Hardware Implementation of Densely Packed Decimal Encoding

Binary Coded Decimal ( BCD ) in which four bits are used for each decimal digit is a widely used encoding for decimal data .Decimal arithmetic and shifting are simplified by using operands in this form, and both rounding to a specified number of digits and conversions to or from characters are trivial. For the storage and simple manipulation of decimal data, BCD remains an appropriate encoding to use. In some situations, however, a more compact representation offers significant advantages. Decimal floating-point numbers in a compact form can be used to implement the requirements of the IEEE 854 standard and meet the increasing demands for decimal arithmetic in applications. An efficient encoding scheme for decimal data is described by Chen and Ho.Chen Ho encoding is a lossless compression of three decimal digits coded in BCD into 10 bits using an algorithm which can be applied or reversed using only simple Boolean operations. Densely Packed Decimal (DPD) is an refinement of the Chen ho encoding. It gives the same compression and speed advantages but is not limited to multiples of three digits. The DPD encoding allows arbitrary-length decimal numbers to be coded efficiently while keeping decimal digit boundaries accessible. This results in efficient decimal arithmetic and makes the efficient and optimized use of available resources such as storage or hardware registers. This thesis embodies the work done to implement the Densely Packed Decimal (DPD) encoding on hardware using digilent board containing VIRTEX-II Pro FPGA

BCD Multiplier

BCD multipliers are the basis of accurate decimal multiplication used in banking systems, scientific calculations, etc. Fractions convert poorly into binary numbers giving rise to conversion error. Therefore, banking industry have been using Binary Coded Decimal numbering system for their banking business transaction to circumvent the error between decimal fraction number to binary. Here we will explore some single-digit Binary Coded Decimal Multiplication units that perform multiplication in hardware for the purpose of future implementation

Design and Implementation of an RNS-based 2D DWT Processor

No abstract availabl

Automated Dynamic Error Analysis Methods for Optimization of Computer Arithmetic Systems

Computer arithmetic is one of the more important topics within computer science and engineering. The earliest implementations of computer systems were designed to perform arithmetic operations and cost if not all digital systems will be required to perform some sort of arithmetic as part of their normal operations. This reliance on the arithmetic operations of computers means the accurate representation of real numbers within digital systems is vital, and an understanding of how these systems are implemented and their possible drawbacks is essential in order to design and implement modern high performance systems. At present the most widely implemented system for computer arithmetic is the IEEE754 Floating Point system, while this system is deemed to the be the best available implementation it has several features that can result in serious errors of computation if not implemented correctly. Lack of understanding of these errors and their effects has led to real world disasters in the past on several occasions. Systems for the detection of these errors are highly important and fast, efficient and easy to use implementations of these detection systems is a high priority. Detection of floating point rounding errors normally requires run-time analysis in order to be effective. Several systems have been proposed for the analysis of floating point arithmetic including Interval Arithmetic, Affine Arithmetic and Monte Carlo Arithmetic. While these systems have been well studied using theoretical and software based approaches, implementation of systems that can be applied to real world situations has been limited due to issues with implementation, performance and scalability. The majority of implementations have been software based and have not taken advantage of the performance gains associated with hardware accelerated computer arithmetic systems. This is especially problematic when it is considered that systems requiring high accuracy will often require high performance. The aim of this thesis and associated research is to increase understanding of error and error analysis methods through the development of easy to use and easy to understand implementations of these techniques

Automated Dynamic Error Analysis Methods for Optimization of Computer Arithmetic Systems

Computer arithmetic is one of the more important topics within computer science and engineering. The earliest implementations of computer systems were designed to perform arithmetic operations and cost if not all digital systems will be required to perform some sort of arithmetic as part of their normal operations. This reliance on the arithmetic operations of computers means the accurate representation of real numbers within digital systems is vital, and an understanding of how these systems are implemented and their possible drawbacks is essential in order to design and implement modern high performance systems. At present the most widely implemented system for computer arithmetic is the IEEE754 Floating Point system, while this system is deemed to the be the best available implementation it has several features that can result in serious errors of computation if not implemented correctly. Lack of understanding of these errors and their effects has led to real world disasters in the past on several occasions. Systems for the detection of these errors are highly important and fast, efficient and easy to use implementations of these detection systems is a high priority. Detection of floating point rounding errors normally requires run-time analysis in order to be effective. Several systems have been proposed for the analysis of floating point arithmetic including Interval Arithmetic, Affine Arithmetic and Monte Carlo Arithmetic. While these systems have been well studied using theoretical and software based approaches, implementation of systems that can be applied to real world situations has been limited due to issues with implementation, performance and scalability. The majority of implementations have been software based and have not taken advantage of the performance gains associated with hardware accelerated computer arithmetic systems. This is especially problematic when it is considered that systems requiring high accuracy will often require high performance. The aim of this thesis and associated research is to increase understanding of error and error analysis methods through the development of easy to use and easy to understand implementations of these techniques

BCD Multiplier

BCD multipliers are the basis of accurate decimal multiplication used in banking systems, scientific calculations, etc. Fractions convert poorly into binary numbers giving rise to conversion error. Therefore, banking industry have been using Binary Coded Decimal numbering system for their banking business transaction to circumvent the error between decimal fraction number to binary. Here we will explore some single-digit Binary Coded Decimal Multiplication units that perform multiplication in hardware for the purpose of future implementation

Stochastic rounding and reduced-precision fixed-point arithmetic for solving neural ordinary differential equations

Although double-precision floating-point arithmetic currently dominates high-performance computing, there is increasing interest in smaller and simpler arithmetic types. The main reasons are potential improvements in energy efficiency and memory footprint and bandwidth. However, simply switching to lower-precision types typically results in increased numerical errors. We investigate approaches to improving the accuracy of reduced-precision fixed-point arithmetic types, using examples in an important domain for numerical computation in neuroscience: the solution of Ordinary Differential Equations (ODEs). The Izhikevich neuron model is used to demonstrate that rounding has an important role in producing accurate spike timings from explicit ODE solution algorithms. In particular, fixed-point arithmetic with stochastic rounding consistently results in smaller errors compared to single precision floating-point and fixed-point arithmetic with round-to-nearest across a range of neuron behaviours and ODE solvers. A computationally much cheaper alternative is also investigated, inspired by the concept of dither that is a widely understood mechanism for providing resolution below the least significant bit (LSB) in digital signal processing. These results will have implications for the solution of ODEs in other subject areas, and should also be directly relevant to the huge range of practical problems that are represented by Partial Differential Equations (PDEs).Comment: Submitted to Philosophical Transactions of the Royal Society

RISC-V Core Instruction Extension Sets M and F

This thesis project presents the hardware design of the components capable of implementing a 5-stages core RV32I, RV32IM with integer multiplication and division expansion, and RV32IMF with partial single-precision floating-point support. These have been developed using Verilog HDL and based on the RISC-V ISA. Furthermore, these designs have been verified and synthesised on "bare-metal" using the FPGA platform from the DE0 development board. In addition, a custom variety of division modules have been produced to offer performance diversity on frequency of operation, resource allocation and number of clock cycles per division operations. The selection of these modules provides implementation options that allow to personalize the product to the customer needs

Number Systems for Deep Neural Network Architectures: A Survey

Deep neural networks (DNNs) have become an enabling component for a myriad of artificial intelligence applications. DNNs have shown sometimes superior performance, even compared to humans, in cases such as self-driving, health applications, etc. Because of their computational complexity, deploying DNNs in resource-constrained devices still faces many challenges related to computing complexity, energy efficiency, latency, and cost. To this end, several research directions are being pursued by both academia and industry to accelerate and efficiently implement DNNs. One important direction is determining the appropriate data representation for the massive amount of data involved in DNN processing. Using conventional number systems has been found to be sub-optimal for DNNs. Alternatively, a great body of research focuses on exploring suitable number systems. This article aims to provide a comprehensive survey and discussion about alternative number systems for more efficient representations of DNN data. Various number systems (conventional/unconventional) exploited for DNNs are discussed. The impact of these number systems on the performance and hardware design of DNNs is considered. In addition, this paper highlights the challenges associated with each number system and various solutions that are proposed for addressing them. The reader will be able to understand the importance of an efficient number system for DNN, learn about the widely used number systems for DNN, understand the trade-offs between various number systems, and consider various design aspects that affect the impact of number systems on DNN performance. In addition, the recent trends and related research opportunities will be highlightedComment: 28 page