splaces. Consequently all the codes are different and it is possible to recover the message from its code. If the channel sequences arenot already sequences of binary digits, they can be ascribed binary numbers in an arbitrary fashion and thebinary code thus translated into signals suitable for the channel. The average number H0 of binary digits used per symbol of original message is easily estimated. We have 1 H0 msps = : N But, 1 1 1 1 1 log ps msps 1 log ps + N 2 p 2 s N N ps and therefore, 1 GN H0 GN + N As Nincreases GNapproaches H, the entropy of the source and H0 approaches H. We see from this that the inefficiency in coding, when only a finite delay of Nsymbols is used, need not be greater than 1 plus the difference between the true entropy Hand the entropy G N Ncalculated for sequences of length N. The per cent excess time needed over the ideal is therefore less than GN 1 1 + , : H HN This method of encoding is substantially the same as one found independently by R. M. Fano.9 His method is to arrange the messages of length Nin order of decreasing probability. Divide this series into twogroups of as nearly equal probability as possible. If the message is in the first group its first binary digitwill be 0, otherwise 1. The groups are similarly divided into subsets of nearly equal probability and theparticular subset determines the second binary digit. This process is continued until each subset containsonly one message. It is easily seen that apart from minor differences (generally in the last digit) this amountsto the same thing as the arithmetic process described above. 10. DISCUSSION AND EXAMPLES In order to obtain the maximum power transfer from a generator to a load, a transformer must in general beintroduced so that the generator as seen from the load has the load resistance. The situation here is roughlyanalogous. The transducer which does the encoding should match the source to the channel in a statisticalsense. The source as seen from the channel through the transducer should have the same statistical structure 9Technical Report No. 65, The Research Laboratory of Electronics, M.I.T., March 17, 1949. 17 =============================================================================== as the source which maximizes the entropy in the channel. The content of Theorem 9 is that, although anexact match is not in general possible, we can approximate it as closely as desired. The ratio of the actualrate of transmission to the capacity Cmay be called the efficiency of the coding system. This is of courseequal to the ratio of the actual entropy of the channel symbols to the maximum possible entropy. In general, ideal or nearly ideal encoding requires a long delay in the transmitter and receiver. In the noiseless case which we have been considering, the main function of this delay is to allow reasonably goodmatching of probabilities to corresponding lengths of sequences. With a good code the logarithm of thereciprocal probability of a long message must be proportional to the duration of the corresponding signal, infact log p1 , C , T must be small for all but a small fraction of the long messages. If a source can produce only one particular message its entropy is zero, and no channel is required. For example, a computing machine set up to calculate the successive digits of produces a definite sequence with no chance element. No channel is required to "transmit" this to another point. One could construct asecond machine to compute the same sequence at the point. However, this may be impractical. In such a casewe can choose to ignore some or all of the statistical knowledge we have of the source. We might considerthe digits of to be a random sequence in that we construct a system capable of sending any sequence of digits. In a similar way we may choose to use some of our statistical knowledge of English in constructinga code, but not all of it. In such a case we consider the source with the maximum entropy subject to thestatistical conditions we wish to retain. The entropy of this source determines the channel capacity whichis necessary and sufficient. In the example the only information retained is that all the digits are chosen from the set 0 1 9. In the 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 the first approximation to Englishand its entropy determines the required channel capacity. As a simple example of some of these results consider a source which produces a sequence of letters chosen from among A, B, C, Dwith probabilities 1 , 1 , 1 , 1 , successive symbols being chosen independently. 2 4 8 8 We have , H 1 log 1 1 log 1 2 log 1 = , 2 2 + 4 4 + 8 8 7 bits per symbol = 4 : Thus we can approximate a coding system to encode messages from this source into binary digits with anaverage of 7 binary digit per symbol. In this case we can actually achieve the limiting value by the following 4 code (obtained by the method of the second proof of Theorem 9): A 0 B 10 C 110 D 111 The average number of binary digits used in encoding a sequence of Nsymbols will be 2 , N1 1 1 2 3 7 N 2 + 4 + = : 8 4 It is easily seen that the binary digits 0, 1 have probabilities 1 , 1 so the Hfor the coded sequences is one 2 2 bit per symbol. Since, on the average, we have 7 binary symbols per original letter, the entropies on a time 4 basis are the same. The maximum possible entropy for the original set is log 4 2, occurring when A, B, C, = Dhave probabilities 1 , 1 , 1 , 1 . Hence the relative entropy is 7 . We can translate the binary sequences into 4 4 4 4 8 the original set of symbols on a two-to-one basis by the following table: 00 A0 01 B0 10 C0 11 D0 18 =============================================================================== This double process then encodes the original message into the same symbols but with an average compres-sion ratio 7 . 8 As a second example consider a source which produces a sequence of A's and B's with probability pfor Aand qfor B. If p qwe have H log pp1 p1 p , = , , plog p1 p1 p p , = = , , e : plog = : p In such a case one can construct a fairly good coding of the message on a 0, 1 channel by sending a specialsequence, say 0000, for the infrequent symbol Aand then a sequence indicating the numberof B's followingit. This could be indicated by the binary representation with all numbers containing the special sequencedeleted. All numbers up to 16 are represented as usual; 16 is represented by the next binary number after 16which does not contain four zeros, namely 17 10001, etc. = It can be shown that as p 0 the coding approaches ideal provided the length of the special sequence is ! properly adjusted. PART II: THE DISCRETE CHANNEL WITH NOISE 11. REPRESENTATION OF A NOISY DISCRETE CHANNEL We now consider the case where the signal is perturbed by noise during transmission or at one or the otherof the terminals. This means that the received signal is not necessarily the same as that sent out by thetransmitter. Two cases may be distinguished. If a particular transmitted signal always produces the samereceived signal, i.e., the received signal is a definite function of the transmitted signal, then the effect may becalled distortion. If this function has an inverse -- no two transmitted signals producing the same receivedsignal -- distortion may be corrected, at least in principle, by merely performing the inverse functionaloperation on the received signal. The case of interest here is that in which the signal does not always undergo the same change in trans- mission. In this case we may assume the received signal Eto be a function of the transmitted signal Sand asecond variable, the noise N. E f S N = ; The noise is considered to be a chance variable just as the message was above. In general it may be repre-sented by a suitable stochastic process. The most general type of noisy discrete channel we shall consideris a generalization of the finite state noise-free channel described previously. We assume a finite number ofstates and a set of probabilities p i j ; : ; This is the probability, if the channel is in state and symbol iis transmitted, that symbol jwill be received and the channel left in state . Thus and range over the possible states, iover the possible transmitted signals and jover the possible received signals. In the case where successive symbols are independently per-turbed by the noise there is only one state, and the channel is described by the set of transition probabilities pi j, the probability of transmitted symbol ibeing received as j. If a noisy channel is fed by a source there are two statistical processes at work: the source and the noise. Thus there are a number of entropies that can be calculated. First there is the entropy H xof the source or of the input to the channel (these will be equal if the transmitter is non-singular). The entropy of theoutput of the channel, i.e., the received signal, will be denoted by H y. In the noiseless case H y H x. = The joint entropy of input and output will be H xy. Finally there are two conditional entropies Hx yand Hy x, the entropy of the output when the input is known and conversely. Among these quantities we have the relations H x y H x Hx y H y Hy x ; = + = + : All of these entropies can be measured on a per-second or a per- symbol basis. 19 =============================================================================== 12. EQUIVOCATION AND CHANNEL CAPACITY If the channel is noisy it is not in general possible to reconstruct the original message or the transmittedsignal with certaintyby any operation on the received signal E. There are, however, ways of transmittingthe information which are optimal in combating noise. This is the problem which we now consider. Suppose there are two possible symbols 0 and 1, and we are transmitting at a rate of 1000 symbols per second with probabilities p 1 0 p1 . Thus our source is producing information at the rate of 1000 bits = = 2 per second. During transmission the noise introduces errors so that, on the average, 1 in 100 is receivedincorrectly (a 0 as 1, or 1 as 0). What is the rate of transmission of information? Certainly less than 1000bits per second since about 1% of the received symbols are incorrect. Our first impulse might be to saythe rate is 990 bits per second, merely subtracting the expected number of errors. This is not satisfactorysince it fails to take into account the recipient's lack of knowledge of where the errors occur. We may carryit to an extreme case and suppose the noise so great that the received symbols are entirely independent ofthe transmitted symbols. The probability of receiving 1 is 1 whatever was transmitted and similarly for 0. 2 Then about half of the received symbols are correct due to chance alone, and we would be giving the systemcredit for transmitting 500 bits per second while actually no information is being transmitted at all. Equally"good" transmission would be obtained by dispensing with the channel entirely and flipping a coin at thereceiving point. Evidently the proper correction to apply to the amount of information transmitted is the amount of this information which is missing in the received signal, or alternatively the uncertainty when we have receiveda signal of what was actually sent. From our previous discussion of entropy as a measure of uncertainty itseems reasonable to use the conditional entropy of the message, knowing the received signal, as a measureof this missing information. This is indeed the proper definition, as we shall see later. Following this ideathe rate of actual transmission, R, would be obtained by subtracting from the rate of production (i.e., theentropy of the source) the average rate of conditional entropy. R H x Hy x = , The conditional entropy Hy xwill, for convenience, be called the equivocation. It measures the average ambiguity of the received signal. In the example considered above, if a 0 is received the a posterioriprobability that a 0 was transmitted is .99, and that a 1 was transmitted is .01. These figures are reversed if a 1 is received. Hence Hy x 99 log 99 0 01 log0 01 = , : : + : : 081 bits/symbol = : or 81 bits per second. We may say that the system is transmitting at a rate 1000 81 919 bits per second. , = In the extreme case where a 0 is equally likely to be received as a 0 or 1 and similarly for 1, the a posterioriprobabilities are 1 , 1 and 2 2 H 1 1 y x log 1 log 1 = , 2 2 + 2 2 1 bit per symbol = or 1000 bits per second. The rate of transmission is then 0 as it should be. The following theorem gives a direct intuitive interpretation of the equivocation and also serves to justify it as the unique appropriate measure. We consider a communication system and an observer (or auxiliarydevice) who can see both what is sent and what is recovered (with errors due to noise). This observer notesthe errors in the recovered message and transmits data to the receiving point over a "correction channel" toenable the receiver to correct the errors. The situation is indicated schematically in Fig. 8. Theorem 10:If the correction channel has a capacity equal to Hy xit is possible to so encode the correction data as to send it over this channel and correct all but an arbitrarily small fraction of the errors.This is not possible if the channel capacity is less than Hy x. 20 =============================================================================== CORRECTION DATA OBSERVER M M0 M SOURCE TRANSMITTER RECEIVER CORRECTING DEVICE Fig. 8 -- Schematic diagram of a correction system. Roughly then, Hy xis the amount of additional information that must be supplied per second at the receiving point to correct the received message. To prove the first part, consider long sequences of received message M0 and corresponding original message M. There will be logarithmically T Hy xof the M's which could reasonably have produced each M0. Thus we have T Hy xbinary digits to send each Tseconds. This can be done with frequency of errors on a channel of capacity Hy x. The second part can be proved by noting, first, that for any discrete chance variables x, y, z Hy x z Hy x ; : The left-hand side can be expanded to give Hy z Hyz x Hy x + Hyz x Hy x Hy z Hy x H z , , : If we identify xas the output of the source, yas the received signal and zas the signal sent over the correctionchannel, then the right-hand side is the equivocation less the rate of transmission over the correction channel.If the capacity of this channel is less than the equivocation the right-hand side will be greater than zero andHyz x 0. But this is the uncertainty of what was sent, knowing both the received signal and the correction signal. If this is greater than zero the frequency of errors cannot be arbitrarily small. Example: Suppose the errors occur at random in a sequence of binary digits: probability pthat a digit is wrongand q 1 pthat it is right. These errors can be corrected if their position is known. Thus the = , correction channel need only send information as to these positions. This amounts to transmittingfrom a source which produces binary digits with probability pfor 1 (incorrect) and qfor 0 (correct).This requires a channel of capacity plog p qlogq , + which is the equivocation of the original system. The rate of transmission Rcan be written in two other forms due to the identities noted above. We have R H x Hy x = , H y Hx y = , H x H y H x y = + , ; : 21 =============================================================================== The first defining expression has already been interpreted as the amount of information sent less the uncer-tainty of what was sent. The second measures the amount received less the part of this which is due to noise.The third is the sum of the two amounts less the joint entropy and therefore in a sense is the number of bitsper second common to the two. Thus all three expressions have a certain intuitive significance. The capacity Cof a noisy channel should be the maximum possible rate of transmission, i.e., the rate when the source is properly matched to the channel. We therefore define the channel capacity by , C Max H x Hy x = , where the maximum is with respect to all possible information sources used as input to the channel. If thechannel is noiseless, Hy x 0. The definition is then equivalent to that already given for a noiseless channel = since the maximum entropy for the channel is its capacity. 13. THE FUNDAMENTAL THEOREM FOR A DISCRETE CHANNEL WITH NOISE It may seem surprising that we should define a definite capacity Cfor a noisy channel since we can neversend certain information in such a case. It is clear, however, that by sending the information in a redundantform the probability of errors can be reduced. For example, by repeating the message many times and by astatistical study of the different received versions of the message the probability of errors could be made verysmall. One would expect, however, that to make this probability of errors approach zero, the redundancyof the encoding must increase indefinitely, and the rate of transmission therefore approach zero. This is byno means true. If it were, there would not be a very well defined capacity, but only a capacity for a givenfrequency of errors, or a given equivocation; the capacity going down as the error requirements are mademore stringent. Actually the capacity Cdefined above has a very definite significance. It is possible to sendinformation at the rate Cthrough the channel with as small a frequency of errors or equivocation as desiredby proper encoding. This statement is not true for any rate greater than C. If an attempt is made to transmitat a higher rate than C, say C R1, then there will necessarily be an equivocation equal to or greater than the + excess R1. Nature takes payment by requiring just that much uncertainty, so that we are not actually gettingany more than Cthrough correctly. The situation is indicated in Fig. 9. The rate of information into the channel is plotted horizontally and the equivocation vertically. Any point above the heavy line in the shaded region can be attained and thosebelow cannot. The points on the line cannot in general be attained, but there will usually be two points onthe line that can. These results are the main justification for the definition of Cand will now be proved. Theorem 11: Let a discrete channel have the capacity Cand a discrete source the entropy per second H. If H Cthere exists a coding system such that the output of the source can be transmitted over the channel with an arbitrarily small frequency of errors (or an arbitrarily small equivocation). If H Cit is possible to encode the source so that the equivocation is less than H C where is arbitrarily small. There is no , + method of encoding which gives an equivocation less than H C. , The method of proving the first part of this theorem is not by exhibiting a coding method having the desired properties, but by showing that such a code must exist in a certain group of codes. In fact we will ATTAINABLE Hy x REGION 1.0 = OPE SL C H x Fig. 9 -- The equivocation possible for a given input entropy to a channel. 22 =============================================================================== average the frequency of errors over this group and show that this average can be made less than . If theaverage of a set of numbers is less than there must exist at least one in the set which is less than . This will establish the desired result. The capacity Cof a noisy channel has been defined as , C Max H x Hy x = , where xis the input and ythe output. The maximization is over all sources which might be used as input tothe channel. Let S0 be a source which achieves the maximum capacity C. If this maximum is not actually achieved by any source let S0 be a source which approximates to giving the maximum rate. Suppose S0 is used asinput to the channel. We consider the possible transmitted and received sequences of a long duration T. Thefollowing will be true: 1. The transmitted sequences fall into two classes, a high probability group with about 2T H x members and the remaining sequences of small total probability. 2. Similarly the received sequences have a high probability set of about 2T H y members and a low probability set of remaining sequences. 3. Each high probability output could be produced by about 2THy x inputs. The probability of all other cases has a small total probability. All the 's and 's implied by the words "small" and "about" in these statements approach zero as we allow Tto increase and S0 to approach the maximizing source. The situation is summarized in Fig. 10 where the input sequences are points on the left and output sequences points on the right. The fan of cross lines represents the range of possible causes for a typicaloutput. E M 2H x T HIGH PROBABILITY 2H y T MESSAGES HIGH PROBABILITY RECEIVED SIGNALS 2Hy x T REASONABLE CAUSES FOR EACH E 2Hx y T REASONABLE EFFECTS FOR EACH M Fig. 10 -- Schematic representation of the relations between inputs and outputs in a channel. Now suppose we have another source producing information at rate Rwith R C. In the period Tthis source will have 2TRhigh probability messages. We wish to associate these with a selection of the possiblechannel inputs in such a way as to get a small frequency of errors. We will set up this association in all 23 =============================================================================== possible ways (using, however, only the high probability group of inputs as determined by the source S0)and average the frequency of errors for this large class of possible coding systems. This is the same ascalculating the frequency of errors for a random association of the messages and channel inputs of durationT. Suppose a particular output y1 is observed. What is the probability of more than one message in the setof possible causes of y x 1? There are 2T Rmessages distributed at random in 2T H points. The probability of a particular point being a message is thus 2T R H x , : The probability that none of the points in the fan is a message (apart from the actual originating message) is x 2T Hy P 1 2T R H x , = , : Now R H x Hy xso R H x Hy x with positive. Consequently , , = , , x 2T Hy P 1 2 THy x T , , = , approaches (as T ) ! 1 2 T , , : Hence the probability of an error approaches zero and the first part of the theorem is proved. The second part of the theorem is easily shown by noting that we could merely send Cbits per second from the source, completely neglecting the remainder of the information generated. At the receiver theneglected part gives an equivocation H x Cand the part transmitted need only add . This limit can also , be attained in many other ways, as will be shown when we consider the continuous case. The last statement of the theorem is a simple consequence of our definition of C. Suppose we can encode a source with H x C ain such a way as to obtain an equivocation Hy x a with positive. Then = + = , R H x C aand = = + H x Hy x C , = + with positive. This contradicts the definition of Cas the maximum of H x Hy x. , Actually more has been proved than was stated in the theorem. If the average of a set of numbers is p p within of of their maximum, a fraction of at most can be more than below the maximum. Since is arbitrarily small we can say that almost all the systems are arbitrarily close to the ideal. 14. DISCUSSION The demonstration of Theorem 11, while not a pure existence proof, has some of the deficiencies of suchproofs. An attempt to obtain a good approximation to ideal coding by following the method of the proof isgenerally impractical. In fact, apart from some rather trivial cases and certain limiting situations, no explicitdescription of a series of approximation to the ideal has been found. Probably this is no accident but isrelated to the difficulty of giving an explicit construction for a good approximation to a random sequence. An approximation to the ideal would have the property that if the signal is altered in a reasonable way by the noise, the original can still be recovered. In other words the alteration will not in general bring itcloser to another reasonable signal than the original. This is accomplished at the cost of a certain amount ofredundancy in the coding. The redundancy must be introduced in the proper way to combat the particularnoise structure involved. However, any redundancy in the source will usually help if it is utilized at thereceiving point. In particular, if the source already has a certain redundancy and no attempt is made toeliminate it in matching to the channel, this redundancy will help combat noise. For example, in a noiselesstelegraph channel one could save about 50% in time by proper encoding of the messages. This is not doneand most of the redundancy of English remains in the channel symbols. This has the advantage, however,of allowing considerable noise in the channel. A sizable fraction of the letters can be received incorrectlyand still reconstructed by the context. In fact this is probably not a bad approximation to the ideal in manycases, since the statistical structure of English is rather involved and the reasonable English sequences arenot too far (in the sense required for the theorem) from a random selection. 24 =============================================================================== As in the noiseless case a delay is generally required to approach the ideal encoding. It now has the additional function of allowing a large sample of noise to affect the signal before any judgment is madeat the receiving point as to the original message. Increasing the sample size always sharpens the possiblestatistical assertions. The content of Theorem 11 and its proof can be formulated in a somewhat different way which exhibits the connection with the noiseless case more clearly. Consider the possible signals of duration Tand supposea subset of them is selected to be used. Let those in the subset all be used with equal probability, and supposethe receiver is constructed to select, as the original signal, the most probable cause from the subset, when aperturbed signal is received. We define N T qto be the maximum number of signals we can choose for the ; subset such that the probability of an incorrect interpretation is less than or equal to q. log N T q Theorem 12:Lim ; C, where Cis the channel capacity, provided that qdoes not equal 0 or = T T ! 1. In other words, no matter how we set out limits of reliability, we can distinguish reliably in time T enough messages to correspond to about CTbits, when Tis sufficiently large. Theorem 12 can be comparedwith the definition of the capacity of a noiseless channel given in Section 1. 15. EXAMPLE OF A DISCRETE CHANNEL AND ITS CAPACITY A simple example of a discrete channel is indicated in Fig. 11. There are three possible symbols. The first isnever affected by noise. The second and third each have probability pof coming through undisturbed, andqof being changed into the other of the pair. We have (letting plog p qlogqand Pand Qbe the = , + p q TRANSMITTED RECEIVED SYMBOLS SYMBOLS q p Fig. 11 -- Example of a discrete channel. probabilities of using the first and second symbols) H x Plog P 2QlogQ = , , Hy x 2Q = : We wish to choose Pand Qin such a way as to maximize H x Hy x, subject to the constraint P 2Q 1. , + = Hence we consider U Plog P 2QlogQ 2Q P 2Q = , , , + + U 1 logP 0 = , , + = P U 2 2 logQ 2 2 0 = , , , + = : Q Eliminating log P log Q = + P Qe Q = = 25 =============================================================================== 1 P Q = = : 2 2 + + The channel capacity is then 2 C log + = : Note how this checks the obvious values in the cases p 1 and p 1 . In the first, 1 and C log 3, = = 2 = = which is correct since the channel is then noiseless with three possible symbols. If p 1 , 2 and = 2 = C log 2. Here the second and third symbols cannot be distinguished at all and act together like one = symbol. The first symbol is used with probability P 1 and the second and third together with probability = 2 1 . This may be distributed between them in any desired way and still achieve the maximum capacity. 2 For intermediate values of pthe channel capacity will lie between log 2 and log 3. The distinction between the second and third symbols conveys some information but not as much as in the noiseless case.The first symbol is used somewhat more frequently than the other two because of its freedom from noise. 16. THE CHANNEL CAPACITY IN CERTAIN SPECIAL CASES If the noise affects successive channel symbols independently it can be described by a set of transitionprobabilities pi j. This is the probability, if symbol iis sent, that jwill be received. The maximum channelrate is then given by the maximum of PipijlogPipijPipijlogpij , + i j i i j ; ; where we vary the Pisubject to Pi 1. This leads by the method of Lagrange to the equations, = ps j ps jlog s 1 2 = = ; ; : : : : j i Pi pi j Multiplying by Psand summing on sshows that C. Let the inverse of ps j(if it exists) be hstso that = s hst psj t j. Then: = hstpsjlogpsjlogPipit Chst , = : s j i s ; Hence: h i Pi pit exp Chsthst psjlog psj = , + i s s j ; or, h i Pi hitexp Chsthstpsjlogpsj = , + : t s s j ; This is the system of equations for determining the maximizing values of Pi, with Cto be determined so that Pi 1. When this is done Cwill be the channel capacity, and the Pithe proper probabilities for the = channel symbols to achieve this capacity. If each input symbol has the same set of probabilities on the lines emerging from it, and the same is true of each output symbol, the capacity can be easily calculated. Examples are shown in Fig. 12. In such a caseHx yis independent of the distribution of probabilities on the input symbols, and is given by pilog pi , where the piare the values of the transition probabilities from any input symbol. The channel capacity is Max H y Hx y Max H y pilogpi , = + : The maximum of H yis clearly log mwhere mis the number of output symbols, since it is possible to make them all equally probable by making the input symbols equally probable. The channel capacity is therefore C log m pilogpi = + : 26 =============================================================================== 1 2 1 2 1 3 1 2 1 3 1 6 1 3 1 2 1 6 1 6 1 6 1 2 1 2 1 6 1 2 1 6 1 3 1 3 1 2 1 3 1 2 1 6 1 3 1 2 1 2 a b c Fig. 12 -- Examples of discrete channels with the same transition probabilities for each input and for each output. In Fig. 12a it would be C log 4 log2 log 2 = , = : This could be achieved by using only the 1st and 3d symbols. In Fig. 12b C log 4 2 log3 1 log6 = , 3 , 3 log 4 log3 1 log2 = , , 3 5 log 1 2 3 = 3 : In Fig. 12c we have C log 3 1 log2 1 log3 1 log6 = , 2 , 3 , 6 3 log = 1 1 1 : 2 2 3 3 6 6 Suppose the symbols fall into several groups such that the noise never causes a symbol in one group to be mistaken for a symbol in another group. Let the capacity for the nth group be Cn(in bits per second)when we use only the symbols in this group. Then it is easily shown that, for best use of the entire set, thetotal probability Pnof all symbols in the nth group should be 2Cn Pn= : 2Cn Within a group the probability is distributed just as it would be if these were the only symbols being used.The channel capacity is C log 2Cn = : 17. AN EXAMPLE OF EFFICIENT CODING The following example, although somewhat unrealistic, is a case in which exact matching to a noisy channelis possible. There are two channel symbols, 0 and 1, and the noise affects them in blocks of seven symbols.A block of seven is either transmitted without error, or exactly one symbol of the seven is incorrect. Theseeight possibilities are equally likely. We have C Max H y Hx y = , 1 7 8 log 1 = 7 + 8 8 4 bits/symbol = 7 : An efficient code, allowing complete correction of errors and transmitting at the rate C, is the following (found by a method due to R. Hamming): 27 =============================================================================== Let a block of seven symbols be X1 X2 X7. Of these X3, X5, X6 and X7 are message symbols and ; ; : : : ; chosen arbitrarily by the source. The other three are redundant and calculated as follows: X4 is chosen to make X4 X5 X6 X7 even = + + + X2 " " " " X2 X3 X6 X7 " = + + + X1 " " " " X1 X3 X5 X7 " = + + + When a block of seven is received and are calculated and if even called zero, if odd called one. The ; binary number then gives the subscript of the Xithat is incorrect (if 0 there was no error). APPENDIX 1 THE GROWTH OF THE NUMBER OF BLOCKS OF SYMBOLS WITH A FINITE STATE CONDITION Let Ni Lbe the number of blocks of symbols of length Lending in state i. Then we have , s N j L Ni L b = , i j i s ; where b1 b2 bmare the length of the symbols which may be chosen in state iand lead to state j. These i j; i j; : : : ; i j are linear difference equations and the behavior as L must be of the type ! Nj A jW L = : Substituting in the difference equation s A bij jW L AiWL, = i s ; or s A bij j AiW, = i s ; s W b , i j i j Ai 0 , = : i s For this to be possible the determinant s D W a bij i j W, i j = j j = , s must vanish and this determines W, which is, of course, the largest real root of D 0. = The quantity Cis then given by log A jW L C Lim logW = L L = ! and we also note that the same growth properties result if we require that all blocks start in the same (arbi-trarily chosen) state. APPENDIX 2 DERIVATION OF H pilog pi = , 1 1 1 Let H A n. From condition (3) we can decompose a choice from smequally likely possi- ; ; : : : ; = n n n bilities into a series of mchoices from sequally likely possibilities and obtain A sm mA s = : 28 =============================================================================== Similarly A tn nA t = : We can choose narbitrarily large and find an mto satisfy sm tn s m1 + : Thus, taking logarithms and dividing by nlog s, m log t m 1 m log t or + , n log s n n n log s where is arbitrarily small. Now from the monotonic property of A n, A sm A tn A sm1 + mA s nA t m 1 A s + : Hence, dividing by nA s, m A t m 1 m A t or + , n A s n n n A s A t logt 2 A t Klogt , = A s log s where Kmust be positive to satisfy (2). ni Now suppose we have a choice from npossibilities with commeasurable probabilities pi where = ni the niare integers. We can break down a choice from nipossibilities into a choice from npossibilitieswith probabilities p1 pnand then, if the ith was chosen, a choice from niwith equal probabilities. Using ; : : : ; condition (3) again, we equate the total choice from nias computed by two methods Klog ni H p1 pn K pilogni = ; : : : ; + : Hence h i H K pilogni pilogni = , ni K pilog K pilog pi = , = , : ni If the piare incommeasurable, they may be approximated by rationals and the same expression must holdby our continuity assumption. Thus the expression holds in general. The choice of coefficient Kis a matterof convenience and amounts to the choice of a unit of measure. APPENDIX 3 THEOREMS ON ERGODIC SOURCES If it is possible to go from any state with P 0 to any other along a path of probability p 0, the system is ergodic and the strong law of large numbers can be applied. Thus the number of times a given path pi jinthe network is traversed in a long sequence of length Nis about proportional to the probability of being ati, say Pi, and then choosing this path, Pi pi jN. If Nis large 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 Pi pi j N : Hence nearly all sequences have a probability pgiven by P N p p ipij = i j 29 =============================================================================== log p and is limited by N log p Pipij log pi j = N or log p Pipijlogpij , : N This proves Theorem 3. Theorem 4 follows immediately from this on calculating upper and lower bounds for n qbased on the possible range of values of pin Theorem 3. In the mixed (not ergodic) case if L piLi = and the entropies of the components are H1 H2 Hnwe have the Theorem:Lim logn q qis a decreasing step function, N N = ' ! s1 s , q Hs in the interval i q i ' = : 1 1 To prove Theorems 5 and 6 first note that FNis monotonic decreasing because increasing Nadds a subscript to a conditional entropy. A simple substitution for pB S in the definition of F i j Nshows that FN NGN N 1 GN1 = , , , 1 and summing this for all Ngives GN Fn. Hence GN FNand GNmonotonic decreasing. Also they = N must approach the same limit. By using Theorem 3 we see that Lim GN H. = N ! APPENDIX 4 MAXIMIZING THE RATE FOR A SYSTEM OF CONSTRAINTS Suppose we have a set of constraints on sequences of symbols that is of the finite state type and can be s represented therefore by a linear graph. Let be the lengths of the various symbols that can occur in `i j s passing from state ito state j. What distribution of probabilities P ifor the different states and p for i j choosing symbol sin state iand going to state jmaximizes the rate of generating information under theseconstraints? The constraints define a discrete channel and the maximum rate must be less than or equal tothe capacity Cof this channel, since if all blocks of large length were 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 Piand s p . i j The rate in question is s s P i p log p N , i j i j = : s s P M i pij`i j s s s Let i j . Evidently for a maximum p kexp . The constraints on maximization are Pi ` = s`i j i j= `i j = 1, j pi j 1, Pi pi j i j 0. Hence we maximize = , = Pipijlog pij , U Pi ipij jPi pij ij = P + + + , i pi j i j ` i U MPi1 log pi j NPi i j + + ` i iPi 0 = , + = : pi j M2 + + 30 =============================================================================== Solving for pi j pi j AiB jD,`ij = : Since p 1 i j 1 A, BjD,`ij = ; i = j j B jD,`ij pi j= : s BsD,`is The correct value of Dis the capacity Cand the B jare solutions of B i j i BjC,` = for then B j pi j C,`ij = Bi Bj Pi C,`ij Pj = Bi or Pi Pj C,`ij= : Bi B j So that if isatisfy iC,`ij j = Pi Bi i = : Both the sets of equations for Biand ican be satisfied since Cis such that C,`ij i j 0 j , j = : In this case the rate is B B P j j i pi jlog C,`ij P B i pi jlog i B C i , = , Pi pi j i j Pipij ij ` ` but PipijlogBjlogBiPjlogBjPilogBi0 , = , = j Hence the rate is Cand as this could never be exceeded this is the maximum, justifying the assumed solution. 31 =============================================================================== PART III: MATHEMATICAL PRELIMINARIES In this final installment of the paper we consider the case where the signals or the messages or both arecontinuously variable, in contrast with the discrete nature assumed heretofore. To a considerable extent thecontinuous case can be obtained through a limiting process from the discrete case by dividing the continuumof messages and signals into a large but finite number of small regions and calculating the various parametersinvolved on a discrete basis. As the size of the regions is decreased these parameters in general approach aslimits the proper values for the continuous case. There are, however, a few new effects that appear and alsoa general change of emphasis in the direction of specialization of the general results to particular cases. We will not attempt, in the continuous case, to obtain our results with the greatest generality, or with the extreme rigor of pure mathematics, since this would involve a great deal of abstract measure theoryand would obscure the main thread of the analysis. A preliminary study, however, indicates that the theorycan be formulated in a completely axiomatic and rigorous manner which includes both the continuous anddiscrete cases and many others. The occasional liberties taken with limiting processes in the present analysiscan be justified in all cases of practical interest. 18. SETS AND ENSEMBLES OF FUNCTIONS We shall have to deal in the continuous case with sets of functions and ensembles of functions. A set offunctions, as the name implies, is merely a class or collection of functions, generally of one variable, time.It can be specified by giving an explicit representation of the various functions in the set, or implicitly bygiving a property which functions in the set possess and others do not. Some examples are: 1. The set of functions: f t sin t = + : Each particular value of determines a particular function in the set. 2. The set of all functions of time containing no frequencies over Wcycles per second. 3. The set of all functions limited in band to Wand in amplitude to A. 4. The set of all English speech signals as functions of time. An ensembleof functions is a set of functions together with a probability measure whereby we may determine the probability of a function in the set having certain properties.1 For example with the set, f t sin t = + ; we may give a probability distribution for , P . The set then becomes an ensemble. Some further examples of ensembles of functions are: 1. A finite set of functions fk t(k 1 2 n) with the probability of fkbeing pk. = ; ; : : : ; 2. A finite dimensional family of functions f 1 2 n; t ; ; : : : ; with a probability distribution on the parameters i: p 1 n ; : : : ; : For example we could consider the ensemble defined by n f a1 an1 n; t aisini t i ; : : : ; ; ; : : : ; = ! + i1 = with the amplitudes aidistributed normally and independently, and the phases idistributed uniformly (from 0 to 2 ) and independently. 1In mathematical terminology the functions belong to a measure space whose total measure is unity. 32 =============================================================================== 3. The ensemble + sin 2W t n f a , i t an ; = 2W t n n , =, p with the ainormal and independent all with the same standard deviation N. This is a representation of "white" noise, band limited to the band from 0 to Wcycles per second and with average power N.2 4. Let points be distributed on the taxis according to a Poisson distribution. At each selected point the function f tis placed and the different functions added, giving the ensemble f t tk + k =, where the tkare the points of the Poisson distribution. This ensemble can be considered as a type ofimpulse or shot noise where all the impulses are identical. 5. The set of English speech functions with the probability measure given by the frequency of occurrence in ordinary use. An ensemble of functions f tis stationaryif the same ensemble results when all functions are shifted any fixed amount in time. The ensemble f t sin t = + is stationary if is distributed uniformly from 0 to 2 . If we shift each function by t1 we obtain f t t1 sin t t1 + = + + sin t = + ' with distributed uniformly from 0 to 2 . Each function has changed but the ensemble as a whole is ' invariant under the translation. The other examples given above are also stationary. An ensemble is ergodicif it is stationary, and there is no subset of the functions in the set with a probability different from 0 and 1 which is stationary. The ensemble sin t + is ergodic. No subset of these functions of probability 0 1 is transformed into itself under all time trans- 6= ; lations. On the other hand the ensemble asin t + with adistributed normally and uniform is stationary but not ergodic. The subset of these functions with abetween 0 and 1 for example is stationary. Of the examples given, 3 and 4 are ergodic, and 5 may perhaps be considered so. If an ensemble is ergodic we may say roughly that each function in the set is typical of the ensemble. More precisely it isknown that with an ergodic ensemble an average of any statistic over the ensemble is equal (with probability1) to an average over the time translations of a particular function of the set.3 Roughly speaking, eachfunction can be expected, as time progresses, to go through, with the proper frequency, all the convolutionsof any of the functions in the set. 2This representation can be used as a definition of band limited white noise. It has certain advantages in that it involves fewer limiting operations than do definitions that have been used in the past. The name "white noise," already firmly entrenched in theliterature, is perhaps somewhat unfortunate. In optics white light means either any continuous spectrum as contrasted with a pointspectrum, or a spectrum which is flat with wavelength(which is not the same as a spectrum flat with frequency). 3This is the famous ergodic theorem or rather one aspect of this theorem which was proved in somewhat different formulations by Birkoff, von Neumann, and Koopman, and subsequently generalized by Wiener, Hopf, Hurewicz and others. The literature onergodic theory is quite extensive and the reader is referred to the papers of these writers for precise and general formulations; e.g.,E. Hopf, "Ergodentheorie," Ergebnisse der Mathematik und ihrer Grenzgebiete,v. 5; "On Causality Statistics and Probability," Journalof Mathematics and Physics,v. XIII, No. 1, 1934; N. Wiener, "The Ergodic Theorem," Duke Mathematical Journal,v. 5, 1939. 33 =============================================================================== Just as we may perform various operations on numbers or functions to obtain new numbers or functions, we can perform operations on ensembles to obtain new ensembles. Suppose, for example, we have anensemble of functions f tand an operator Twhich gives for each function f ta resulting function g t: g t T f t = : Probability measure is defined for the set g tby means of that for the set f t. The probability of a certain subset of the g tfunctions is equal to that of the subset of the f tfunctions which produce members of the given subset of gfunctions under the operation T. Physically this corresponds to passing the ensemblethrough some device, for example, a filter, a rectifier or a modulator. The output functions of the deviceform the ensemble g t. A device or operator Twill be called invariant if shifting the input merely shifts the output, i.e., if g t T f t = implies g t t1 T f t t1 + = + for all f tand all t1. It is easily shown (see Appendix 5 that if Tis invariant and the input ensemble is stationary then the output ensemble is stationary. Likewise if the input is ergodic the output will also beergodic. A filter or a rectifier is invariant under all time translations. The operation of modulation is not since the carrier phase gives a certain time structure. However, modulation is invariant under all translations whichare multiples of the period of the carrier. Wiener has pointed out the intimate relation between the invariance of physical devices under time translations and Fourier theory.4 He has shown, in fact, that if a device is linear as well as invariant Fourieranalysis is then the appropriate mathematical tool for dealing with the problem. An ensemble of functions is the appropriate mathematical representation of the messages produced by a continuous source (for example, speech), of the signals produced by a transmitter, and of the perturbingnoise. Communication theory is properly concerned, as has been emphasized by Wiener, not with operationson particular functions, but with operations on ensembles of functions. A communication system is designednot for a particular speech function and still less for a sine wave, but for the ensemble of speech functions. 19. BAND LIMITED ENSEMBLES OF FUNCTIONS If a function of time f tis limited to the band from 0 to Wcycles per second it is completely determined by giving its ordinates at a series of discrete points spaced 1 seconds apart in the manner indicated by the 2W following result.5 Theorem 13:Let f tcontain no frequencies over W. Then sin 2W t n f t X , n = 2W t n , , where n Xn f = : 2W 4Communication theory is heavily indebted to Wiener for much of its basic philosophy and theory. His classic NDRC report, The Interpolation, Extrapolation and Smoothing of Stationary Time Series(Wiley, 1949), contains the first clear-cut formulation ofcommunication theory as a statistical problem, the study of operations on time series. This work, although chiefly concerned with thelinear prediction and filtering problem, is an important collateral reference in connection with the present paper. We may also referhere to Wiener's Cybernetics(Wiley, 1948), dealing with the general problems of communication and control. 5For a proof of this theorem and further discussion see the author's paper "Communication in the Presence of Noise" published in the Proceedings of the Institute of Radio Engineers,v. 37, No. 1, Jan., 1949, pp. 10­21. 34 =============================================================================== In this expansion f tis represented as a sum of orthogonal functions. The coefficients Xnof the various terms can be considered as coordinates in an infinite dimensional "function space." In this space eachfunction corresponds to precisely one point and each point to one function. A function can be considered to be substantially limited to a time Tif all the ordinates Xnoutside this interval of time are zero. In this case all but 2TWof the coordinates will be zero. Thus functions limited toa band Wand duration Tcorrespond to points in a space of 2TWdimensions. A subset of the functions of band Wand duration Tcorresponds to a region in this space. For example, the functions whose total energy is less than or equal to Ecorrespond to points in a 2TWdimensional sphere p with radius r 2W E. = An ensembleof functions of limited duration and band will be represented by a probability distribution p x1 xnin the corresponding ndimensional space. If the ensemble is not limited in time we can consider ; : : : ; the 2TWcoordinates in a given interval Tto represent substantially the part of the function in the interval Tand the probability distribution p x1 xnto give the statistical structure of the ensemble for intervals of ; : : : ; that duration. 20. ENTROPY OF A CONTINUOUS DISTRIBUTION The entropy of a discrete set of probabilities p1 pnhas been defined as: ; : : : ; H pilogpi = , : In an analogous manner we define the entropy of a continuous distribution with the density distributionfunction p xby: Z H p xlog p x dx = , : , With an ndimensional distribution p x1 xnwe have ; : : : ; Z Z H p x1 xnlog p x1 xn dx1 dxn = , ; : : : ; ; : : : ; : If we have two arguments xand y(which may themselves be multidimensional) the joint and conditionalentropies of p x yare given by ; Z Z H x y p x ylog p x y dx dy ; = , ; ; and Z Z p x y ; Hx y p x ylog dx dy = , ; p x Z Z p x y H ; y x p x ylog dx dy = , ; p y where Z p x p x y dy = ; Z p y p x y dx = ; : The entropies of continuous distributions have most (but not all) of the properties of the discrete case. In particular we have the following: 1. If xis limited to a certain volume vin its space, then H xis a maximum and equal to log vwhen p x is constant (1 v) in the volume. = 35 =============================================================================== 2. With any two variables x, ywe have H x y H x H y ; + with equality if (and only if) xand yare independent, i.e., p x y p x p y (apart possibly from a ; = set of points of probability zero). 3. Consider a generalized averaging operation of the following type: Z p0 y a x y p x dx = ; with Z Z a x y dx a x y dy 1 a x y 0 ; = ; = ; ; : Then the entropy of the averaged distribution p0 yis equal to or greater than that of the original distribution p x. 4. We have H x y H x Hx y H y Hy x ; = + = + and Hx y H y : 5. Let p xbe a one-dimensional distribution. The form of p xgiving a maximum entropy subject to the condition that the standard deviation of xbe fixed at is Gaussian. To show this we must maximize Z H x p xlog p x dx = , with Z Z 2 p x x2 dx and 1 p x dx = = as constraints. This requires, by the calculus of variations, maximizing Z p xlog p x p x x2 p x dx , + + : The condition for this is 1 log p x x2 0 , , + + = and consequently (adjusting the constants to satisfy the constraints) 1 p x e x2 2 2 , = p = : 2 Similarly in ndimensions, suppose the second order moments of p x1 xnare fixed at Ai j: ; : : : ; Z Z Ai j xix j p x1 xn dx1 dxn = ; : : : ; : Then the maximum entropy occurs (by a similar calculation) when p x1 xnis the ndimensional ; : : : ; Gaussian distribution with the second order moments Ai j. 36 =============================================================================== 6. The entropy of a one-dimensional Gaussian distribution whose standard deviation is is given by p H x log 2 e = : This is calculated as follows: 1 p x e x2 2 2 , = p = 2 x2 p log p x log 2 , = + 2 2 Z H x p xlog p x dx = , Z Z x2 p p xlog 2 dx p x dx = + 2 2 2 p log 2 = + 2 2 p p log 2 log e = + p log 2 e = : Similarly the ndimensional Gaussian distribution with associated quadratic form ai jis given by 1 a i j2 j j p x 1 1 xn exp aijxixj ; : : : ; = , 2 n2 2 = and the entropy can be calculated as 1 H log 2 e n2 = a, i j 2 = j j where ai jis the determinant whose elements are ai j. j j 7. If xis limited to a half line (p x 0 for x 0) and the first moment of xis fixed at a: = Z a p x x dx = ; 0 then the maximum entropy occurs when 1 p x e x a , = = a and is equal to log ea. 8. There is one important difference between the continuous and discrete entropies. In the discrete case the entropy measures in an absoluteway the randomness of the chance variable. In the continuouscase the measurement is relative to the coordinate system. If we change coordinates the entropy willin general change. In fact if we change to coordinates y1 ynthe new entropy is given by Z Z x x H y p x1 xn J log p x1 xn J dy1 dyn = ; : : : ; ; : : : ; y y , where J xis the Jacobian of the coordinate transformation. On expanding the logarithm and chang- y ing the variables to x1 xn, we obtain: Z Z x H y H x p x1 xnlog J dx1 dxn = , ; : : : ; : : : : y 37 =============================================================================== Thus the new entropy is the old entropy less the expected logarithm of the Jacobian. In the continuouscase the entropy can be considered a measure of randomness relative to an assumed standard, namelythe coordinate system chosen with each small volume element dx1 dxngiven equal weight. When we change the coordinate system the entropy in the new system measures the randomness when equalvolume elements dy1 dynin the new system are given equal weight. In spite of this dependence on the coordinate system the entropy concept is as important in the con-tinuous case as the discrete case. This is due to the fact that the derived concepts of information rateand channel capacity depend on the differenceof two entropies and this difference does notdependon the coordinate frame, each of the two terms being changed by the same amount. The entropy of a continuous distribution can be negative. The scale of measurements sets an arbitraryzero corresponding to a uniform distribution over a unit volume. A distribution which is more confinedthan this has less entropy and will be negative. The rates and capacities will, however, always be non-negative. 9. A particular case of changing coordinates is the linear transformation y j aijxi = : i In this case the Jacobian is simply the determinant a 1 , i j and j j H y H x log ai j = + j j: In the case of a rotation of coordinates (or any measure preserving transformation) J 1 and H y = = H x. 21. ENTROPY OF AN ENSEMBLE OF FUNCTIONS Consider an ergodic ensemble of functions limited to a certain band of width Wcycles per second. Let p x1 xn ; : : : ; be the density distribution function for amplitudes x1 xnat nsuccessive sample points. We define the ; : : : ; entropy of the ensemble per degree of freedom by 1 Z Z H0 Lim p x1 xnlog p x1 xn dx1 dxn = , ; : : : ; ; : : : ; : : : : n n ! We may also define an entropy Hper second by dividing, not by n, but by the time Tin seconds for nsamples. Since n 2TW, H 2W H0. = = With white thermal noise pis Gaussian and we have p H0 log 2 eN = ; H Wlog 2 eN = : For a given average power N, white noise has the maximum possible entropy. This follows from the maximizing properties of the Gaussian distribution noted above. The entropy for a continuous stochastic process has many properties analogous to that for discrete pro- cesses. In the discrete case the entropy was related to the logarithm of the probabilityof long sequences,and to the numberof reasonably probable sequences of long length. In the continuous case it is related ina similar fashion to the logarithm of the probability densityfor a long series of samples, and the volumeofreasonably high probability in the function space. More precisely, if we assume p x1 xncontinuous in all the xifor all n, then for sufficiently large n ; : : : ; log p H0 , n 38 =============================================================================== for all choices of x1 xnapart from a set whose total probability is less than , with and arbitrarily ; : : : ; small. This follows form the ergodic property if we divide the space into a large number of small cells. The relation of Hto volume can be stated as follows: Under the same assumptions consider the n dimensional space corresponding to p x1 xn. Let Vn qbe the smallest volume in this space which ; : : : ; includes in its interior a total probability q. Then logVn q Lim H0 = n n ! provided qdoes not equal 0 or 1. These results show that for large nthere is a rather well-defined volume (at least in the logarithmic sense) of high probability, and that within this volume the probability density is relatively uniform (again in thelogarithmic sense). In the white noise case the distribution function is given by 1 1 p x1 xn exp x2 ; : : : ; = , 2 N n2 i: = 2N Since this depends only on x2 the surfaces of equal probability density are spheres and the entire distri- i p bution has spherical symmetry. The region of high probability is a sphere of radius nN. As n the ! p probability of being outside a sphere of radius n N approaches zero and 1 times the logarithm of the + n p volume of the sphere approaches log 2 eN. In the continuous case it is convenient to work not with the entropy Hof an ensemble but with a derived quantity which we will call the entropy power. This is defined as the power in a white noise limited to thesame band as the original ensemble and having the same entropy. In other words if H0 is the entropy of anensemble its entropy power is 1 N1 exp 2H0 = : 2 e In the geometrical picture this amounts to measuring the high probability volume by the squared radius of asphere having the same volume. Since white noise has the maximum entropy for a given power, the entropypower of any noise is less than or equal to its actual power. 22. ENTROPY LOSS IN LINEAR FILTERS Theorem 14:If an ensemble having an entropy H1 per degree of freedom in band Wis passed through a filter with characteristic Y fthe output ensemble has an entropy 1 Z H 2 2 H1 log Y f d f = + j j : W W The operation of the filter is essentially a linear transformation of coordinates. If we think of the different frequency components as the original coordinate system, the new frequency components are merely the oldones multiplied by factors. The coordinate transformation matrix is thus essentially diagonalized in termsof these coordinates. The Jacobian of the transformation is (for nsine and ncosine components) n J Y f2 i = j j i1 = where the fiare equally spaced through the band W. This becomes in the limit 1 Z exp log Y f2 d f j j : W W Since Jis constant its average value is the same quantity and applying the theorem on the change of entropywith a change of coordinates, the result follows. We may also phrase it in terms of the entropy power. Thusif the entropy power of the first ensemble is N1 that of the second is 1 Z N 2 1 exp log Y f d f j j : W W 39 =============================================================================== TABLE I ENTROPY ENTROPY GAIN POWER POWER GAIN IMPULSE RESPONSE FACTOR IN DECIBELS 1 1 1 sin2 t2 , ! 8 69 = e2 , : t2 2 = ! 0 1 1 1 2 2 4 sint cos t , ! 5 33 2 , : e t3 , t2 ! 0 1 1 1 3 cos t 1 cos t sint , , ! 0 411 3 87 6 : , : t4 , 2t2 + t3 ! 0 1 1 p 1 2 2 2 J1 t , ! 2 67 e , : 2 t ! 0 1 1 1 1 8 69 cos 1 t cos t : , , e2 , t2 ! 0 1 The final entropy power is the initial entropy power multiplied by the geometric mean gain of the filter. Ifthe gain is measured in db, then the output entropy power will be increased by the arithmetic mean dbgainover W. In Table I the entropy power loss has been calculated (and also expressed in db) for a number of ideal gain characteristics. The impulsive responses of these filters are also given for W 2 , with phase assumed = to be 0. The entropy loss for many other cases can be obtained from these results. For example the entropy power factor 1 e2 for the first case also applies to any gain characteristic obtain from 1 by a measure = , ! preserving transformation of the axis. In particular a linearly increasing gain G , or a "saw tooth" ! ! = ! characteristic between 0 and 1 have the same entropy loss. The reciprocal gain has the reciprocal factor.Thus 1 has the factor e2. Raising the gain to any power raises the factor to this power. =! 23. ENTROPY OF A SUM OF TWO ENSEMBLES If we have two ensembles of functions f tand g twe can form a new ensemble by "addition." Suppose the first ensemble has the probability density function p x1 xnand the second q x1 xn. Then the ; : : : ; ; : : : ; 40 =============================================================================== density function for the sum is given by the convolution: Z Z r x1 xn p y1 yn q x1 y1 xn yn dy1 dyn ; : : : ; = ; : : : ; , ; : : : ; , : Physically this corresponds to adding the noises or signals represented by the original ensembles of func-tions. The following result is derived in Appendix 6. Theorem 15:Let the average power of two ensembles be N1 and N2 and let their entropy powers be N1 and N2. Then the entropy power of the sum, N3, is bounded by N1 N2 N3 N1 N2 + + : White Gaussian noise has the peculiar property that it can absorb any other noise or signal ensemble which may be added to it with a resultant entropy power approximately equal to the sum of the white noisepower and the signal power (measured from the average signal value, which is normally zero), provided thesignal power is small, in a certain sense, compared to noise. Consider the function space associated with these ensembles having ndimensions. The white noise corresponds to the spherical Gaussian distribution in this space. The signal ensemble corresponds to anotherprobability distribution, not necessarily Gaussian or spherical. Let the second moments of this distributionabout its center of gravity be ai j. That is, if p x1 xnis the density distribution function ; : : : ; Z Z ai j p xi i x j j dx1 dxn = , , where the iare the coordinates of the center of gravity. Now ai jis a positive definite quadratic form, and we can rotate our coordinate system to align it with the principal directions of this form. ai jis then reducedto diagonal form bii. We require that each biibe small compared to N, the squared radius of the sphericaldistribution. In this case the convolution of the noise and signal produce approximately a Gaussian distribution whose corresponding quadratic form is N bii + : The entropy power of this distribution is h i1 n = N bii + or approximately h i1 n = N n b n1 , ii N = + 1 : N bii = + : n The last term is the signal power, while the first is the noise power. PART IV: THE CONTINUOUS CHANNEL 24. THE CAPACITY OF A CONTINUOUS CHANNEL In a continuous channel the input or transmitted signals will be continuous functions of time f tbelonging to a certain set, and the output or received signals will be perturbed versions of these. We will consideronly the case where both transmitted and received signals are limited to a certain band W. They can thenbe specified, for a time T, by 2TWnumbers, and their statistical structure by finite dimensional distributionfunctions. Thus the statistics of the transmitted signal will be determined by P x1 xn P x ; : : : ; = 41 =============================================================================== and those of the noise by the conditional probability distribution Px y y P y 1 xn 1 n x ; : : : ; = : ; :::; The rate of transmission of information for a continuous channel is defined in a way analogous to that for a discrete channel, namely R H x Hy x = , where H xis the entropy of the input and Hy xthe equivocation. The channel capacity Cis defined as the maximum of Rwhen we vary the input over all possible ensembles. This means that in a finite dimensionalapproximation we must vary P x P x1 xnand maximize = ; : : : ; Z ZZ P x y P xlog P x dx P x ylog ; dx dy , + ; : P y This can be written Z Z P x y P x ylog ; dx dy ; P x P y Z Z Z using the fact that P x ylog P x dx dy P xlog P x dx. The channel capacity is thus expressed as ; = follows: 1 ZZ P x y C Lim Max P x ylog ; dx dy = ; : T P x T P x P y ! It is obvious in this form that Rand Care independent of the coordinate system since the numerator P x y and denominator in log ; will be multiplied by the same factors when xand yare transformed in P x P y any one-to-one way. This integral expression for Cis more general than H x Hy x. Properly interpreted , (see Appendix 7) it will always exist while H x Hy xmay assume an indeterminate form in some , , cases. This occurs, for example, if xis limited to a surface of fewer dimensions than nin its ndimensionalapproximation. If the logarithmic base used in computing H xand Hy xis two then Cis the maximum number of binary digits that can be sent per second over the channel with arbitrarily small equivocation, just as inthe discrete case. This can be seen physically by dividing the space of signals into a large number ofsmall cells, sufficiently small so that the probability density Px yof signal xbeing perturbed to point yis substantially constant over a cell (either of xor y). If the cells are considered as distinct points the situation isessentially the same as a discrete channel and the proofs used there will apply. But it is clear physically thatthis quantizing of the volume into individual points cannot in any practical situation alter the final answersignificantly, provided the regions are sufficiently small. Thus the capacity will be the limit of the capacitiesfor the discrete subdivisions and this is just the continuous capacity defined above. On the mathematical side it can be shown first (see Appendix 7) that if uis the message, xis the signal, yis the received signal (perturbed by noise) and vis the recovered message then H x Hy x H u Hv u , , regardless of what operations are performed on uto obtain xor on yto obtain v. Thus no matter how weencode the binary digits to obtain the signal, or how we decode the received signal to recover the message,the discrete rate for the binary digits does not exceed the channel capacity we have defined. On the otherhand, it is possible under very general conditions to find a coding system for transmitting binary digits at therate Cwith as small an equivocation or frequency of errors as desired. This is true, for example, if, when wetake a finite dimensional approximating space for the signal functions, P x yis continuous in both xand y ; except at a set of points of probability zero. An important special case occurs when the noise is added to the signal and is independent of it (in the probability sense). Then Px yis a function only of the difference n y x, = , Px y Q y x = , 42 =============================================================================== and we can assign a definite entropy to the noise (independent of the statistics of the signal), namely theentropy of the distribution Q n. This entropy will be denoted by H n. Theorem 16:If the signal and noise are independent and the received signal is the sum of the transmitted signal and the noise then the rate of transmission is R H y H n = , ; i.e., the entropy of the received signal less the entropy of the noise. The channel capacity is C Max H y H n = , : P x We have, since y x n: = + H x y H x n ; = ; : Expanding the left side and using the fact that xand nare independent H y Hy x H x H n + = + : Hence R H x Hy x H y H n = , = , : Since H nis independent of P x, maximizing Rrequires maximizing H y, the entropy of the received signal. If there are certain constraints on the ensemble of transmitted signals, the entropy of the receivedsignal must be maximized subject to these constraints. 25. CHANNEL CAPACITY WITH AN AVERAGE POWER LIMITATION A simple application of Theorem 16 is the case when the noise is a white thermal noise and the transmittedsignals are limited to a certain average power P. Then the received signals have an average power P N + where Nis the average noise power. The maximum entropy for the received signals occurs when they alsoform a white noise ensemble since this is the greatest possible entropy for a power P Nand can be obtained + by a suitable choice of transmitted signals, namely if they form a white noise ensemble of power P. Theentropy (per second) of the received ensemble is then H y Wlog 2 e P N = + ; and the noise entropy is H n Wlog 2 eN = : The channel capacity is P N C H y H n Wlog + = , = : N Summarizing we have the following: Theorem 17:The capacity of a channel of band Wperturbed by white thermal noise power Nwhen the average transmitter power is limited to Pis given by P N + C Wlog = : N This means that by sufficiently involved encoding systems we can transmit binary digits at the rate P N Wlog + 2 bits per second, with arbitrarily small frequency of errors. It is not possible to transmit at a N higher rate by any encoding system without a definite positive frequency of errors. To approximate this limiting rate of transmission the transmitted signals must approximate, in statistical properties, a white noise.6 A system which approaches the ideal rate may be described as follows: Let 6This and other properties of the white noise case are discussed from the geometrical point of view in "Communication in the Presence of Noise," loc. cit. 43 =============================================================================== M 2ssamples of white noise be constructed each of duration T. These are assigned binary numbers from = 0 to M 1. At the transmitter the message sequences are broken up into groups of sand for each group , the corresponding noise sample is transmitted as the signal. At the receiver the Msamples are known andthe actual received signal (perturbed by noise) is compared with each of them. The sample which has theleast R.M.S. discrepancy from the received signal is chosen as the transmitted signal and the correspondingbinary number reconstructed. This process amounts to choosing the most probable (a posteriori) signal.The number Mof noise samples used will depend on the tolerable frequency of errors, but for almost all selections of samples we have log M T P N ; + Lim Lim Wlog = ; 0 T T N ! ! so that no matter how small is chosen, we can, by taking Tsufficiently large, transmit as near as we wish P N to TWlog + binary digits in the time T. N P N Formulas similar to C Wlog + for the white noise case have been developed independently = N by several other writers, although with somewhat different interpretations. We may mention the work ofN. Wiener,7 W. G. Tuller,8 and H. Sullivan in this connection. In the case of an arbitrary perturbing noise (not necessarily white thermal noise) it does not appear that the maximizing problem involved in determining the channel capacity Ccan be solved explicitly. However,upper and lower bounds can be set for Cin terms of the average noise power Nthe noise entropy power N1.These bounds are sufficiently close together in most practical cases to furnish a satisfactory solution to theproblem. Theorem 18:The capacity of a channel of band Wperturbed by an arbitrary noise is bounded by the inequalities P N1 P N Wlog + C Wlog + N1 N1 where P average transmitter power = N average noise power = N1 entropy power of the noise. = Here again the average power of the perturbed signals will be P N. The maximum entropy for this + power would occur if the received signal were white noise and would be Wlog 2 e P N. It may not + be possible to achieve this; i.e., there may not be any ensemble of transmitted signals which, added to theperturbing noise, produce a white thermal noise at the receiver, but at least this sets an upper bound to H y. We have, therefore C Max H y H n = , Wlog 2 e P N Wlog 2 eN1 + , : This is the upper limit given in the theorem. The lower limit can be obtained by considering the rate if wemake the transmitted signal a white noise, of power P. In this case the entropy power of the received signalmust be at least as great as that of a white noise of power P N1 since we have shown in in a previous + theorem that the entropy power of the sum of two ensembles is greater than or equal to the sum of theindividual entropy powers. Hence Max H y Wlog 2 e P N1 + 7Cybernetics, loc. cit.8"Theoretical Limitations on the Rate of Transmission of Information," Proceedings of the Institute of Radio Engineers,v. 37, No. 5, May, 1949, pp. 468­78. 44 =============================================================================== and C Wlog 2 e P N1 Wlog 2 eN1 + , P N1 + Wlog = : N1 As Pincreases, the upper and lower bounds approach each other, so we have as an asymptotic rate P N Wlog + : N1 If the noise is itself white, N N1 and the result reduces to the formula proved previously: = P C Wlog 1 = + : N If the noise is Gaussian but with a spectrum which is not necessarily flat, N1 is the geometric mean of the noise power over the various frequencies in the band W. Thus 1 Z N1 exp log N f d f = W W where N fis the noise power at frequency f. Theorem 19:If we set the capacity for a given transmitter power Pequal to P N + , C Wlog = N1 then is monotonic decreasing as Pincreases and approaches 0 as a limit. Suppose that for a given power P1 the channel capacity is P1 N 1 Wlog + , : N1 This means that the best signal distribution, say p x, when added to the noise distribution q x, gives a received distribution r ywhose entropy power is P1 N 1 . Let us increase the power to P1 Pby + , + adding a white noise of power Pto the signal. The entropy of the received signal is now at least H y Wlog 2 e P1 N 1 P = + , + by application of the theorem on the minimum entropy power of a sum. Hence, since we can attain theHindicated, the entropy of the maximizing distribution must be at least as great and must be monotonic decreasing. To show that 0 as P consider a signal which is white noise with a large P. Whatever ! ! the perturbing noise, the received signal will be approximately a white noise, if Pis sufficiently large, in thesense of having an entropy power approaching P N. + 26. THE CHANNEL CAPACITY WITH A PEAK POWER LIMITATION In some applications the transmitter is limited not by the average power output but by the peak instantaneouspower. The problem of calculating the channel capacity is then that of maximizing (by variation of theensemble of transmitted symbols) H y H n , p subject to the constraint that all the functions f tin the ensemble be less than or equal to S, say, for all t. A constraint of this type does not work out as well mathematically as the average power limitation. The S most we have obtained for this case is a lower bound valid for all , an "asymptotic" upper bound (valid N S S for large ) and an asymptotic value of Cfor small. N N 45 =============================================================================== Theorem 20:The channel capacity Cfor a band Wperturbed by white thermal noise of power Nis bounded by 2 S C Wlog ; e3 N S where Sis the peak allowed transmitter power. For sufficiently large N 2 S N + C Wlog e 1 + N S where is arbitrarily small. As 0 (and provided the band Wstarts at 0) ! N . S C Wlog 1 1 + ! : N S We wish to maximize the entropy of the received signal. If is large this will occur very nearly when N we maximize the entropy of the transmitted ensemble. The asymptotic upper bound is obtained by relaxing the conditions on the ensemble. Let us suppose that the power is limited to Snot at every instant of time, but only at the sample points. The maximum entropy ofthe transmitted ensemble under these weakened conditions is certainly greater than or equal to that under theoriginal conditions. This altered problem can be solved easily. The maximum entropy occurs if the different p p samples are independent and have a distribution function which is constant from Sto S. The entropy , + can be calculated as Wlog 4S: The received signal will then have an entropy less than Wlog 4S 2 eN1 + + S with 0 as and the channel capacity is obtained by subtracting the entropy of the white noise, ! ! N Wlog 2 eN: 2 S N + Wlog 4S 2 eN1 Wlog 2 eN Wlog e 1 + + , = + : N This is the desired upper bound to the channel capacity. To obtain a lower bound consider the same ensemble of functions. Let these functions be passed through an ideal filter with a triangular transfer characteristic. The gain is to be unity at frequency 0 and declinelinearly down to gain 0 at frequency W. We first show that the output functions of the filter have a peak sin 2 W t power limitation Sat all times (not just the sample points). First we note that a pulse going into 2 W t the filter produces 1 sin2 W t 2 W t2 in the output. This function is never negative. The input function (in the general case) can be thought of asthe sum of a series of shifted functions sin 2 W t a 2 W t p where a, the amplitude of the sample, is not greater than S. Hence the output is the sum of shifted functions of the non-negative form above with the same coefficients. These functions being non-negative, the greatest p positive value for any tis obtained when all the coefficients ahave their maximum positive values, i.e., S. p In this case the input function was a constant of amplitude Sand since the filter has unit gain for D.C., the output is the same. Hence the output ensemble has a peak power S. 46 =============================================================================== The entropy of the output ensemble can be calculated from that of the input ensemble by using the theorem dealing with such a situation. The output entropy is equal to the input entropy plus the geometricalmean gain of the filter: Z W Z W W f2 log G2 d f log , d f 2W = = , : 0 0 W Hence the output entropy is 4S Wlog 4S 2W Wlog , = e2 and the channel capacity is greater than 2 S Wlog : e3 N S We now wish to show that, for small (peak signal power over average white noise power), the channel N capacity is approximately S C Wlog 1 = + : N . S S More precisely C Wlog 1 1 as 0. Since the average signal power Pis less than or equal + ! ! N N S to the peak S, it follows that for all N P S C Wlog 1 Wlog 1 + + : N N S Therefore, if we can find an ensemble of functions such that they correspond to a rate nearly Wlog 1 + Nand are limited to band Wand peak Sthe result will be proved. Consider the ensemble of functions of the p p following type. A series of tsamples have the same value, either Sor S, then the next tsamples have + , p p the same value, etc. The value for a series is chosen at random, probability 1 for Sand 1 for S. If 2 + 2 , this ensemble be passed through a filter with triangular gain characteristic (unit gain at D.C.), the output ispeak limited to S. Furthermore the average power is nearly Sand can be made to approach this by taking t sufficiently large. The entropy of the sum of this and the thermal noise can be found by applying the theoremon the sum of a noise and a small signal. This theorem will apply if S p t N S is sufficiently small. This can be ensured by taking small enough (after tis chosen). The entropy power N will be S Nto as close an approximation as desired, and hence the rate of transmission as near as we wish + to S N Wlog + : N PART V: THE RATE FOR A CONTINUOUS SOURCE 27. FIDELITY EVALUATION FUNCTIONS In the case of a discrete source of information we were able to determine a definite rate of generatinginformation, namely the entropy of the underlying stochastic process. With a continuous source the situationis considerably more involved. In the first place a continuously variable quantity can assume an infinitenumber of values and requires, therefore, an infinite number of binary digits for exact specification. Thismeans that to transmit the output of a continuous source with exact recoveryat the receiving point requires, 47 =============================================================================== in general, a channel of infinite capacity (in bits per second). Since, ordinarily, channels have a certainamount of noise, and therefore a finite capacity, exact transmission is impossible. This, however, evades the real issue. Practically, we are not interested in exact transmission when we have a continuous source, but only in transmission to within a certain tolerance. The question is, can weassign a definite rate to a continuous source when we require only a certain fidelity of recovery, measured ina suitable way. Of course, as the fidelity requirements are increased the rate will increase. It will be shownthat we can, in very general cases, define such a rate, having the property that it is possible, by properlyencoding the information, to transmit it over a channel whose capacity is equal to the rate in question, andsatisfy the fidelity requirements. A channel of smaller capacity is insufficient. It is first necessary to give a general mathematical formulation of the idea of fidelity of transmission. Consider the set of messages of a long duration, say Tseconds. The source is described by giving theprobability density, in the associated space, that the source will select the message in question P x. A given communication system is described (from the external point of view) by giving the conditional probabilityPx ythat if message xis produced by the source the recovered message at the receiving point will be y. The system as a whole (including source and transmission system) is described by the probability function P x y ; of having message xand final output y. If this function is known, the complete characteristics of the systemfrom the point of view of fidelity are known. Any evaluation of fidelity must correspond mathematicallyto an operation applied to P x y. This operation must at least have the properties of a simple ordering of ; systems; i.e., it must be possible to say of two systems represented by P1 x yand P2 x ythat, according to ; ; our fidelity criterion, either (1) the first has higher fidelity, (2) the second has higher fidelity, or (3) they haveequal fidelity. This means that a criterion of fidelity can be represented by a numerically valued function: , v P x y ; whose argument ranges over possible probability functions P x y. ; , We will now show that under very general and reasonable assumptions the function v P x y can be ; written in a seemingly much more specialized form, namely as an average of a function x yover the set ; of possible values of xand y: Z Z , v P x y P x y x y dx dy ; = ; ; : To obtain this we need only assume (1) that the source and system are ergodic so that a very long samplewill be, with probability nearly 1, typical of the ensemble, and (2) that the evaluation is "reasonable" in thesense that it is possible, by observing a typical input and output x1 and y1, to form a tentative evaluationon the basis of these samples; and if these samples are increased in duration the tentative evaluation will,with probability 1, approach the exact evaluation based on a full knowledge of P x y. Let the tentative ; evaluation be x y. Then the function x yapproaches (as T ) a constant for almost all x ywhich ; ; ! ; are in the high probability region corresponding to the system: , x y v P x y ; ! ; and we may also write Z Z x y P x y x y dx dy ; ! ; ; since Z Z P x y dx dy 1 ; = : This establishes the desired result. The function x yhas the general nature of a "distance" between xand y.9 It measures how undesirable ; it is (according to our fidelity criterion) to receive ywhen xis transmitted. The general result given abovecan be restated as follows: Any reasonable evaluation can be represented as an average of a distance functionover the set of messages and recovered messages xand yweighted according to the probability P x yof ; getting the pair in question, provided the duration Tof the messages be taken sufficiently large. The following are simple examples of evaluation functions: 9It is not a "metric" in the strict sense, however, since in general it does not satisfy either x y y xor x y y z x z. ; = ; ; + ; ; 48 =============================================================================== 1. R.M.S. criterion. , 2 v x t y t = , : In this very commonly used measure of fidelity the distance function x yis (apart from a constant ; factor) the square of the ordinary Euclidean distance between the points xand yin the associatedfunction space. 1 Z T 2 x y x t y t dt ; = , : T0 2. Frequency weighted R.M.S. criterion. More generally one can apply different weights to the different frequency components before using an R.M.S. measure of fidelity. This is equivalent to passing thedifference x t y tthrough a shaping filter and then determining the average power in the output. , Thus let e t x t y t = , and Z f t e k t d = , , then 1 Z T x y f t2 dt ; = : T0 3. Absolute error criterion. 1 Z T x y x t y t dt ; = , : T0 4. The structure of the ear and brain determine implicitly an evaluation, or rather a number of evaluations, appropriate in the case of speech or music transmission. There is, for example, an "intelligibility"criterion in which x yis equal to the relative frequency of incorrectly interpreted words when ; message x tis received as y t. Although we cannot give an explicit representation of x yin these ; cases it could, in principle, be determined by sufficient experimentation. Some of its properties followfrom well-known experimental results in hearing, e.g., the ear is relatively insensitive to phase and thesensitivity to amplitude and frequency is roughly logarithmic. 5. The discrete case can be considered as a specialization in which we have tacitly assumed an evaluation based on the frequency of errors. The function x yis then defined as the number of symbols in the ; sequence ydiffering from the corresponding symbols in xdivided by the total number of symbols inx. 28. THE RATE FOR A SOURCE RELATIVE TO A FIDELITY EVALUATION We are now in a position to define a rate of generating information for a continuous source. We are givenP xfor the source and an evaluation vdetermined by a distance function x ywhich will be assumed ; continuous in both xand y. With a particular system P x ythe quality is measured by ; Z Z v x y P x y dx dy = ; ; : Furthermore the rate of flow of binary digits corresponding to P x yis ; Z Z P x y R P x ylog ; dx dy = ; : P x P y We define the rate R1 of generating information for a given quality v1 of reproduction to be the minimum ofRwhen we keep vfixed at v1 and vary Px y. That is: Z Z P x y R ; 1 Min P x ylog dx dy = ; Px y P x P y 49 =============================================================================== subject to the constraint: Z Z v1 P x y x y dx dy = ; ; : This means that we consider, in effect, all the communication systems that might be used and that transmit with the required fidelity. The rate of transmission in bits per second is calculated for each oneand we choose that having the least rate. This latter rate is the rate we assign the source for the fidelity inquestion. The justification of this definition lies in the following result: Theorem 21:If a source has a rate R1 for a valuation v1 it is possible to encode the output of the source and transmit it over a channel of capacity Cwith fidelity as near v1 as desired provided R1 C. This is not possible if R1 C. The last statement in the theorem follows immediately from the definition of R1 and previous results. If it were not true we could transmit more than Cbits per second over a channel of capacity C. The first partof the theorem is proved by a method analogous to that used for Theorem 11. We may, in the first place,divide the x yspace into a large number of small cells and represent the situation as a discrete case. This ; will not change the evaluation function by more than an arbitrarily small amount (when the cells are verysmall) because of the continuity assumed for x y. Suppose that P1 x yis the particular system which ; ; minimizes the rate and gives R1. We choose from the high probability y's a set at random containing 2 R T 1+ members where 0 as T . With large Teach chosen point will be connected by a high probability ! ! line (as in Fig. 10) to a set of x's. A calculation similar to that used in proving Theorem 11 shows that withlarge Talmost all x's are covered by the fans from the chosen ypoints for almost all choices of the y's. Thecommunication system to be used operates as follows: The selected points are assigned binary numbers.When a message xis originated it will (with probability approaching 1 as T ) lie within at least one ! of the fans. The corresponding binary number is transmitted (or one of them chosen arbitrarily if there areseveral) over the channel by suitable coding means to give a small probability of error. Since R1 Cthis is possible. At the receiving point the corresponding yis reconstructed and used as the recovered message. The evaluation v0 for this system can be made arbitrarily close to v 1 1 by taking Tsufficiently large. This is due to the fact that for each long sample of message x tand recovered message y tthe evaluation approaches v1 (with probability 1). It is interesting to note that, in this system, the noise in the recovered message is actually produced by a kind of general quantizing at the transmitter and not produced by the noise in the channel. It is more or lessanalogous to the quantizing noise in PCM. 29. THE CALCULATION OF RATES The definition of the rate is similar in many respects to the definition of channel capacity. In the former Z Z P x y R Min P x ylog ; dx dy = ; Px y P x P y Z Z with P xand v1 P x y x y dx dyfixed. In the latter = ; ; Z Z P x y ; C Max P x ylog dx dy = ; P x P x P y with Px yfixed and possibly one or more other constraints (e.g., an average power limitation) of the form R R K P x y x y dx dy. = ; ; A partial solution of the general maximizing problem for determining the rate of a source can be given. Using Lagrange's method we consider Z Z P x y ; P x ylog P x y x y x P x y dx dy ; + ; ; + ; : P x P y 50 =============================================================================== The variational equation (when we take the first variation on P x y) leads to ; P x y ; y x B x e, = where is determined to give the required fidelity and B xis chosen to satisfy Z B x e x y , ; dx 1 = : This shows that, with best encoding, the conditional probability of a certain cause for various received y, Py xwill decline exponentially with the distance function x ybetween the xand yin question. ; In the special case where the distance function x ydepends only on the (vector) difference between x ; and y, x y x y ; = , we have Z B x e x y , , dx 1 = : Hence B xis constant, say , and P x y , y x e, = : Unfortunately these formal solutions are difficult to evaluate in particular cases and seem to be of little value.In fact, the actual calculation of rates has been carried out in only a few very simple cases. If the distance function x yis the mean square discrepancy between xand yand the message ensemble ; is white noise, the rate can be determined. In that case we have R Min H x Hy x H x MaxHy x = , = , with N x y2. But the Max Hy xoccurs when y xis a white noise, and is equal to W1 log 2 eNwhere = , , W1 is the bandwidth of the message ensemble. Therefore R W1 log 2 eQ W1 log 2 eN = , Q W1 log = N where Qis the average message power. This proves the following: Theorem 22:The rate for a white noise source of power Qand band W1 relative to an R.M.S. measure of fidelity is Q R W1 log = N where Nis the allowed mean square error between original and recovered messages. More generally with any message source we can obtain inequalities bounding the rate relative to a mean square error criterion. Theorem 23:The rate for any source of band W1 is bounded by Q1 Q W1 log R W1 log N N where Qis the average power of the source, Q1 its entropy power and Nthe allowed mean square error. The lower bound follows from the fact that the Max H 2 y x for a given x y Noccurs in the white , = noise case. The upper bound results if we place points (used in the proof of Theorem 21) not in the best way p but at random in a sphere of radius Q N. , 51 =============================================================================== ACKNOWLEDGMENTS The writer is indebted to his colleagues at the Laboratories, particularly to Dr. H. W. Bode, Dr. J. R. Pierce,Dr. B. McMillan, and Dr. B. M. Oliver for many helpful suggestions and criticisms during the course of thiswork. Credit should also be given to Professor N. Wiener, whose elegant solution of the problems of filteringand prediction of stationary ensembles has considerably influenced the writer's thinking in this field. APPENDIX 5 Let S1 be any measurable subset of the gensemble, and S2 the subset of the fensemble which gives S1under the operation T. Then S1 T S2 = : Let H be the operator which shifts all functions in a set by the time . Then HS1 HT S2 T HS2 = = since Tis invariant and therefore commutes with H. Hence if m Sis the probability measure of the set S m HS1 m T HS2 m HS2 = = m S2 m S1 = = where the second equality is by definition of measure in the gspace, the third since the fensemble isstationary, and the last by definition of gmeasure again. To prove that the ergodic property is preserved under invariant operations, let S1 be a subset of the g ensemble which is invariant under H, and let S2 be the set of all functions fwhich transform into S1. Then HS1 HT S2 T HS2 S1 = = = so that HS2 is included in S2 for all . Now, since m HS2 m S1 = this implies HS2 S2 = for all with m S2 0 1. This contradiction shows that S1 does not exist. 6= ; APPENDIX 6 The upper bound, N3 N1 N2, is due to the fact that the maximum possible entropy for a power N1 N2 + + occurs when we have a white noise of this power. In this case the entropy power is N1 N2. + To obtain the lower bound, suppose we have two distributions in ndimensions p xiand q xiwith entropy powers N1 and N2. What form should pand qhave to minimize the entropy power N3 of theirconvolution r xi: Z r xi p yi q xi yi dyi = , : The entropy H3 of ris given by Z H3 r xilog r xi dxi = , : We wish to minimize this subject to the constraints Z H1 p xilog p xi dxi = , Z H2 q xilog q xi dxi = , : 52 =============================================================================== We consider then Z U r xlog r x p xlog p x q xlog q x dx = , + + Z U 1 logr x r x 1 log p x p x 1 logq x q x dx = , + + + + + : If p xis varied at a particular argument xi si, the variation in r xis = r x q xi si = , and Z U q xi silog r xi dxi log p si 0 = , , , = and similarly when qis varied. Hence the conditions for a minimum are Z q xi silog r xi dxi log p si , = , Z p xi silog r xi dxi log q si , = , : If we multiply the first by p siand the second by q siand integrate with respect to siwe obtain H3 H1 = , H3 H2 = , or solving for and and replacing in the equations Z H1 q xi silog r xi dxi H3 log p si , = , Z H2 p xi silog r xi dxi H3 log q si , = , : Now suppose p xiand q xiare normal A n2 = i j j j p x 1 i exp Aijxixj = , 2 n2 2 = B n2 = i j j j q x 1 i exp Bijxixj = , : 2 n2 2 = Then r xiwill also be normal with quadratic form Ci j. If the inverses of these forms are ai j, bi j, ci jthen ci j ai j bi j = + : We wish to show that these functions satisfy the minimizing conditions if and only if ai j Kbi jand thus = give the minimum H3 under the constraints. First we have n 1 log r x 1 i log Ci j Cijxixj = j j , 2 2 2 Z n 1 q x 1 1 i silog r xi dxi log Ci j CijsisjCijbij , = j j , , : 2 2 2 2 This should equal H3 n 1 log A 1 i j Aijsisj j j , H 2 1 2 2 H1 H1 which requires Ai j Ci j. In this case Ai j Bi jand both equations reduce to identities. = = H3 H2 53 =============================================================================== APPENDIX 7 The following will indicate a more general and more rigorous approach to the central definitions of commu-nication theory. Consider a probability measure space whose elements are ordered pairs x y. The variables ; x, yare to be identified as the possible transmitted and received signals of some long duration T. Let us callthe set of all points whose xbelongs to a subset S1 of xpoints the strip over S1, and similarly the set whoseybelong to S2 the strip over S2. We divide xand yinto a collection of non-overlapping measurable subsetsXiand Yiapproximate to the rate of transmission Rby 1 P Xi Yi R ; 1 P Xi Yilog = ; T P X P Y i i i where P Xi is the probability measure of the strip over Xi P Yi is the probability measure of the strip over Yi P Xi Yi is the probability measure of the intersection of the strips ; : A further subdivision can never decrease R1. For let X1 be divided into X1 X0 X00 and let = 1 + 1 P Y1 a P X1 b c = = + P X0 b P X0 Y1 d 1 = 1; = P X00 c P X00 Y1 e 1 = 1 ; = P X1 Y1 d e ; = + : Then in the sum we have replaced (for the X1, Y1 intersection) d e d e + d elog by dlog elog + + : a b c ab ac + It is easily shown that with the limitation we have on b, c, d, e, d e d e + ddee + b c bdce + and consequently the sum is increased. Thus the various possible subdivisions form a directed set, withRmonotonic increasing with refinement of the subdivision. We may define Runambiguously as the leastupper bound for R1 and write it 1 ZZ P x y R P x ylog ; dx dy = ; : T P x P y This integral, understood in the above sense, includes both the continuous and discrete cases and of coursemany others which cannot be represented in either form. It is trivial in this formulation that if xand uarein one-to-one correspondence, the rate from uto yis equal to that from xto y. If vis any function of y(notnecessarily with an inverse) then the rate from xto yis greater than or equal to that from xto vsince, inthe calculation of the approximations, the subdivisions of yare essentially a finer subdivision of those forv. More generally if yand vare related not functionally but statistically, i.e., we have a probability measurespace y v, then R x v R x y. This means that any operation applied to the received signal, even though ; ; ; it involves statistical elements, does not increase R. Another notion which should be defined precisely in an abstract formulation of the theory is that of "dimension rate," that is the average number of dimensions required per second to specify a member ofan ensemble. In the band limited case 2Wnumbers per second are sufficient. A general definition can beframed as follows. Let f tbe an ensemble of functions and let T f t f t be a metric measuring ; 54 =============================================================================== the "distance" from fto fover the time T(for example the R.M.S. discrepancy over this interval.) LetN Tbe the least number of elements fwhich can be chosen such that all elements of the ensemble ; ; apart from a set of measure are within the distance of at least one of those chosen. Thus we are covering the space to within apart from a set of small measure . We define the dimension rate for the ensemble by the triple limit log N T Lim Lim Lim ; ; = : 0 0 T Tlog ! ! ! This is a generalization of the measure type definitions of dimension in topology, and agrees with the intu-itive dimension rate for simple ensembles where the desired result is obvious. 55 =============================================================================== ************ DDooccuummeenntt OOuuttlliinnee ************ * A Mathematical Theory of Communication * Introduction * Part I: Discrete Noiseless Systems o The Discrete Noiseless Channel o The Discrete Source of Information o The Series of Approximations to English o Graphical Representations of a Markoff Process o Ergodic and Mixed Sources o Choice, Uncertainty and Entropy o Representation of the Encoding and Decoding Operation o The Fundamental Theorem of a Noiseless Channel o Discussion and Examples * Part II: The Discrete Channel with Noise o Representation of a Noisy Discrete Channel o The Fundamental Theorem for a Discrete Channel with Noise o Discussion o Example of a Discrete Channel and its Capacity o The Channel Capacity in Certain Special Cases o An Example of Efficient Coding o A1. The Growth of the Number of Blocks of Symbols with a Finite State Condition o A2. The Derivation of Entropy o A3. Theorems on Ergodic Sources o A4. Maximizing the Rate for a System of Constraints * Part III: Mathematical Prelininaries o Sets and Ensembles of Functions o Band Limited Ensembles of Functions o Entropy of a Continuous Distribution o Entropy of an Ensemble of Functions o Entropy Loss in Linear Filters o Entropy of a Sum of Two Ensembles * Part IV: The Continuous Channel o The Capacity of a Continuous Channel o Channel Capacity with an Average Power Limitation o The Channel Capacity with a Peak Power Limitation * Part V: The Rate for a Continuous Source o Fidelity Evaluation Functions o The Rate for a Source Relative to a Fidelity Evaluation o The Calculation of Rates o A5 o A6 o A7 ===============================================================================