On digit-recurrence division algorithms for self-timed circuits

The optimization of algorithms for self-timed or asynchronous circuits requires specific solutions. Due to the variable-time capabilities of asynchronous circuits, the average computation time should be optimized and not only the worst case of the signal propagation. If efficient algorithms and implementations are known for asynchronous addition and multiplication, only straightforward algorithms have been studied for division. This paper compares several digit-recurrence division algorithms (speed, area and circuit activity for estimating the power consumption). The comparison is based on simulations of the different operators described at the gate level. This work shows that the best solutions for asynchronous circuits are quite different from those used in synchronous circuits

Reliable and Fault-Resilient Schemes for Efficient Radix-4 Complex Division

Complex division is commonly used in various applications in signal processing and control theory including astronomy and nonlinear RF measurements. Nevertheless, unless reliability and assurance are embedded into the architectures of such structures, the suboptimal (and thus erroneous) results could undermine the objectives of such applications. As such, in this thesis, we present schemes to provide complex number division architectures based on (Sweeney, Robertson, and Tocher) SRT-division with fault diagnosis mechanisms. Different fault resilient architectures are proposed in this thesis which can be tailored based on the eventual objectives of the designs in terms of area and time requirements, among which we pinpoint carefully the schemes based on recomputing with shifted operands (RESO) to be able to detect both natural and malicious faults and with proper modification achieve high throughputs. The design also implements a minimized look up table approach which favors in error detection based designs and provides high fault coverage with relatively-low overhead. Additionally, to benchmark the effectiveness of the proposed schemes, extensive fault diagnosis assessments are performed for the proposed designs through fault simulations and FPGA implementations; the design is implemented on Xilinx Spartan-VI and Xilinx Virtex-VI FPGA families

High sample-rate Givens rotations for recursive least squares

The design of an application-specific integrated circuit of a parallel array processor is considered for recursive least squares by QR decomposition using Givens rotations, applicable in adaptive filtering and beamforming applications. Emphasis is on high sample-rate operation, which, for this recursive algorithm, means that the time to perform arithmetic operations is critical. The algorithm, architecture and arithmetic are considered in a single integrated design procedure to achieve optimum results. A realisation approach using standard arithmetic operators, add, multiply and divide is adopted. The design of high-throughput operators with low delay is addressed for fixed- and floating-point number formats, and the application of redundant arithmetic considered. New redundant multiplier architectures are presented enabling reductions in area of up to 25%, whilst maintaining low delay. A technique is presented enabling the use of a conventional tree multiplier in recursive applications, allowing savings in area and delay. Two new divider architectures are presented showing benefits compared with the radix-2 modified SRT algorithm. Givens rotation algorithms are examined to determine their suitability for VLSI implementation. A novel algorithm, based on the Squared Givens Rotation (SGR) algorithm, is developed enabling the sample-rate to be increased by a factor of approximately 6 and offering area reductions up to a factor of 2 over previous approaches. An estimated sample-rate of 136 MHz could be achieved using a standard cell approach and O.35pm CMOS technology. The enhanced SGR algorithm has been compared with a CORDIC approach and shown to benefit by a factor of 3 in area and over 11 in sample-rate. When compared with a recent implementation on a parallel array of general purpose (GP) DSP chips, it is estimated that a single application specific chip could offer up to 1,500 times the computation obtained from a single OP DSP chip

Design and Implementation of a Radix-100 Division Unit

In this thesis, a novel radix-100 divider is designed and implemented. The proposed divider is 3% faster then the current decimal dividers

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

A Study on Efficient Designs of Approximate Arithmetic Circuits

Approximate computing is a popular field where accuracy is traded with energy. It can benefit applications such as multimedia, mobile computing and machine learning which are inherently error resilient. Error introduced in these applications to a certain degree is beyond human perception. This flexibility can be exploited to design area, delay and power efficient architectures. However, care must be taken on how approximation compromises the correctness of results. This research work aims to provide approximate hardware architectures with error metrics and design metrics analyzed and their effects in image processing applications. Firstly, we study and propose unsigned array multipliers based on probability statistics and with approximate 4-2 compressors, full adders and half adders. This work deals with a new design approach for approximation of multipliers. The partial products of the multiplier are altered to introduce varying probability terms. Logic complexity of approximation is varied for the accumulation of altered partial products based on their probability. The proposed approximation is utilized in two variants of 16-bit multipliers. Synthesis results reveal that two proposed multipliers achieve power savings of 72% and 38% respectively compared to an exact multiplier. They have better precision when compared to existing approximate multipliers. Mean relative error distance (MRED) figures are as low as 7.6% and 0.02% for the proposed approximate multipliers, which are better than the previous state-of-the-art works. Performance of the proposed multipliers is evaluated with geometric mean filtering application, where one of the proposed models achieves the highest peak signal to noise ratio (PSNR). Second, approximation is proposed for signed Booth multiplication. Approximation is introduced in partial product generation and partial product accumulation circuits. In this work, three multipliers (ABM-M1, ABM-M2, and ABM-M3) are proposed in which the modified Booth algorithm is approximated. In all three designs, approximate Booth partial product generators are designed with different variations of approximation. The approximations are performed by reducing the logic complexity of the Booth partial product generator, and the accumulation of partial products is slightly modified to improve circuit performance. Compared to the exact Booth multiplier, ABM-M1 achieves up to 15% reduction in power consumption with an MRED value of 7.9 × 10-4. ABM-M2 has power savings of up to 60% with an MRED of 1.1 × 10-1. ABM-M3 has power savings of up to 50% with an MRED of 3.4 × 10-3. Compared to existing approximate Booth multipliers, the proposed multipliers ABM-M1 and ABM-M3 achieve up to a 41% reduction in power consumption while exhibiting very similar error metrics. Image multiplication and matrix multiplication are used as case studies to illustrate the high performance of the proposed approximate multipliers. Third, distributed arithmetic based sum of products units approximation is analyzed. Sum of products units are key elements in many digital signal processing applications. Three approximate sum of products models which are based on distributed arithmetic are proposed. They are designed for different levels of accuracy. First model of approximate sum of products achieves an improvement up to 64% on area and 70% on power, when compared to conventional unit. Other two models provide an improvement of 32% and 48% on area and 54% and 58% on power, respectively, with a reduced error rate compared to the first model. Third model achieves MRED and normalized mean error distance (NMED) as low as 0.05% and 0.009%. Performance of approximate units is evaluated with a noisy image smoothing application, where the proposed models are capable of achieving higher PSNR than existing state of the art techniques. Fourth, approximation is applied in division architecture. Two approximation models are proposed for restoring divider. In the first design, approximation is performed at circuit level, where approximate divider cells are utilized in place of exact ones by simplifying the logic equations. In the second model, restoring divider is analyzed strategically and number of restoring divider cells are reduced by finding the portions of divisor and dividend with significant information. An approximation factor $p$ is used in both designs. In model 1, the design with p=8 has a 58% reduction in both area and power consumption compared to exact design, with a Q-MRED of 1.909 × 10-2 and Q-NMED of 0.449 × 10-2. The second model with an approximation factor p=4 has 54% area savings and 62% power savings compared to exact design. The proposed models are found to have better error metrics compared to existing designs, with better performance at similar error values. A change detection image processing application is used for real time assessment of proposed and existing approximate dividers and one of the models achieves a PSNR of 54.27 dB

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

High-radix division and square-root with speculation

The speed of high-radix digit-recurrence dividers and square-root units is mainly determined by the complexity of the result-digit selection. We present a scheme in which a simpler function speculates the result digit, and, when this speculation is incorrect, a rollback or a partial advance is performed. This results in operations with a shorter cycle time and a variable number of cycles. The scheme can be used in separate division and square-root units, or in a combined one. Several designs were realized and compared in terms of execution time and area. The fastest unit considered is a radix-512 divider with a partial advance of six bits.Peer ReviewedPostprint (published version

Design of Unsigned Approximate Hybrid Dividers based on Restoring Array and Logarithmic Dividers

Approximate computer arithmetic has been extensively studied due to its advantages to further reduce power consumption and increase performance at reduced accuracy. Although a number of approximate adders and multipliers have been studied, only a few approximate dividers have been proposed. A logarithmic divider (LD) has low complexity and accuracy, while an exact array divider (EXD) has a high complexity. Therefore, in this paper, an approximate hybrid divider (AXHD) is proposed. It takes advantage of both LD and EXD to achieve a tradeoff between hardware performance and accuracy. Exact restoring divider cells are used to generate the most significant bits (MSBs) of the quotient for attaining a high accuracy while the other quotient digits are generated by using a LD as an approximate scheme to improve figures of merit such as power consumption, area and delay. To further save hardware resources, a so-called eliminated approximate hybrid divider (E-AXHD) based on AXHD is also proposed. In this improved design, a reduced width divider is used to replace the EXD in AXHD. Specifically, for a 16-by-8 design, n=(n + 1) array division is used to replace the n=8 array division (n < 8). The proposed AXHD and E-AXHD are evaluated and analyzed using error and hardware metrics. The proposed designs are also compared with EXD, LD and previous approximate dividers. The results show that the proposed designs outperform previous approximate dividers by considering both energy and error. The proposed hybrid dividers are of particular interest for error tolerant applications such as image processing and machine learning

Radix-16 signed-digit division

Journal ArticleFor use in the context of a linearly scalable arithmetic architecture supporting high/variable precision arithmetic operations (integer or fractional), a two-stage algorithm for fixed point, radix-16 signed-digit division is presented. The algorithm uses two limited precision radix-4 quotient digit selection stages to produce the full radix-16 quotient digit.The algorithm requires a two digit estimate of the (initial) partial remainder and a three digit estimate of the divisor to correctly select each successive quotient digit. The normalization of redundant signed-digit numbers requires accommodation of some fuzziness at one end of the range of numeric values that are considered normalized. A set of general equations for determining the ranges of normalized signed-digit numbers is derived. Another set of general equations for determining the precisions of estimates of the divisor and dividend required in a limited precision SRT model signed-digit division are derived. These two sets of equations permit design tradeoff analyses to be made with respect to the complexity of the model division. The specific case of a two-stage radix-16 signed-digit division is presented. The staged division algorithm used can be extended to other radices as long as the signed-digit number representation used has certain properties