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»

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»

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»

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»

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»

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»

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»

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»

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»

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»

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»

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»

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»

In this paper, networks of finite-state machines, rather than individual machines, are discussed. The investigation centers around cascade networks, where the output of one machine serves as an input to another. It is shown how, by means of connection matrices, the characteristics of such a network can be obtained from those of the component machines, and how a specified machine can be decomposed ... 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»

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»

A special-purpose, binary computer called CORDIC (COordinate Rotation DIgital Computer) contains a unique arithmetic unit composed of three shift registers, three adder-subtractors, and suitable interconnections for efficiently performing calculations involving trigonometric functions. A technique is formulated for using the CORDIC arithmetic unit to convert between angles expressed in degrees and... View full abstract»

The vertex frame is similar to an n-dimensional cube or Tychonoff frame that has been cut and unfolded into the shape of a Karnaugh map. The methods of use are similar to those for Karnaugh maps. Minimal disjunctive and conjunctive normal formulas are found for problems with or without don't-care cases. The selection graph, a linear graph, is used to enhance the prime antecedent (= prime implicant... View full abstract»

The problem of economical synthesis of circuits for digital computers leads to the problem of finding Boolean expressions of minimal length equivalent to a Boolean expression f. Previous authors restricted themselves to ``sum of products'' expressions; dualizing this gives ``products of sums.'' The next more efficient step is to find minimal ``sums of products of sums'' expressions. In this paper,... View full abstract»

This paper is an exposition of the theory of regular expressions and its applications to sequential circuits. The results of several authors are presented in a unified manner, pointing out the similarities and differences in the various treatments of the subject. Whenever possible, the terminology and notation of sequential circuit theory are used. The topics presented include: the relation of reg... View full abstract»

The reflected binary or Gray code has been used chiefly in analog-to-digital conversion devices because its code sequences, representing any two consecutive integral numbers, differ in only one digit. This paper presents a method for performing the arithmetic operations of addition, subtraction, multiplication, and division using a modified reflected binary code. The modification for integral numb... View full abstract»

This paper begins with Mealy's model of a sequential machine and introduces a ``connection matrix'' which describes the machine completely. The ``equivalence'' of states of such a machine may be analyzed systematically by an iterative technique, the validity of which is rigorously established. Once equivalence is completely analyzed, it is a simple matter to write the connection matrix for the sim... View full abstract»