A Mathematical Theory of Communication

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INFORMATION,SOURCE TRANSMITTER RECEIVER DESTINATION. SIGNAL RECEIVED,MESSAGE MESSAGE, Fig 1 Schematic diagram of a general communication system. a decimal digit is about 3 13 bits A digit wheel on a desk computing machine has ten stable positions and. therefore has a storage capacity of one decimal digit In analytical work where integration and differentiation. are involved the base e is sometimes useful The resulting units of information will be called natural units. Change from the base a to base b merely requires multiplication by logb a. By a communication system we will mean a system of the type indicated schematically in Fig 1 It. consists of essentially five parts, 1 An information source which produces a message or sequence of messages to be communicated to the. receiving terminal The message may be of various types a A sequence of letters as in a telegraph. of teletype system b A single function of time f t as in radio or telephony c A function of. time and other variables as in black and white television here the message may be thought of as a. function f x y t of two space coordinates and time the light intensity at point x y and time t on a. pickup tube plate d Two or more functions of time say f t g t h t this is the case in three. dimensional sound transmission or if the system is intended to service several individual channels in. multiplex e Several functions of several variables in color television the message consists of three. functions f x y t g x y t h x y t defined in a three dimensional continuum we may also think. of these three functions as components of a vector field defined in the region similarly several. black and white television sources would produce messages consisting of a number of functions. of three variables f Various combinations also occur for example in television with an associated. audio channel, 2 A transmitter which operates on the message in some way to produce a signal suitable for trans. mission over the channel In telephony this operation consists merely of changing sound pressure. into a proportional electrical current In telegraphy we have an encoding operation which produces. a sequence of dots dashes and spaces on the channel corresponding to the message In a multiplex. PCM system the different speech functions must be sampled compressed quantized and encoded. and finally interleaved properly to construct the signal Vocoder systems television and frequency. modulation are other examples of complex operations applied to the message to obtain the signal. 3 The channel is merely the medium used to transmit the signal from transmitter to receiver It may be. a pair of wires a coaxial cable a band of radio frequencies a beam of light etc. 4 The receiver ordinarily performs the inverse operation of that done by the transmitter reconstructing. the message from the signal, 5 The destination is the person or thing for whom the message is intended.
We wish to consider certain general problems involving communication systems To do this it is first. necessary to represent the various elements involved as mathematical entities suitably idealized from their. physical counterparts We may roughly classify communication systems into three main categories discrete. continuous and mixed By a discrete system we will mean one in which both the message and the signal. are a sequence of discrete symbols A typical case is telegraphy where the message is a sequence of letters. and the signal a sequence of dots dashes and spaces A continuous system is one in which the message and. signal are both treated as continuous functions e g radio or television A mixed system is one in which. both discrete and continuous variables appear e g PCM transmission of speech. We first consider the discrete case This case has applications not only in communication theory but. also in the theory of computing machines the design of telephone exchanges and other fields In addition. the discrete case forms a foundation for the continuous and mixed cases which will be treated in the second. half of the paper,PART I DISCRETE NOISELESS SYSTEMS. 1 T HE D ISCRETE N OISELESS C HANNEL, 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 elementary. symbols S1, Sn can be transmitted from one point to another Each of the symbols Si is assumed to have. a certain duration in time ti seconds not necessarily the same for different Si for example the dots and. dashes in telegraphy It is not required that all possible sequences of the Si be capable of transmission on. the system certain sequences only may be allowed These will be possible signals for the channel Thus. in telegraphy suppose the symbols are 1 A dot consisting of line closure for a unit of time and then line. open for a unit of time 2 A dash consisting of three time units of closure and one unit open 3 A letter. space consisting of say three units of line open 4 A word space of six units of line open We might place. the restriction on allowable sequences that no spaces follow each other for if two letter spaces are adjacent. it is identical with a word space The question we now consider is how one can measure the capacity of. such a channel to transmit information, In the teletype case where all symbols are of the same duration and any sequence of the 32 symbols. is allowed the answer is easy Each symbol represents five bits of information If the system transmits n. symbols per second it is natural to say that the channel has a capacity of 5n bits per second This does not. mean that the teletype channel will always be transmitting information at this rate this is the maximum. possible rate and whether or not the actual rate reaches this maximum depends on the source of information. which feeds the channel as will appear later, In the more general case with different lengths of symbols and constraints on the allowed sequences we.
make the following definition, Definition The capacity C of a discrete channel is given by. where N T is 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 symbols. S1 Sn are allowed and these symbols have durations t1. tn What is the channel capacity If N t, represents the number of sequences of duration t we 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. Sn and 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 t to X0t where X0 is the largest real solution of the characteristic equation. X t1 X t2 X tn 1,and therefore, In case there are restrictions on allowed sequences we may still often obtain a difference equation of this.
type and find C from 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 C is log 0 where 0 is the positive root of 1 2 4 5 7 8 10 Solving this we find. 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. can be transmitted different subsets for the different states When one of these has been transmitted the. state changes to a new state depending both on the old state and the particular symbol transmitted The. telegraph case is a simple example of this There are two states depending on whether or not a space was. the 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. 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 shown. that if the conditions on allowed sequences can be described in this form C will exist and can be calculated. in accordance with the following result, Theorem 1 Let bi j be the duration of the sth symbol which is allowable in state i and leads to state j. Then the channel capacity C is equal to logW where W is the largest real root of the determinant equation. W i j i j 0,where i j 1 if i j and is zero otherwise. For example in the telegraph case Fig 2 the determinant is. On expansion this leads to the equation given above for this case. 2 T HE D ISCRETE S OURCE OF I NFORMATION, We have seen that under very general conditions the logarithm of the number of possible signals in a discrete.
channel increases linearly with time The capacity to transmit information can be specified by giving this. rate of increase the number of bits per second required to specify the particular signal used. We now consider the information source How is an information source to be described mathematically. and how much information in bits per second is produced in a given source The main point at issue is the. effect of statistical knowledge about the source in reducing the required capacity of the channel by the use. of proper encoding of the information In telegraphy for example the messages to be transmitted consist of. sequences of letters These sequences however are not completely random In general they form sentences. and have the statistical structure of say English The letter E occurs more frequently than Q the sequence. TH more frequently than XP etc The existence of this structure allows one to make a saving in time or. channel capacity by properly encoding the message sequences into signal sequences This is already done. to a limited extent in telegraphy by using the shortest channel symbol a dot for the most common English. letter E while the infrequent letters Q X Z are represented by longer sequences of dots and dashes This. idea is carried still further in certain commercial codes where common words and phrases are represented. by four or five letter code groups with a considerable saving in average time The standardized greeting. and anniversary telegrams now in use extend this to the point of encoding a sentence or two into a relatively. short sequence of numbers, We can think of a discrete source as generating the message symbol by symbol It will choose succes. sive symbols according to certain probabilities depending in general on preceding choices as well as the. particular symbols in question A physical system or a mathematical model of a system which produces. such a sequence of symbols governed by a set of probabilities is known as a stochastic process 3 We may. consider a discrete source therefore to be represented by a stochastic process Conversely any stochastic. process which produces a discrete sequence of symbols chosen from a finite set may be considered a discrete. source This will include such cases as, 1 Natural written languages such as English German Chinese. 2 Continuous information sources that have been rendered discrete by some quantizing process For. example the quantized speech from a PCM transmitter or a quantized television signal. 3 Mathematical cases where we merely define abstractly a stochastic process which generates a 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 typical. BDCBCECCCADCBDDAAECEEA,A 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. AAACDCBDCEAADADACEDA,E 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 choice. depends only on the preceding letter and not on ones before that The statistical structure can. then be described by a set of transition probabilities pi j the probability that letter i is followed. by letter j The indices i and j range over all the possible symbols A second equivalent way of. specifying 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. 3 See for example S Chandrasekhar Stochastic Problems in Physics and Astronomy Reviews of Modern Physics v 15 No 1. January 1943 p 1, 4 Kendall and Smith Tables of Random Sampling Numbers Cambridge 1939.
pi j and the digram probabilities p i j are related by the following formulas. p i p i j p j i p j p j i,p i j p i pi j,pi j p i p i j 1. As a specific example suppose there are three letters A B C with the probability tables. HE recent development of various methods of modulation such as PCM and PPM which exchange bandwidth for signal to noise ratio has intensi ed the interest in a general theory of communication A basis for such a theory is contained in the important papers of Nyquist1 and Hartley2 on this subject In the

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