This book presents a new math approach for logic designs, using fixed-point decimal and binary calculations. Problems resulting from rounding off and 2’s complement operations are avoided. The techniques introduced in the book have resulted in more efficient designs with lower gate count and lower cost.
In computers, calculations are executed with fixed-point and floating-point binary numbers. The floating-point numbers have an advantage over fixed-point numbers. They are implemented with signed magnitude binary numbers and have an almost infinity range of positive and negative values. They are symmetrical with respect to integer zero. On the other hand, beside their limited range, fixed-point binary numbers are well suited for logic circuit implementation. But the binary fixed-point numbers and integers are not symmetrical with respect to integer zero. The need to compute 2’s complement values results into extra work and gate count. The solution is to apply averaging and symmetry to fixed-point numbers. In 3D graphics, averaging makes for better solutions by removing many problems and corner cases.
For example, the non-restoring division produces different results depending on the four sign combinations of dividend and divisor. The symmetry of averaged numbers provides for simple solutions. Other applications include implementation of adders, Booth Wallace multiplier and function tables, such as sine, cosine, logarithmic and exponential functions.
As a bonus, the author describes his experience as a designer and architect of the Link DIG, the fastest real-time 3D computer that was producing 60 anti-aliased images/sec. The DIG was used for pilot training in the Space Shuttle, F-111, B-52 and AH-64 Apache mission simulators.
For more info about DIG image, click on “+ More About DIG” link below image.
This image was produced in 1978 by a 3D Digital Image Generator (DIG) from Link APO (Advanced Product Operation of Link Flight Simulation) in Sunnyvale CA. This image of a KC-10 Tanker refueling an F-111 Fighter, consists of up to 3400 small triangles (or 10000 triangle edges), computed one at a time by the DIG and then put together in the displayed image. The DIG could produce 60 such high quality images per second (field update rate). Also, these images had edge smoothing, that is Anti-Aliasing with “No Jaggies”. The smooth surfaces on the planes surface are simulated with intensity gradients inside of the triangles (Gouraud Shading). The DIG was 5 to 10 time faster than competitive systems at that time.
For the NASA Space program, the DIG-1 from Link APO in Sunnyvale CA has provided the Space Shuttle astronauts with SMS training capabilities that came 5 years early.
When Link delivered the first DIG to NASA in 1977, the DIG was the fastest Computer Image Generator in the world. It was 5 years ahead of the closest competition, the CT-5 (Continuous-Tone 5) from E&S in Utah, delivered in 1982. No other system was available to prepare the Space Shuttle astronauts just in time for their first scheduled flights with Columbia on April 12, 1981. Michel Rohner was a significant contributor to the Link DIG Program in Sunnyvale. He re-designed the Frame Calculator (DIG Front -End) to achieve a 5 to 10x speed improvement from the prototype. By providing the fastest real-time image generator at that time, Michel probably made a significant contribution to the SMS in the space program.
Four DIG systems were delivered for the Space Shuttle Mission Simulator (SMS), in Houston, TX. The SMS has been used by all the Space Shuttle astronauts for realistic training before each mission in space. Before each mission the astronauts spent countless hours training in the SMS, simulating all the possible scenarios and critical situations. The images produced by the DIG provided the best training available at that time.
Following the delivery of the 1st DIG to NASA, from 1977 to late 1980’s, Link APO got orders for up to 70 DIG systems from the Air Force and the Army, at a cost of around $2M/DIG. Link delivered DIGs for the following programs: 4 for NASA SMS, 1 for NASA Ames (vertical simulator), 10 for F-111 Fighter, 12 for B-52 Bomber, 1 for the F-117 Stealth Fighter, 40 for Helicopter Simulators (including AH-64 Apache, AH-1 Cobra, UH-60 Black Hawk and CH-47 Chinook). Besides the DIGs, the Link Division in Binghamton sold training platform with most of the DIG systems. These DIG systems included: aircraft cockpit, several display windows, and motion simulation platform with 6 degrees of freedom.
Among the DIG’s key contributors at Link APO: Judit K. (Judy) Florence, Robert (Bob) Lotz, Michel Rohner, Johnson Yan, Walt Rossman, Dave Marsh, Ray Osofsky, John Sullivan, Dan Weaver, Dennis Yeo, Nancy Gormezano, Doug Allen, Mike Picardo, Dave Hinkle, Gary Daniel, Andrew Rosman, John Haire, James Chang, Gene Munce, Mike Rosemire, Tom Fries, Bob Gonsalves, Rob Mapes, Thom Belevsky, Tom O’Connel and Bob Withcomb.