IRE Transactions on Electronic Computers

The COordinate Rotation DIgital Computer(CORDIC) is a special-purpose digital computer for real-time airborne computation. In this computer, a unique computing technique is employed which is especially suitable for solving the trigonometric relationships involved in plane coordinate rotation and conversion from rectangular to polar coordinates. CORDIC is an entire-transfer computer; it contains a ... View full abstract»

Algorithms are presented for 1) converting a state graph describing the behavior of an automaton to a regular expression describing the behavior of the same automaton (section 2), and 2) for converting a regular expression into a state graph (sections 3 and 4). These algorithms are justified by theorems, and examples are given. The first section contains a brief introduction to state graphs and th... View full abstract»

This paper describes a class of number representations which are called signed-digit representations. Signed-digit representations limit carry-propagation to one position to the left during the operations of addition and subtraction in digital computers. Carry-propagation chains are eliminated by the use of redundant representations for the operands. Redundancy in the number representation allows ... View full abstract»

The algorithm described in this paper is the outcome of an endeavor to answer the following question: Is it possible to find procedures which would enable a computer to solve efficiently path-connection problems inherent in logical drawing, wiring diagramming, and optimal route finding? The results are highly encouraging. Within our framework, we are able to solve the following types of problems: ... View full abstract»

A large, extremely fast digital adder with sum selection and multiple-radix carry is described. Boolean expressions for the operation are included. The amount of hardware and the logical delay for a 100-bit ripple-carry adder and a carry-select adder are compared. The adder system described increases the speed of the addition process by reducing the carry-propagation time to the minimum commensura... View full abstract»

Numerous physical situations related to data processing are shown to be modeled by a mathematical entity called a quasi-machine. The situations described include 1) single inputs producing multiple outputs, 2) machines yielding no outputs upon insertion of certain inputs, 3) the retention of the last n outputs only, 4) ``erase left'' on tape, 5) different input routines doing the same work, and 6)... View full abstract»

A method of computer multiplication and division is proposed which uses binary logarithms. The logarithm of a binary number may be determined approximately from the number itself by simple shifting and counting. A simple add or subtract and shift operation is all that is required to multiply or divide. Since the logarithms used are approximate there can be errors in the result. An error analysis i... View full abstract»

Conditional-sum addition is a new mechanism for parallel, high-speed addition of digitally-represented numbers. Its design is based on the computation of ``conditional'' sums and carries that result from the assumption of all the possible distributions of carries for various groups of columns. A rapid-sequence mode of operation provides an addition rate that is invariant with the lengths of the su... View full abstract»

After a very brief summary of the various well-known methods of expediting carry-propagation in binary arithmetic units, the paper discusses and develops the ``anticipated-carry'' or ``carry-skip'' technique originally due in decimal form to Babbage, much used in mechanical calculators and lately revived for use in binary units. Various degrees of refinement are possible. It appears that for a giv... View full abstract»

Recent developments in computer technology have produced devices (parametrons, Esaki diodes) that act logically as binary majority-decision elements. Conventional design techniques fail to utilize fully the logical properties of these devices. The resulting designs are extravagant with respect to the number of components used and the operating time required. This paper reviews the conventional tec... View full abstract»

The problem of synthesizing switching networks out of linear-input (threshold) elements is studied for the class of symmetric switching functions. Tight bounds are derived for the number of elements required in a minimal realization, and a method of synthesis is presented which yields economical networks. Minimal networks result for all symmetric functions of no more than about twelve variables, a... View full abstract»

Existing large scale binary computers typically must allow for the maximum full length carry time in each addition. It has been shown that average carry sequences are significantly shorter than this maximum, on the average only five stages for a 40 digit addition. A method is described to realize the implied 8 to 1 time saving by deriving an actual ``carry completion'' signal. Experimental results... View full abstract»

A method is described which permits the encoding of arbitrary geometric configurations so as to facilitate their analysis and manipulation by means of a digital computer. It is shown that one can determine through the use of relatively simple numerical techniques whether a given arbitrary plane curve is open or closed, whether it is singly or multiply connected, and what area it encloses. Further,... View full abstract»

After a brief survey of the basic Atlas machine, the paper describes an automatic system which in principle can be applied to any combination of two storage systems so that the combination can be regarded by the machine user as a single level. The actual system described relates to a fast core store-drum combination. The effect of the system on instruction times is illustrated, and the tape transf... View full abstract»

The character recognition problem, usually resulting from characters being corrupted by printing deterioration and/or inherent noise of the devices, is considered from the viewpoint of statistical decision theory. The optimization consists of minimizing the expected risk for a weight function which is preassigned to measure the consequences of system decisions As an alternative minimization of the... View full abstract»

The electronic computer field continues to expand at a rapid rate. New developments, applications, and publications are too numerous for one person to keep informed of all aspects. In this review a complete report will not be attempted, but some highlights and selected references will be reported. View full abstract»

The parametric subharmonic oscillator operates by energy transfer from the pump frequency to the oscillator frequency through a nonlinear energy storage element-in the present case, the nonlinear capacitance of a semiconductor diode. This paper examines both the requirements on the diode for satisfactory performance in this circuit and the limitations on oscillator performance which arise from the... View full abstract»

This paper describes a class of division methods best suited for use in digital computers with facilities for floating point arithmetic. The division methods may be contrasted with conventional division procedures by considering the nature of each quotient digit as generated during the division process. In restoring division, each quotient digit has one of the values 0,1, . . . , r -1, for an arbi... View full abstract»

A technique is presented which yields a minimal-state machine satisfying a given set of behavioral specifications. The machine is constructed in the same manner as has commonly been done in the past in synthesizing a ``primitive flow table.'' This contribution consists, not in describing a new method of synthesizing machines, but in showing that a particular instance of an established method yield... View full abstract»

The success of computers in doing routine work formerly done by people suggests that a computer capable of adjusting itself to a changing environment might be desirable. Such a characteristic might be especially valuable to the telephone industry which must service large numbers of people having changing needs and desires. As a step in this direction a relay machine which plays a penny-matching ga... View full abstract»

A novel number system called the residue number system is developed from the linear congruence viewpoint. The residue number system is of particular interest because the arithmetic operations of addition, subtraction and multiplication may be executed in the same period of time without the need for carry. The main difficulties of the residue code pertain to the determination of the relative magnit... View full abstract»

As an attempt to investigate a general theory of real-time computability in digital computers, a subclass of Turing machines is formally introduced together with some classes of functions that are computable by them in real time. Then the existence is established of a class of recursive functions that are not computable in real time by use of a class of machines, no matter how general we make the ... View full abstract»

Sequential table look-up algorithms are proposed for the evaluation of ln x and exp x. Tables of pre-computed constants are utilized in transforming the argument into a range where the function may be approximated by a simpler polynomial. In the case of ln x, x is transformed so that it falls into a close neighborhood of 1; in the case of exp x, x is transformed to a close neighborhood of 0. These... View full abstract»

The complexity of an electronic switching circuit is defined in a sufficiently general way so that most definitions which are presently used may be included. If ø(p, q) is the complexity of a p input q output circuit which has been minimized then we may define E(p, q) as the maximum of ø(p, q) over all p input, q output circuits. In spite of the generality of the definition of complexity one may o... View full abstract»

This paper contains some applications of an elementary study of unate truth functions. One application is a method of deciding when a truth function is linearly separated, i. e., is expressible as a linear polynomial inequality in its arguments (letting 1 represent truth and 0 represent falsity). Other applications are to contact nets and to rectifier nets. Much of the material of this paper, alth... View full abstract»

Techniques are developed for the logical design of magnetic core circuits to produce arbitrary single-output combinational switching functions. The approach is based on the relationship of a single magnetic core circuit to a linearly separable switching function. A synthesis procedure is developed which uses a pair of logical primitives, AND with NOT and OR with NOT, which are similar to the STROK... View full abstract»

An algebra suited to logical design with majority-decision elements (parametrons, Esaki diodes, etc.) is developed axiomatically. The utility of the new algebra is demonstrated by resolving sample problems. View full abstract»

In this paper, the problem of determining economical state assignments for finite-state sequential machines is studied. The fundamental idea in this study is to find methods for selection of these assignments in which each binary variable describing the new state depends on as few variables of the old state as possible. In general, these variable assignments in which the dependence is reduced yiel... View full abstract»

The object of this paper is to find state assignments for the internal states of a sequential machine such that the logical equations representing the machine are relatively simple. This is done by finding assignments for which the computation of a particular state variable depends only on the previous values of a small subset of the variables. The chief tool is the concept of a partition pair, wh... View full abstract»

This paper describes a method for synthesizing a switching function directly from its: ruth table. A switching function is defined as any mapping of a set of binary input combinations onto 0 and 1. Hence, the procedures apply equally well to the don't care cases. The method rests on the concept alogically passive function (LPF). Roughly speaking, an LPF is a truth table which can be realized with ... View full abstract»

Logical cryogenic circuits differ from the circuitry used with other devices as paths for both the required function and its denial must be provided. Present techniques for logical design of cryogenic circuits either rely on the experience of the designer to achieve a minimal circuit or are derived from analogous relay circuits. An algorithm to develop minimal (or near minimal) circuits by collaps... View full abstract»

Several higher-order difference methods are investigated and compared for the problem of finding the natural frequencies of the lateral vibration of a beam. All of the methods considered are applicable to either digital or analog computers, although particular reference is made to the analog computer. The methods considered in most detail use the same basic central difference approximation, the va... View full abstract»

A method is described for the analog computation of the solution of linear two-point boundary value problems, i.e., the problem of a linear ordinary differential equation with linear two-point boundary conditions. The nonhomogeneous problem, in particular, is considered; the differential equal for y on the real interval 0≦x≦1 is taken to be nonhomogeneous. The method for its solution is based upon... View full abstract»

A data-processing technique is described in which a function is evaluated by rapid interrogation of the given data for the presence of combinations of data variables giving rise to values of the function which are of interest. Testing unlabeled measurements from several observers to find data sets which satisfy a test function is discussed as an example. For many problems the required time is much... View full abstract»

This paper is primarily concerned with finding, in the most efficient possible way, the set of all solutions to a cyclic prime implicant table. (A solution is a set of rows such that every column contains at least one marked entry in a row belonging to the set and such that no row can be deleted from the set without destroying this property.) Extensive use is made of the relationship between this ... View full abstract»

In this paper a circuit is developed which provides a train of uniform pulses logarithmically spaced in time. As applied to computing the system offers new methods for calculating products, quotients, powers, and roots. A pulse-interval error of less than one per cent (rms) was obtained with an experimental circuit. View full abstract»

A data-handling unit is described which permits binary or decimal arithmetic to be performed on data fields of any length from one to sixty-four bits. Within the field, character structure can be further specified: these processing entities, called bytes, may be from one to eight bits long. Fields may be stored with or without algebraic sign. On all operations, the relative offset or shift between... View full abstract»

The improvement of reliability and availability through redundancy of entire machines rather than of components is investigated. An attempt is made to break down the cost of operating a digital computer, and to determine the relationship between cost and system failure. Three specific cases are discussed. Case 1: Where n machines are operated independently, processing the same input data. The outp... View full abstract»

One of the major limitations of the electronic analog computer is its inability to perform directly an integration with respect to a dependent variable. This paper reviews the usual methods of overcoming this limitation, describes the results on an attempt to use Padé time-delay units in generalized integration, and presents the development of a new analog integrator based on a simple numerical in... View full abstract»

An important field of application for computers is in real-time systems simulation. This requires the generation of non-linear functions, obtaining the sums and products of these functions and solving systems of nonlinear differential equations. A new type of analog computer suitable for systems simulation is described which combines the desirable features of the digital and analog computers in it... View full abstract»

The most important property of the codes derived in this paper is that two numbers, i and j, have coded forms, C(i) and C(j) that when added in a conventional binary adder, give a sum C(i)+C(j) that differs from C(i+j), the code for the sum, by (at most) an additive constant. This makes possible the detection and/or correction of errors committed by the arithmetic element of a computer. In additio... View full abstract»

The recent discovery of the tunnel diode with band-widths extending into the kilomegacycle region has prompted investigation of their use in the logic and control portions of high-speed computers. Considerations of diode uniformity requirements, stability problems and power supply requirements has led to a monostable type of logical circuit. The switching properties of this circuit are analyzed an... View full abstract»

According to the model discussed in this paper, a pattern recognizer is said to consist of two parts: a receptor, which generates a set of measurements of the physical sample to be recognized, and a categorizer, which assigns each set of measurements to one of a finite number of categories. The rule of operation of the categorizer is called the ``recognition function.'' The optimization of the rec... View full abstract»

The representation of a wide variety of nonlinear functions by means of the interconnection of unstabilized operational amplifiers is discussed. The nonlinear functions described include rectification, saturation functions, coulomb friction, dead space, and starting friction. The use of operational amplifiers alone to produce square waves and triangular waves, as well as gating operations, is also... View full abstract»

Due to the physical unrealizability of electronic adding and integrating circuits with ideal characteristics, errors will be introduced in the solution of differential equations obtained by the use of electrical analog computers. Numerical errors in the solution will be introduced by fluctuations in the value of plate and of grid supply voltages, changes in the values of circuit components, and ch... View full abstract»

The counters without short-time internal memory, conceived by Bigelow, disclosed by Ware, and extended by Brown, are discussed from the standpoint of their respective logic. It is shown that the (self-instructed) bidirectional1 counter of Brown has a more rigorous logic than the unidirectional counter of Ware; the operation of Brown's bidirectional counter being subjected to the only restriction t... View full abstract»

A transistor-driven magnetic-core memory which has a capacity of 1024 18-bit words has been built and is being studied. Both the read and write operations employ the coincident-current technique. The memory-drive currents are developed by transistors and the desired memory location is selected by magnetic-core selection switches. Eighteen thousand, four hundred and thirty-two memory cores are used... View full abstract»