Contains research objectives and reports on one research project.Lincoln Laboratory (Purchase Order DDL-B187)Department of the ArmyDepartment of the NavyDepartment of the Air Force under Contract AF19(122)-45

Elias, Peter

Fano, R. M.

Huffman, D. A.

Unger, S. H.

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XII. PROCESSING AND TRANSMISSION OF INFORMATIONProf. P. Elias W. G. Breckenridge R. S. MarcusProf. R. M. Fano R. R. Capraro L. S. OnyshkevychProf. D. A. Huffman E. Ferretti E. T. SchoenProf. C. E. Shannon J. V. Harrington S. H. UngerDr. M. V. Cerrillo K. Joannou M. L. WetherellDr. M. P. Schitzenberger R. M. Lerner J. M. WozencraftDr. T. Von Randow (visiting) W. A. YoungbloodRESEARCH OBJECTIVESThe interest of this group ranges over a wide variety of problems in the processingand transmission of information.One current activity is the statistical investigation of information sources. The mainobjective is to estimate the rate at which the sources generate information and to deter-mine how to encode their output economically, in order to decrease the channel capa-city required for transmission. The group is currently continuing such an investigationon pictures as information sources by recording the pictures in digital form, for analysisand processing on a digital computer.A second current activity is the investigation of channels for information transmis-sion. This includes study of the fundamental theoretical relationships between trans-mission rate, error probability, delay, and equipment complexity, and the constructionof suitable coding procedures, especially for binary channels.In the design of coding and decoding devices, binary switching circuits play a criti-cal role. Viewed at its terminals, a switching circuit can be considered to be a trans-ducer that transforms an input stream of binary digits into a related output stream. Inthe process, information may be lost and its coded form changed drastically. In thissense a switching circuit is a rather general information channel. Research in thisarea deals with relationships among the number of memory elements necessary in acircuit, the number of signal feedback loops and the complexity of the signals that flowin them, and the terminal effect of nonideal behavior of the elements of the circuit.A further intimate relationship between sequential switching circuits and informa-tion theory becomes apparent if we consider that an n-state circuit, in effect, classi-fies each of the infinite number of possible input sequences into one of n different sets.Thus the minimization of the complexity of a circuit corresponds essentially to the class-fication of an infinite number of sequence-patterns into a minimum number of sets tobe recognized.P. Elias, R. M. Fano, D. A. HuffmanA. A NEW MODE OF OPERATION FOR SEQUENTIAL SWITCHING CIRCUITSA new mode of operation for asynchronous sequential switching circuits which, undercertain conditions, may be economically advantageous, compared with the usual modeconsidered in reference 1 and in other sources, is described.Figure XII-la shows the usual form of a sequential switching circuit. The signalpaths leading from the output end of the function generator back to the input end will beThis research was supported in part by Purchase Order DDL-B187 with LincolnLaboratory, which is supported by the Department of the Army, the Department of theNavy, and the Department of the Air Force under Contract AF19(122)-458 with M.I. T.108(XII. PROCESSING AND TRANSMISSION OF INFORMATION)SYSTEM o SYSTEMFUNCTION *INPUT GENERATOR OUTPUT(a)SYSTEM SYSTEMINPUT FUNCTION OUTPUTGENERATOR(b)Fig. XII-1. Forms of sequential switching circuit:(a) usual form; (b) double-delay form.called "state branches," and the signals entering and leaving these branches will becalled state-branch "excitations" and "responses." A single inertial delay element ineach state branch (see Fig. XII-1) is sufficient for proper operation of such a circuit,provided that the flow matrix does not include critical race conditions (1). (It might bewell to explain that an "inertial" delay element of magnitude D is characterized by thefact that its output will change only after an input change has persisted for time D. Alldelays considered here will be inertial.)The necessity for avoiding critical races leads, in general, to row assignments thatrequire more state branches than would otherwise be needed (2). In this respect theproposed method, which will be called the "double-delay mode," offers an opportunityfor economy. As shown in Fig. XII- lb, two cascaded inertial delays with values A andB are placed in each state branch, and if the magnitudes of these delays can be main-tained within certain limits, flow-table row assignments can be made without having toavoid races. Since an amplifier is usually needed in each state branch, reducing thenumber of state branches leads to a saving in amplifiers, as well as to a possible reduc-tion in the complexity of the combinational circuits that comprise the function generator.In order to achieve these advantages, a larger number of delay elements must be used,and the values of the delay elements must be held within specified tolerances.109(XII. PROCESSING AND TRANSMISSION OF INFORMATION)Consider a typical flow-table transition that begins with the system in a stablestate S i . Following an input change, the system enters an unstable state S 2 , andthen the internal state (determined by the state-branch responses) changes, so that thesystem enters a new stable state S 3 . In a double-delay system (see Fig. XII-lb) sucha transition takes place in the following manner:1. The system input changes, thereby altering the state of the system from S1 toS 2 , in which the excitations of the state branches correspond to S3. This may entailhaving several state branches become unstable simultaneously.2. After a time lapse that is at least equal to A m + B (the subscripts m and Mwill denote minimum and maximum values of the parameters concerned), the responseof one of the state branches will change as the new excitation signal flows through theA- and B-delays of that branch, thus initiating the interval that we shall call the "errorinterval." If several state branches become unstable in S , we cannot predict the onethat will be the first to change its response, unless we know the relative values of thedelays involved. Hence the system may enter any of several states at the beginning ofthe error interval, and we can expect some of the state-branch excitations to be alteredto values other than those corresponding to S 3 . However, if the delay values are withinthe proper limits, the S 3 excitation signals will appear at the A-delay outputs before theerror interval begins. This will happen if A m + B > A3. The error interval (in which some of the state-branch excitations may be incor-rect) terminates when all of the state-branch responses have assumed values corre-sponding to S 3 . At this point, the system is in S 3 , a stable state, and all of the state-branch excitations will take on the S 3 values which they had prior to the beginning of theerror interval. None of the A-delay outputs will change during this period, providedthat it is of sufficiently short duration; that is, if (AM + BM) - (A + B ) < A .The A-delays serve as buffers to absorb differences in the values of the state-branchdelays. This buffering action is possible because of the inertial nature of the A-delays,which, incidentally, can be placed either before or after the B-delays in the statebranches without changing the operation of the system. The B-delays need not beinertial.By using the two inequalities stated above, constraints can be placed on the relativevalues of the A- and B-delays and on the permissible tolerances. Let A and B be thespecified values of the two types of delay element, and define the normalized toler-ances EA and EB so that AM = A(1 + EA), Am = A(1 - EA), BM = B(1 + EB), and BmB(l - EB). ThusA M +A A - AA= M m E =M m2 ' 2A '110(XII. PROCESSING AND TRANSMISSION OF INFORMATION)Manipulating the inequalities and definitions in a straightforward manner, we obtainthe following relationships:i - EBE - (1)A 3 +EBB _ 2 (2)A- 3+EBThese equations can be used to obtain equivalent constraints in other forms.Note that, according to Eq. 1, E B can vary from zero to one, but that EA rangesonly from zero to one-third. In other words, regardless of how precise the B-delaysare made, the maximum allowed variation in A is one-third. If E A = E B = E, then Eis approximately 24 per cent.The two equations also show that the value of B should be somewhere between one-half and two-thirds the value of A, depending upon the tolerances. Either A or B canbe chosen without reference to the constraints.S. H. UngerReferences1. D. A. Huffman, Technical Report 274, Research Laboratory of Electronics, M.I.T.,Jan. 10, 1954.2. D. A. Huffman, Technical Report 293, Research Laboratory of Electronics, M, I. T.,March 14, 1955.111

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