The Birth of Cordic

The very earliest history of the CORDIC computing technique—a highly efficient method to compute elementary functions—is presented. The CORDIC technique was born out of necessity, the incentive being the replacement of the analog navigation computer of the B-58 aircraft by a high accuracy, high-performance digital computer. The revolutionary development of the CORDIC technique is presented, along with details of the very first implementations: the CORDIC I prototype and the CORDIC II airborne digital navigation computer. In early 1956 the aeroelectronics department of Convair, Fort Worth, was given the task of determining the feasibility of replacing the analog computer-driven navigation system of the B-58 bomber (see Fig. 1) with a digital computer. This replacement effort was deemed necessary because of the limited accuracy of analog computing elements. At that time, digitalization of the B-58 navigation system was considered a formidable task without any assurance of worthwhile results. Transistors were new and limited to a 250 KHz logic rate. The main challenge was the real-time calculation of the complicated navigation equations required for determining present position on a spherical earth. By then, digital differential analyzers had been developed that were capable of efficiently solving continuous navigation problems, but they could not produce solutions in real-time during flights near the North Pole. Also, they were too slow in providing solutions for the discontinuous problems of fix-taking from either startracking or radar ground sightings. Therefore, an entireword transfer type of computation was definitely necessary. However, the existing trigonometric algorithms necessary for navigation were too time-consuming for the real-time requirements of the B-58. Most navigation system specialists agreed that the existing B-58 navigation computer was an ingenuous device utilizing analog resolvers to compute, in real time, the complex trigonometric relationships necessary for navigation over a spherical earth. Each resolver was capable of performing either a rotation of input coordinates or inversely determining the magnitude and angle of the vector defined by the input coordinates—also called vectoring—as shown in the diagram of Fig. 2. Trying to solve navigation problems without resolver capabilities is an extremely complicated problem. Resolver capability allows solution flow diagrams to be drawn with interconnected resolvers as was shown in the original CORDIC paper [1]. For example, in great circle navigation, the solution of course angle and distance to destination requires a network of only 5 interconnected resolvers. 1. Digitalization of the Analog Resolver At that time there were no known digital operators equivalent to the analog resolver. Digital computation of navigation problems required either series expansions, approximations or table look-up of individual trigonometric functions. At Convair, the first brain-storming effort toward digitalization was directed toward either encoding the sine and cosine functions or some intermediate function on optical encoders. This effort was soon abandoned. Afterwards, I began looking for some clue to solving this problem in the trigonometric equations tabulated in my 1946 edition of the Handbook of Chemistry and Physics. This led to the massaging of the basic angle addition equations to obtain the following interesting equations. If tan(φ) = 2−n , then: Kn R sin(θ ± φ) = R sin(θ)± 2−n R cos(θ) (1) Kn R cos(θ ± φ) = R cos(θ)∓ 2−n R sin(θ) , (2) where Kn = √ 1+ 2−2n .