Abimael Martell
2543abe371
Justified print breaks words at syllables; after paragraph lines are joined
with spaces those breaks survive as "de- fendant" — thousands of them in a
long document — and, when an emphasis span was split with the word, as
"Bap-** **tist".
Whether the hyphen belongs in the word cannot be decided locally ("de-
fendant" is one word, "Third- Party" is a hyphenated compound), so the
document is used as its own dictionary. For each break:
1. fragments appear joined elsewhere ("defendant") -> join plain
2. appear hyphenated elsewhere ("six-month"), or the
continuation is capitalized ("Hinds- Radix") -> keep the hyphen
3. both fragments are words the document uses and the
continuation has 4+ letters ("commercial- type") -> keep the hyphen
4. no evidence -> leave untouched
The policy contains zero hard-coded words: vocabulary evidence,
capitalization, and two length invariants. The 4-letter floor keeps
suspended hyphens intact in any language ("mid- and long-term", "klein- und
mittelgroß", "kuva- tai video") because conjunctions are near-universally
1-3 letters. Fragments over 40 combined characters are fused reading-order
noise and are never joined. The vocabulary is collected after scrubbing the
break pairs themselves and excludes fenced code blocks; table rows and code
blocks are never rewritten. Split emphasis spans rejoin inside their
markers. Runs under the existing fix_hyphenation option (default on).
On a 1,370-page justified legal reporter this rejoins ~8,000 broken words
(98.8% of breaks; evidence-less ones stay visibly intact); word recall
against a reference extraction rises from 97.8% to 99.1%. No "six-month" ->
"sixmonth" class errors, and no fused-column corruption by construction:
no rule joins without evidence.
Regression-checked against a ~200-document corpus with semantic scoring
against an OCR baseline: zero regressions. Three in-repo fixture snapshots
regenerated with each diff inspected. 17 unit tests cover every rule, the
vocabulary scrubbing and code-block exclusion, the length gates, chained
breaks, mismatched emphasis markers, accented and Cyrillic words, German
and Finnish suspended hyphens, and the table/code skips.
44 lines
7.1 KiB
Markdown
44 lines
7.1 KiB
Markdown
Reprinted with corrections from *The Bell System Technical Journal,* Vol. 27, pp. 379–423, 623–656, July, October, 1948.
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## A Mathematical Theory of Communication
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### By C. E. SHANNON
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INTRODUCTION
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HE recent development of various methods of modulation such as PCM and PPM which exchange
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# Tbandwidth for signal-to-noise ratio has intensified the interest in a general theory of communication. A
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basis for such a theory is contained in the important papers of Nyquist¹ and Hartley² on this subject. In the present paper we will extend the theory to include a number of new factors, in particular the effect of noise in the channel, and the savings possible due to the statistical structure of the original message and due to the nature of the final destination of the information. The fundamental problem of communication is that of reproducing at one point either exactly or ap- proximately a message selected at another point. Frequently the messages have *meaning*; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one *selected from a set* of possible messages. The system must be designed to operate for each possible selection, not just the one which will actually be chosen since this is unknown at the time of design. If the number of messages in the set is finite then this number or any monotonic function of this number can be regarded as a measure of the information produced when one message is chosen from the set, all choices being equally likely. As was pointed out by Hartley the most natural choice is the logarithmic function. Although this definition must be generalized considerably when we consider the influence of the statistics of the message and when we have a continuous range of messages, we will in all cases use an essentially logarithmic measure. The logarithmic measure is more convenient for various reasons:
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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 2 logarithm of this number. Doubling the time roughly squares the number of possible messages, or doubles the logarithm, etc.
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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, that two punched cards should have twice the capacity of one for information storage, and two identical channels twice the capacity of one for transmitting information.
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3. It is mathematically more suitable. Many of the limiting operations are simple in terms of the logarithm but would require clumsy restatement in terms of the number of possibilities. The choice of a logarithmic base corresponds to the choice of a unit for measuring information. If the
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base 2 is used the resulting units may be called binary digits, or more briefly *bits,* a word suggested by
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J. W. Tukey. A device with two stable positions, such as a relay or a flip-flop circuit, can store one bit of information. *N* such devices can store*N* bits, since the total number of possible states is 2
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*N* and log₂2 *N* = *N*. If the base 10 is used the units may be called decimal digits. Since
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log₂*M* = log₁₀*M*= log₁₀2 = 3:32 log₁₀*M*;
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1 Nyquist, 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. 2 Hartley, R. V. L., “Transmission of Information,” *Bell System Technical Journal,* July 1928, p. 535.
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INFORMATION SOURCE TRANSMITTER RECEIVER DESTINATION
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SIGNAL RECEIVED SIGNAL MESSAGE MESSAGE
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NOISE SOURCE
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Fig. 1 — Schematic diagram of a general communication system.
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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 log*ba*. By a communication system we will mean a system of the type indicated schematically in Fig. 1. It consists of essentially five parts:
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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.
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2. A *transmitter* which operates on the message in some way to produce a signal suitable for transmission 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.
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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.
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4. The *receiver* ordinarily performs the inverse operation of that done by the transmitter, reconstructing the message from the signal.
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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
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necessary to represent the various elements involved as mathematical entities, suitably idealized from their
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