Multi-operand Decimal Adder Trees for FPGAs

The research and development of hardware designs for decimal arithmetic is currently going under an intense activity. For most part, the methods proposed to implement fixed and floating point operations are intended for ASIC designs. Thus, a direct mapping or adaptation of these techniques into a FPGA could be far from an optimal solution. Only a few studies have considered new methods more suitable for FPGA implementations. A basic operation that has not received enough attention in this context is multi-operand BCD addition. For example, it is of interest for low latency implementations of decimal fixed and floating point multipliers and decimal fused multiply-add units. We have explored the most representative proposals for multi-operand BCD addition and found that the resultant implementations in FPGAs are still very inefficient in terms of both area and latency when compared to their binary counterparts. In this paper we present a new method for fast and efficient implementation of multi-operand BCD addition in current FPGA devices. In particular, our proposal maps quite well into the slice structure of the Xilinx Virtex-5/Virtex-6 families and it is highly pipelineable. The synthesis results for a Virtex-6 device indicate that our implementations halve the area and latency of previous proposals, presenting area and delay figures close to those of optimal binary adder trees.La recherche sur l'implantation en matériel de l'arithmétique décimale est actuellement très active, la plupart des travaux portant sur des opérateurs pour les processeurs, en virgule fixe ou flottante. Mais les techniques développées pour un circuit intégré n'aboutissent pas forcément à une implémentation optimale dans un FPGA. Il n'y a que peu d'études ciblant explicitement les FPGA. Cet article s'intéresse dans ce contexte, à l'addition BCD multi-opérande, au cœur de multiplieurs et de multiplieurs-accumulateurs à faible latence. Nous étudions les architectures proposées pour cette opération décimale, et nous observons que, sur FPGA, leur performance (surface et latence) est très inférieure à celle des opérations binaire à précision comparable. Nous présentons donc dans cet article une nouvelle technique d'addition BCD multi-opérandes qui s'avère plus efficace que les propositions précédentes sur les FPGA actuels. Elle s'adapte particulièrement bien à la structure fine des FPGA Xilinx Virtex-5/Virtex-6, et se prête bien au pipeline. Les résultats de synthèse montrent que notre implémentation divise par deux la surface et la latence par rapport aux propositions précédentes, les ramenant à des valeurs comparables à celles des meilleurs additionneurs multi-opérandes binaires

High-speed radix-10 multiplication using partial shifter adder tree-based convertor

A radix-10 multiplication is the foremost frequent operations employed by several monetary business and user-oriented applications, decimal multiplier using in state of art digital systems are significantly good but can be upgraded with time delay and area optimization. This work is proposed a more area and time delay optimized new design of overloaded decimal digit set (ODDS) architecture-based radix-10 multiplier for signed numbers. Binary coded decimal (BCD) to binary followed by binary multiplication and finally binary to BCD conversion are 3 major modules employed in radix-10 multiplication. This paperwork presents a replacement technique for binary coded decimal (BCD) to binary and vice-versa convertors in radix-10 multiplication. A novel addition tree structure called as partial shifter adder (PSA) tree-based approach has been developed for BCD to binary conversion, and it is used to add partially generated products. To meet our major concern i.e. speed, we need particular high-speed multiplication, hence the proposed PSA based radix-10 multiplier is using vertical cross binary multiplication and concurrent shifter-based addition method. The design has been tested on 45nm technology-based Zynq-7 field programmable gate array (FPGA) devices with a 6-input lookup table (LUTs). A combinational implementation maps quite well into the slice structure of the Xilinx Zynq-7 families field programmable gate array. The synthesis results for a Zynq-7 device indicate that our design outperforms in terms of the area and time delay

Decimal Floating-point Fused Multiply Add with Redundant Number Systems

The IEEE standard of decimal floating-point arithmetic was officially released in 2008. The new decimal floating-point (DFP) format and arithmetic can be applied to remedy the conversion error caused by representing decimal floating-point numbers in binary floating-point format and to improve the computing performance of the decimal processing in commercial and financial applications. Nowadays, many architectures and algorithms of individual arithmetic functions for decimal floating-point numbers are proposed and investigated (e.g., addition, multiplication, division, and square root). However, because of the less efficiency of representing decimal number in binary devices, the area consumption and performance of the DFP arithmetic units are not comparable with the binary counterparts. IBM proposed a binary fused multiply-add (FMA) function in the POWER series of processors in order to improve the performance of floating-point computations and to reduce the complexity of hardware design in reduced instruction set computing (RISC) systems. Such an instruction also has been approved to be suitable for efficiently implementing not only stand-alone addition and multiplication, but also division, square root, and other transcendental functions. Additionally, unconventional number systems including digit sets and encodings have displayed advantages on performance and area efficiency in many applications of computer arithmetic. In this research, by analyzing the typical binary floating-point FMA designs and the design strategy of unconventional number systems, ``a high performance decimal floating-point fused multiply-add (DFMA) with redundant internal encodings" was proposed. First, the fixed-point components inside the DFMA (i.e., addition and multiplication) were studied and investigated as the basis of the FMA architecture. The specific number systems were also applied to improve the basic decimal fixed-point arithmetic. The superiority of redundant number systems in stand-alone decimal fixed-point addition and multiplication has been proved by the synthesis results. Afterwards, a new DFMA architecture which exploits the specific redundant internal operands was proposed. Overall, the specific number system improved, not only the efficiency of the fixed-point addition and multiplication inside the FMA, but also the architecture and algorithms to build up the FMA itself. The functional division, square root, reciprocal, reciprocal square root, and many other functions, which exploit the Newton's or other similar methods, can benefit from the proposed DFMA architecture. With few necessary on-chip memory devices (e.g., Look-up tables) or even only software routines, these functions can be implemented on the basis of the hardwired FMA function. Therefore, the proposed DFMA can be implemented on chip solely as a key component to reduce the hardware cost. Additionally, our research on the decimal arithmetic with unconventional number systems expands the way of performing other high-performance decimal arithmetic (e.g., stand-alone division and square root) upon the basic binary devices (i.e., AND gate, OR gate, and binary full adder). The proposed techniques are also expected to be helpful to other non-binary based applications

Fast decimal floating-point division

A new implementation for decimal floating-point (DFP) division is introduced. The algorithm is based on high-radix SRT division The SRT division algorithm is named after D. Sweeney, J. E. Robertson, and T. D. Tocher. with the recurrence in a new decimal signed-digit format. Quotient digits are selected using comparison multiples, where the magnitude of the quotient digit is calculated by comparing the truncated partial remainder with limited precision multiples of the divisor. The sign is determined concurrently by investigating the polarity of the truncated partial remainder. A timing evaluation using a logic synthesis shows a significant decrease in the division execution time in contrast with one of the fastest DFP dividers reported in the open literatureHooman Nikmehr, Braden Phillips and Cheng-Chew Li

HIGH-SPEED CO-PROCESSORS BASED ON REDUNDANT NUMBER SYSTEMS

There is a growing demand for high-speed arithmetic co-processors for use in applications with computationally intensive tasks. For instance, Fast Fourier Transform (FFT) co-processors are used in real-time multimedia services and financial applications use decimal co-processors to perform large amounts of decimal computations. Using redundant number systems to eliminate word-wide carry propagation within interim operations is a well-known technique to increase the speed of arithmetic hardware units. Redundant number systems are mostly useful in applications where many consecutive arithmetic operations are performed prior to the final result, making it advantageous for arithmetic co-processors. This thesis discusses the implementation of two popular arithmetic co-processors based on redundant number systems: namely, the binary FFT co-processor and the decimal arithmetic co-processor. FFT co-processors consist of several consecutive multipliers and adders over complex numbers. FFT architectures are implemented based on fixed-point and floating-point arithmetic. The main advantage of floating-point over fixed-point arithmetic is the wide dynamic range it introduces. Moreover, it avoids numerical issues such as scaling and overflow/underflow concerns at the expense of higher cost. Furthermore, floating-point implementation allows for an FFT co-processor to collaborate with general purpose processors. This offloads computationally intensive tasks from the primary processor. The first part of this thesis, which is devoted to FFT co-processors, proposes a new FFT architecture that uses a new Binary-Signed Digit (BSD) carry-limited adder, a new floating-point BSD multiplier and a new floating-point BSD three-operand adder. Finally, a new unit labeled as Fused-Dot-Product-Add (FDPA) is designed to compute AB+CD+E over floating-point BSD operands. The second part of the thesis discusses decimal arithmetic operations implemented in hardware using redundant number systems. These operations are popularly used in decimal floating-point co-processors. A new signed-digit decimal adder is proposed along with a sequential decimal multiplier that uses redundant number systems to increase the operational frequency of the multiplier. New redundant decimal division and square-root units are also proposed. The architectures proposed in this thesis were all implemented using Hardware-Description-Language (Verilog) and synthesized using Synopsys Design Compiler. The evaluation results prove the speed improvement of the new arithmetic units over previous pertinent works. Consequently, the FFT and decimal co-processors designed in this thesis work with at least 10% higher speed than that of previous works. These architectures are meant to fulfill the demand for the high-speed co-processors required in various applications such as multimedia services and financial computations

Introduction to Logic Circuits & Logic Design with VHDL

The overall goal of this book is to fill a void that has appeared in the instruction of digital circuits over the past decade due to the rapid abstraction of system design. Up until the mid-1980s, digital circuits were designed using classical techniques. Classical techniques relied heavily on manual design practices for the synthesis, minimization, and interfacing of digital systems. Corresponding to this design style, academic textbooks were developed that taught classical digital design techniques. Around 1990, large-scale digital systems began being designed using hardware description languages (HDL) and automated synthesis tools. Broad-scale adoption of this modern design approach spread through the industry during this decade. Around 2000, hardware description languages and the modern digital design approach began to be taught in universities, mainly at the senior and graduate level. There were a variety of reasons that the modern digital design approach did not penetrate the lower levels of academia during this time. First, the design and simulation tools were difficult to use and overwhelmed freshman and sophomore students. Second, the ability to implement the designs in a laboratory setting was infeasible. The modern design tools at the time were targeted at custom integrated circuits, which are cost- and time-prohibitive to implement in a university setting. Between 2000 and 2005, rapid advances in programmable logic and design tools allowed the modern digital design approach to be implemented in a university setting, even in lower-level courses. This allowed students to learn the modern design approach based on HDLs and prototype their designs in real hardware, mainly field programmable gate arrays (FPGAs). This spurred an abundance of textbooks to be authored teaching hardware description languages and higher levels of design abstraction. This trend has continued until today. While abstraction is a critical tool for engineering design, the rapid movement toward teaching only the modern digital design techniques has left a void for freshman- and sophomore-level courses in digital circuitry. Legacy textbooks that teach the classical design approach are outdated and do not contain sufficient coverage of HDLs to prepare the students for follow-on classes. Newer textbooks that teach the modern digital design approach move immediately into high-level behavioral modeling with minimal or no coverage of the underlying hardware used to implement the systems. As a result, students are not being provided the resources to understand the fundamental hardware theory that lies beneath the modern abstraction such as interfacing, gate-level implementation, and technology optimization. Students moving too rapidly into high levels of abstraction have little understanding of what is going on when they click the “compile and synthesize” button of their design tool. This leads to graduates who can model a breadth of different systems in an HDL but have no depth into how the system is implemented in hardware. This becomes problematic when an issue arises in a real design and there is no foundational knowledge for the students to fall back on in order to debug the problem

RADIX-10 PARALLEL DECIMAL MULTIPLIER

This paper introduces novel architecture for Radix-10 decimal multiplier. The new generation of highperformance decimal floating-point units (DFUs) is demanding efficient implementations of parallel decimal multiplier. The parallel generation of partial products is performed using signed-digit radix-10 recoding of the multiplier and a simplified set of multiplicand multiples. The reduction of partial products is implemented in a tree structure based on a new algorithm decimal multioperand carry-save addition that uses a unconventional decimal-coded number systems. We further detail these techniques and it significantly improves the area and latency of the previous design, which include: optimized digit recoders, decimal carry-save adders (CSA’s) combining different decimal-coded operands, and carry free adders implemented by special designed bit counters

Introduction to Logic Circuits & Logic Design with Verilog

The overall goal of this book is to fill a void that has appeared in the instruction of digital circuits over the past decade due to the rapid abstraction of system design. Up until the mid-1980s, digital circuits were designed using classical techniques. Classical techniques relied heavily on manual design practices for the synthesis, minimization, and interfacing of digital systems. Corresponding to this design style, academic textbooks were developed that taught classical digital design techniques. Around 1990, large-scale digital systems began being designed using hardware description languages (HDL) and automated synthesis tools. Broad-scale adoption of this modern design approach spread through the industry during this decade. Around 2000, hardware description languages and the modern digital design approach began to be taught in universities, mainly at the senior and graduate level. There were a variety of reasons that the modern digital design approach did not penetrate the lower levels of academia during this time. First, the design and simulation tools were difficult to use and overwhelmed freshman and sophomore students. Second, the ability to implement the designs in a laboratory setting was infeasible. The modern design tools at the time were targeted at custom integrated circuits, which are cost- and time-prohibitive to implement in a university setting. Between 2000 and 2005, rapid advances in programmable logic and design tools allowed the modern digital design approach to be implemented in a university setting, even in lower-level courses. This allowed students to learn the modern design approach based on HDLs and prototype their designs in real hardware, mainly fieldprogrammable gate arrays (FPGAs). This spurred an abundance of textbooks to be authored, teaching hardware description languages and higher levels of design abstraction. This trend has continued until today. While abstraction is a critical tool for engineering design, the rapid movement toward teaching only the modern digital design techniques has left a void for freshman- and sophomore-level courses in digital circuitry. Legacy textbooks that teach the classical design approach are outdated and do not contain sufficient coverage of HDLs to prepare the students for follow-on classes. Newer textbooks that teach the modern digital design approach move immediately into high-level behavioral modeling with minimal or no coverage of the underlying hardware used to implement the systems. As a result, students are not being provided the resources to understand the fundamental hardware theory that lies beneath the modern abstraction such as interfacing, gate-level implementation, and technology optimization. Students moving too rapidly into high levels of abstraction have little understanding of what is going on when they click the “compile and synthesize” button of their design tool. This leads to graduates who can model a breadth of different systems in an HDL but have no depth into how the system is implemented in hardware. This becomes problematic when an issue arises in a real design and there is no foundational knowledge for the students to fall back on in order to debug the problem

IEEE Compliant Double-Precision FPU and 64-bit ALU with Variable Latency Integer Divider

Together the arithmetic logic unit (ALU) and floating-point unit (FPU) perform all of the mathematical and logic operations of computer processors. Because they are used so prominently, they fall in the critical path of the central processing unit - often becoming the bottleneck, or limiting factor for performance. As such, the design of a high-speed ALU and FPU is vital to creating a processor capable of performing up to the demanding standards of today\u27s computer users. In this paper, both a 64-bit ALU and a 64-bit FPU are designed based on the reduced instruction set computer architecture. The ALU performs the four basic mathematical operations - addition, subtraction, multiplication and division - in both unsigned and two\u27s complement format, basic logic operations and shifting. The division algorithm is a novel approach, using a comparison multiples based SRT divider to create a variable latency integer divider. The floating-point unit performs the double-precision floating-point operations add, subtract, multiply and divide, in accordance with the IEEE 754 standard for number representation and rounding. The ALU and FPU were implemented in VHDL, simulated in ModelSim, and constrained and synthesized using Synopsys Design Compiler (2006.06). They were synthesized using TSMC 0.1 3nm CMOS technology. The timing, power and area synthesis results were recorded, and, where applicable, compared to those of the corresponding DesignWare components.The ALU synthesis reported an area of 122,215 gates, a power of 384 mW, and a delay of 2.89 ns - a frequency of 346 MHz. The FPU synthesis reported an area 84,440 gates, a delay of 2.82 ns and an operating frequency of 355 MHz. It has a maximum dynamic power of 153.9 mW