160

While from (16) for: = 16, -y = 9 we obtain

24 212- < b3(d+1) < - (23)25 20

(Compare (23) with Tung's [6] result.)It is interesting to note that the range transformation

schemes are analogous to the Harvard iterative scheme[7] (see also Wallace [8]) in which the divisor is suc-cessively multiplied so as to be transformed to a numberarbitrarily close to unity, as desired. The only difference,as explained in [1], is that this initial range transforma-tion together with a nonrestoring division scheme ismore economical, since the numerator need not bemultiplied more than once. However, the choice of asuitable multiplier initially poses a problem and thishas to be considered to evaluate the relative merits.Some studies have been made by the author regardingoptimal precision requirement and choice of the multi-pliers for the direct as well as iterative schemes. (See[9], [12].)In this context, it is worthwhile noting that the inter-

val width or range of b in (16) contracts as y increases,with a zero width for y= 3 -1. As observed elsewhere[9], the range transformation of numbers involvesgreater computational labor if the range width de-creases; thus Tung's algorithm becomes unsuitable asredundancy increases. Also, in Tung's algorithm thequotient digit obtained can have a magnitude (f - 1),larger than y, thereby requiring facility for storing atwo-digit quotient and a facility for two-digit multipli-cation for computing the partial remainder. A differentanalysis made by the author to remove these difficultieshas given rise to more efficient algorithms; these areavailable in [10] and [11], and will appear elsewhere.

REFERENCES[1] S. K. Nandi and E. V. Krishnamurthy, "A simple technique for

digital division," Commun. ACM, vol. 10, pp. 299-301, May1967.

[2] E. V. Krishnamurthy, "On a divide-and-correct method forvariable precision division," Commun. A CM, vol. 8, pp. 179-18 1,March 1965.

[3] E. V. Krishnamurthy and S. K. Nandi, "On the normalizationrequirement of divisor in divide and correct methods," Commun.ACM, vol. 10, pp. 809-813, December 1967.

[4] A. Svoboda, "An algorithm for division," Information ProcessingMachines (Prague, Czechoslovakia), no. 9, 1963.

[5] A. Avizienis, "Signed-digit number representations for fastparallel arithmetic," IRE Trans. Electronic Computers, vol. EC-10, pp. 389-400, September 1961.

[6] C. Tung, "A division algorithm for signed-digit arithmetic,"IEEE Trans. Computers (Short Notes), vol. C-17, pp. 887-889,September 1968.

[7] R. K. Richards, Arithmetic Operations in Digital Computers.London: Van Nostrand, 1955.

[8] C. S. Wallace, "A suggestion for fast multiplier," IEEE Trans.Electronic Computers, vol. EC-13, pp. 14-17, February 1964.

[9] E. V. Krishnamurthy and B. P. Sarkar, "Algorithm for multiple-precision range transformation," Sankhyd (B), vol. 30, parts 1and 3, pp. 33-46, June 1968.

[10] E. V. Krishnamurthy, "A more efficient range-transformationalgorithm for signed-digit division," Internatl. J. of Control, tobe published.

[11] "'Carry-borrow free sequential quotient generation withsegmented signed-digit operands," Internatl. J. of Control, tobe published.

[12] , "On optimal iterative schemes for high-speed division,"IEEE Trans. Computers, vol. C-19, March 1970.

IEEE/TRANSACTIONS ON COMPUTERS, FEBRUARY 1970

I t Pictorial Output with a Line Printer

I. MACTEIDL.Abstract-An improved method for the production of pictorial

output on a line printer is described. A reasonable black-whitecontrast ratio is obtained by overprinting up to eight characters, andpseudorandom noise is used to smooth out discontinuities in therange of print densities.

Index Terms-Linear interpolation, line printers, overprinting,pictorial output, picture coding, picture output.

The possible character positions on line printer out-put may be regarded as cells in a two-dimensional array.By choosing the character (or combination of over-printed characters) printed in each cell on the basis ofaverage print density, a pictorial representation of anydesired two-dimensional data may be generated. Themaximum number of printed characters in a line is arestriction to the horizontal size of the picture unless itis assembled from several strips. Pictures produced inthis manner will be inferior to those produced by special-purpose hardware, but the convenience and ready avail-ability of a line printer will in many cases outweigh anyloss of quality.

Perry and Mendelsohn [1] describe such a methodof picture generation with a line printer. They use pairsof adjacent character positions in each printed line asbasic density cells, and overprint a maximum of twocharacters. We have found that a greater degree ofoverprinting yields a worthwhile increase in maximumdensity (and thus contrast) without an undue penaltyin printout time on the IBM 1403 line printer (1100 1pmmax.) attached to the Australian National University'sIBM 360/50 computer. Using a maximum of eight over-printed characters, the output shown in Fig. 2(b) tookapproximately one minute to generate on this printer.The use of individual (rather than pairs of) characterpositions as basic density cells results in finer resolutionand allows more information to be represented on eachline. With the 1403 printer set to ten characters per inchhorizontally and eight lines per inch vertically, notice-able distortion results from the difference in horizontaland vertical scales. This distortion is reduced by linearlyinterpolating output lines between rows of the dataarray, printing only eight lines for every ten rows in thedata array.A difficulty in determining the density codes (i.e.,

overprinted character combinations) to be employed isthat it is not possible, given a conventional characterset, to choose combinations such that there is a smoothtransition from white to black, without perceptible dis-continuities. The sudden transition from white (blank)to the next lightest combination (comma) is noticeablein the examples of pictorial output given in [1]. Theproblem of representing numerical data with density

Manuscript received April 18, 1969; revised August 27, 1969.The author is with the Department of Engineering Physics,

Australian National University, A.C.T. 2600, Australia.

SHORT NOTES ~~~~~ui-R,/2 ,DV 0 +tRf/2

D] D, DI: Di1, D, AVAILABLEI -- - I I I - CODESO DP Dd Ti I

lightest darkest PRINTDENSITY

Fig. 1. Statistical selection of density code. Dd=desired density;D,=density value of next code below Dd; Di+i=density valueof next code above Dd; Ri=ith density range-=D,+-D,; T,=ithdecision threshold = (Di+Di+i)/2; Us =uniform random dis-tribution of mean zero and range ±Ri/2; Du= a particular valuechosen from Us; Dp= desired density perturbed by addition of Du.

values chosen from a large but unevenly distributed set,e.g., the set of overprinted character combinations, isrelated to that of coding pictorial input with a small butevenly distributed set of numerical values. FollowingRoberts' technique for encoding pictures with a smallset of symbols [2], the particular density code to beprinted in each cell is chosen statistically, such that theaverage density of several neighboring cells is usuallycloser to the desired density than are any of the indi-vidual density codes. Roberts' examples illustrate theeffectiveness of this technique.

Referring to Fig. 1, the following method is used todecide which density code is to be printed in each posi-tion.

1) The closest available density codes both belowand above the desired density are determined.

2) A pseudorandom number is chosen from a uniformdistribution with zero mean and a range equal tothe density difference of the closest availablecodes, and added to the desired density.

3) The next code above the desired density is printedif the perturbed density is greater than the aver-age value of the two closest codes, otherwise thenext code below is printed. For the example givenin Fig. 1, code Di would be printed.

The probability of a particular code being printed in agiven cell thus depends on how close its density value isto the desired density. Schroeder [3] reports the use ofthis technique in the computer production of pictorialoutput on a microfilm plotter.The quality of output which may be obtained is

illustrated in Fig. 2. A scanner was used to convert theoriginal photograph shown in Fig. 2(a) into a 160X 128array of density values. The pictorial output shown inFig. 2(b) was generated from this array. The output is128 lines (interpolated on the 160 rows of data) deepby 128 density cells (i.e., character positions) wide. Thetwenty-one basic codes employed, with their approxi-mate density values on a normalized scale, are given inTable I. These values were obtained with medium printpressure and a well-inked ribbon, using a type QN printtrain in our IBM 1403 line printer. Intermediate codes,together with their normalized density values, were

(a)

(b)Fig. 2. Example of output quality. (a) Original photograph.

(b) Line printer output.

161

IEEE TRANSACTIONS ON COMPUTERS, FEBRUARY 1970

TABLE IDENSITY CODES EMPLOYED

Overprinted Character Estimated DensityCombinations Values

blank 0.00.150.22

+ 0.250.29

1 0.33Z 0.37X 0.40A 0.42M 0.450- 0.530= 0.560 + 0.60O +' 0.640 +. 0.670 + - 0.79OX'.X - 0.85OX' .H C 0.89OX' .HB 0.93OX' .HBV 0.970 X'. H B V A 1.00

determined by visual comparison with other codes ratherthan by measurements of inked-in areas. As a result, thevalues given in Table I tend to be linear on a scale ofperceived rather than measured density.

REFERENCES[11 B. Perry and M. L. Mendelsohn, "Picture generation with a

standard line printer," Commun. ACM, vol. 7, pp. 311-313, May1964.

[21 L. G. Roberts, "Picture coding using pseudo-random noise,"IRE Trans. Information Theory, vol. IT-8, pp. 145-154, February1962.

[31 M. R. Schroeder, "Images from computers and microfilmplotters," Commun. ACM, vol. 12, pp. 95-101, February 1969.

The Necessity of Closed Circuit Loops in MinimalCombinational CircuitsWILLIAM H. KAUTZ

Abstract-A cellular-logic approach is used to generate a familyof multiple-output combinational switching circuits containing closedloops (of the type that normally generate sequential behavior) andcomposed of simple gates. These networks contain fewer gates thanany loop-free realizations. Some members of the family are oscilla-tory, while others are stable with multiple stable states, but the out-puts remain quiescent in both cases. This result appears to haverepercussions on some of the well-known optimality results of switch-ing theory.

Index Terms-Cellular logic, closed loops, combinational net-works, switching theory.

It is generally assumed in switching theory that 1)combinational circuits composed of gate-type elementsare those that contain no closed loops, and 2) the pres-ence of a closed loop implies sequential behavior. How-ever, Short [1 ] provided an example of a combinational

Manuscript received May 2, 1968; revised August 18, 1969.This research was sponsored by the U. S. Air Force Cambridge Re-search Laboratories, Office of Aerospace Research, under ContractF19628-68-C-0262.

The author is with Stanford Research Institute, Menlo Park,Calif. 94025.

Fig. 1. Minimal transfer-contact network with a circuit loop [1].

TA-7258-9

Fig. 2. A one-dimensional cellular cascade with feedback.

transfer-contact network that contains a closed loopand uses fewer contacts than any other loop-free realiza-tion within a certain class-namely, the class of transfer-contact networks in which no pole (armature) is directlyconnected to another. This example is shown in Fig. 1.His example also establishes the fact that a computerprogram that consists entirely of binary decisions [2]must in general contain closed loops if the total numberof such decisions is to be minimized.Huffman [3 ] has recently found examples of combina-

tional circuits composed of AND, OR, and NOT elementsthat are minimal in the number of NOTS (although notin the number of ANDs and ORS) only if loops are allowed.In fact, in some of Huffman's networks, signals aroundthe closed loop may even oscillate, although the outputsof the network remain quiescent.

In order to generate a class of combinational circuitswith loops, and yet having nontrivial terminal behavior,we may consider the one-dimensional unilateral cellularcascade shown in Fig. 2. Note that the cascade outputof the mth cell is fed back as the cascade input of thefirst cell. For the moment, the input, output, and inter-cellular lines will be assumed to be single conductors, sothat each cell (the ith cell, say) realizes two functions,9i and zi, of two variables, yi and xi. All cells are assumedto be identical. To avoid a degenerate network, the logicof a typical cell must satisfy the following conditions.

1) In order for the loop to be potentially closed, yetdependent upon the input variables and capable ofsupporting both 0 and 1 signals, the cascade out-put 9i must be dependent upon both yi-specifi-cally as yi in a unate fashion-and x,.

2) In order for the cell outputs to be nontrivial func-tions, zi must be dependent upon yi.

3) In order that the cell outputs remain stable whenthe loop is closed, then zi must be independent ofyj whenever xi has a value such that 9i = yi.

162