Concrete mathematics, a foundation for computer science, by Ronald L. Graham, Donald E. Knuth and Oren Patashnik. Pp 625. £24·95. 1989. ISBN 0-201-14236-8 (Addison-Wesley)

The main characteristic of Reconfigurable Computing is the presence of hardware that can be reconfigured to implement specific functionality more suitable for specially tailored hardware than on a simple uniprocessor. Reconfigurable computing systems join microprocessors and programmable hardware in order to take advantage of the combined strengths of hardware and software and have been used in applications ranging from embedded systems to high performance computing. Many of the fundamental theories have been identified and used by the Hardware/Software Co-Design research field. Although the same background ideas are shared in both areas, they have different goals and use different approaches.This book is intended as an introduction to the entire range of issues important to reconfigurable computing, using FPGAs as the context, or "computing vehicles" to implement this powerful technology. It will take a reader with a background in the basics of digital design and software programming and provide them with the knowledge needed to be an effective designer or researcher in this rapidly evolving field.

· Treatment of FPGAs as computing vehicles rather than glue-logic or ASIC substitutes
· Views of FPGA programming beyond Verilog/VHDL
· Broad set of case studies demonstrating how to use FPGAs in novel and efficient ways

Reconfigurable Computing: The Theory and Practice of FPGA-Based Computation

Year 2009 marks the completion of 50 years of the invention of CORDIC (coordinate rotation digital computer) by Jack E. Volder. The beauty of CORDIC lies in the fact that by simple shift-add operations, it can perform several computing tasks such as the calculation of trigonometric, hyperbolic and logarithmic functions, real and complex multiplications, division, square-root, solution of linear systems, eigenvalue estimation, singular value decomposition, QR factorization and many others. As a consequence, CORDIC has been utilized for applications in diverse areas such as signal and image processing, communication systems, robotics and 3-D graphics apart from general scientific and technical computation. In this article, we present a brief overview of the key developments in the CORDIC algorithms and architectures along with their potential and upcoming applications.

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50 Years of CORDIC: Algorithms, Architectures, and Applications

Figures of merit connecting processing capabilities with power dissipated (OpS/Watt, Joule/bit, etc.) are becoming dominant factors in choosing technologies for implementing the next generation of computing and communication network systems. Superconductivity is viewed as a technology capable of achieving higher energy efficiencies than other technologies. Static power dissipation of standard RSFQ logic, associated with dc bias resistors, is responsible for most of the circuit power dissipation. In this paper, we review and compare different superconductor digital technology approaches and logic families addressing this problem. We present a novel energy-efficient single flux quantum logic family, ERSFQ/eSFQ. We also discuss energy-efficient approaches for output data interface and overall cryosystem design.

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Energy-Efficient Single Flux Quantum Technology

Large-scale computing system characteristics vary by application class, but power and energy use has become a major problem for all classes. Superconducting computing may be able to serve the needs of these systems significantly better than conventional technology. Recent developments in single flux quantum circuit technology for digital logic include variants with greatly improved energy efficiency. Concepts were investigated for computing systems capable of performance in the range from 1 to 1000 PFLOP/s. The concept systems were constrained to use existing commercial cryogenic refrigerators and Nb superconducting technology. In order to meet the performance goals, cache and main memory capable of operating at cryogenic temperatures will be required. Superconducting computing is shown to be potentially competitive on the basis of power and energy efficiency if key component technologies can meet specific goals. Potential advantages of superconducting computing are identified as well as areas requiring further development.

Energy-Efficient Superconducting Computing—Power Budgets and Requirements

Proposes two redundant CORDIC (coordinate rotation digital computer) methods with a constant scale factor for sine and cosine computation, called the double rotation method and the correcting rotation method. In both methods, the CORDIC is accelerated by the use of a redundant binary number representation, as in the previously proposed redundant CORDIC. In the proposed methods, since the number of rotation-extensions performed for each angle is a constant, the scale factor is a constant independent of the operand. Hence, one does not need to calculate the scale factor during the computation, and can make a more efficient sine and cosine generator than that based on the previous redundant CORDIC. >

Redundant CORDIC methods with a constant scale factor for sine and cosine computation

A high speed multiplier and divider for MOS LSI based on a new algorithm is presented. When we implement the multiplier and the divider in LSI, the features such as high speed operation, small number of transistors and easy layout are the most important factors. A computational algorithm using a redundant binary representation has several excellent features such as high speed addition operations. We improved the algorithm and the method of implementation, and designed an advanced multiplier and divider with the above mentioned features. We expect mat our multiplier and divider are excellent compared with multipliers using the Booth algorithm and the Wallace tree, and with divider using the SRT method, respectively.

Design of high speed MOS multiplier and divider using redundant binary representation

We describe the recent progress on a Nb nine-layer fabrication process for large-scale single flux quantum (SFQ) circuits. A device fabricated in this process is composed of an active layer including Josephson junctions (JJ) at the top, passive transmission line (PTL) layers in the middle, and a DC power layer at the bottom. We describe the process conditions and the fabrication equipment. We use both diagnostic chips and shift register (SR) chips to improve the fabrication process. The diagnostic chip was designed to evaluate the characteristics of basic elements such as junctions, contacts, resisters, and wiring, in addition to their defect evaluations. The SR chip was designed to evaluate defects depending on the size of the SFQ circuits. The results of a long-term evaluation of the diagnostic and SR chips showed that there was fairly good correlation between the defects of the diagnostic chips and yields of the SRs. We could obtain a yield of 100% for SRs including 70,000 JJs. These results show that considerable progress has been made in reducing the number of defects and improving reliability. key words: superconducting fabrication technology, Nb/AlOx/Nb Josephson junction, single flux quantum, planarization, shift register

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Nb 9-Layer Fabrication Process for Superconducting Large-Scale SFQ Circuits and Its Process Evaluation

A 16-bit /spl times/ 16-bit multiplier for 2 two's-complement binary numbers based on a new algorithm is described. This multiplier has been fabricated on an LSI chip using a standard n-E/D MOS process technology with a 2.7-/spl mu/m design rule. This multiplier is characterized by use of a binary tree of redundant binary adders. In the new algorithm, n-bit multiplication is performed in a time proportional to log/SUB 2/ n and the physical design of the multiplier is constructed of a regular cellular array. This new algorithm has been proposed by N. Takagi et al. (1982, 1983). The 16-bit/spl times/16-bit multiplier chip size is 5.8 /spl times/ 6.3 mm/SUP 2/ using the new layout for a binary adder tree. The chip contains about 10600 transistors, and the longest logic path includes 46 gates. The multiplication time was measured as 120 ns. It is estimated that a 32-bit /spl times/ 32-bit multiplication time is about 140 ns.

A high-speed multiplier using a redundant binary adder tree

We describe a general approach decomposing a function into a sum of functions, each with a smaller input site than the original. Hence we can map such functions with essentially the same precision using small ROM tables and adders. We derive an easy method to compute the worst case error for many elementary functions and an error bound for the rest. Important applications are reciprocals, logarithms, exponentials and others. >

Naofumi Takagi

Papers

Redundant CORDIC methods with a constant scale factor for sine and cosine computation

Design of high speed MOS multiplier and divider using redundant binary representation

Nb 9-Layer Fabrication Process for Superconducting Large-Scale SFQ Circuits and Its Process Evaluation

A high-speed multiplier using a redundant binary adder tree

Function evaluation by table look-up and addition