read

and therefore C logX0

In case there are restrictions on allowed sequences we may still often obtain a difference equation of this type and findC 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. HenceC is log 0 where 0 is the positive root of 1 2 4 5 7 8 10. 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 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

DASH

DOT

DASH

DOT

LETTER SPACE

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 b s i j be the duration of the sth symbol which is allowable in state i and leads to state j.

Then the channel capacityC is equal to logW whereW is the largest real root of the determinant equation:

∑ s W b s

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:

1 W 2 W 4

W 3 W 6 W 2 W 4 1 0

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 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

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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 example. B D C B C E C C C A D C B D D A A E C E E A 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: A A A C D C B D C E A A D A D A C E D A 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

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.

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pi j and the digram probabilities p i j are related by the following formulas:

p i ∑ j p i j ∑

j p j i ∑

j p j p j i

p i j p i pi j

∑ j pi j ∑

i p i ∑

i j p i j 1

As a specific example suppose there are three letters A, B, C with the probability tables:

pi j j A B C

A 0 4 5

1 5

i B 1 2

1 2 0

C 1 2

2 5

1 10

i p i

A 9 27

B 16 27

C 2 27

p i j j A B C

A 0 4 15

1 15

i B 8 27

8 27 0

C 1 27

4 135

1 135

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 B B 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 of a letter would depend on the preceding two letters but not on the message before that point. A set of trigram frequencies p i j k or equivalently a set of transition probabilities pi j k would 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 in or of transition probabilities pi1 i2 in 1 in is required to specify the statistical structure.

(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 with associated 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 typical message might be: DAB EE A BEBE DEED DEB ADEE ADEE EE DEB BEBE BEBE BEBE ADEE BED DEED DEED 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, but the description may be simpler in terms of the word structure and probabilities. We may also generalize 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 artificial languages. The zero-order approximation is obtained by choosing all letters with the same probability and independently. The first-order approximation is obtained by choosing successive letters independently but each 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 with probability .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 Urgent by Fletcher Pratt, Blue Ribbon Books, 1939. Word frequen- cies are tabulated in Relative Frequency of English Speech Sounds, G. Dewey, Harvard University Press, 1923.

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digrams such as TH, ED, etc. In the second-order approximation, digram structure is introduced. After a letter is chosen, the next one is chosen in accordance with the frequencies with which the various letters follow 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 two letters.

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 OOBTTVA NAH 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 their appropriate frequencies.

REPRESENTING AND SPEEDILY IS AN GOOD APT OR COME CAN DIFFERENT NAT- URALHEREHE THEA INCAME THE TOOF TOEXPERTGRAYCOMETO FURNISHES THE 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- ACTEROF THIS POINT IS THEREFOREANOTHERMETHODFORTHELETTERS THAT THE 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 their construction. 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 more words can easily be placed in sentences without unusual or strained constructions. The particular sequence of ten words “attack on an English writer that the character of this” is not at all unreasonable. It appears then that 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.

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To construct (3) for example, one opens a book at random and selects a letter at random on the page. This letter 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 the succeeding letter recorded, etc. A similar process was used for (4), (5) and (6). It would be interesting if further 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 processes and have been extensively studied in the literature.6 The general case can be described as follows: There exist a finite number of possible “states” of a system; S1 S2 Sn. In addition there is a set of transition probabilities; pi j the probability that if the system is in state Si it 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 transition from 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 B

C

D

E

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

A A

B

B

BC

C

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 pairs of 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 a Markoff process. Among the possible discrete Markoff processes there is a group with special properties of significance in communication theory. This special class consists of the “ergodic” processes and we shall call the corresponding sources ergodic sources. Although a rigorous definition of an ergodic process is somewhat involved, the general idea is simple. In an ergodic process every sequence produced by the process

6For a detailed treatment see M. Fréchet, Méthode des fonctions arbitraires. Théorie des événements en chaı̂ne dans le cas d’un nombre fini d’états possibles. Paris, Gauthier-Villars, 1938.

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