Optimization of Quantum Cost for Low Energy Reversible Signed/Unsigned Multiplier Using Urdhva-Tiryakbhyam Sutra

One of the elementary operations in computing systems is multiplication. Therefore, high-speed and low-power multipliers design is mandatory for efficient computing systems. In designing low-energy dissipation circuits, reversible logic is more efficient than irreversible logic circuits but at the cost of higher complexity. This paper introduces an efficient signed/unsigned 4 × 4 reversible Vedic multiplier with minimum quantum cost. The Vedic multiplier is considered fast as it generates all partial product and their sum in one step. This paper proposes two reversible Vedic multipliers with optimized quantum cost and garbage output. First, the unsigned Vedic multiplier is designed based on the Urdhava Tiryakbhyam (UT) Sutra. This multiplier consists of bitwise multiplication and adder compressors. Compared with Vedic multipliers in the literature, the proposed design has a quantum cost of 111 with a reduction of 94% compared to the previous design. It has a garbage output of 30 with optimization of the best-compared design. Second, the proposed unsigned multiplier is expanded to allow the multiplication of signed numbers as well as unsigned numbers. Two signed Vedic multipliers are presented with the aim of obtaining more optimization in performance parameters. DesignI has separate binary two’s complement (B2C) and MUX circuits, while DesignII combines binary two’s complement and MUX circuits in one circuit. DesignI shows the lowest quantum cost, 231, regarding state-of-the-art. DesignII has a quantum cost of 199, reducing to 86.14% of DesignI. The functionality of the proposed multiplier is simulated and verified using XILINX ISE 14.2.     

基于RS码的可重构有限域乘法器的设计与实现

为了提高伽罗华有限域乘法器的通用性,降低实现的复杂度,采用自然基算法,用简单的逻辑门电路实现乘法运算过程。提出可重构的迭代计算结构,能满足域长m为3～8的乘法器,并用FPGA实现。结果表明,可重构有限域乘法器能够满足多种标准RS码的乘法运算的需要。     

带时间参数的测试产生

进延测试对于高速集成电路非常重要。本文介绍一个带时间参烽的时延测试产生系统。该系统使用一个时刻逻辑值来表示一个波形,并将输入波形限制为只有唯一的一个输入在０时刻有跳变,其它输入为稳定的０或１,从而实现了波形敏化条件下的时延测试产生,与以往的不考察时间因素的时延测试产生系统相比,带时间参烽的测试产生提高了故障覆盖率,并且更接近于电路的实际。     

Design and implementation of low power and high speed multiplier using quaternary carry look-ahead adder

Need of Digital Signal Processing (DSP) systems which is embedded and portable has been increasing as a result of the speed growth of semiconductor technology. Multiplier is a most crucial part in almost every DSP application. So, the low power, high speed multipliers is needed for high speed DSP. Array multiplier is one of the fast multiplier because it has regular structure and it can be designed very easily. Array multiplier is used for multiplication of unsigned numbers by using full adders and half adders. It depends on the previous computations of partial sum to produce the final output. Hence, delay is more to produce the output. In the previous work, Complementary Metal Oxide Semiconductor (CMOS) Carry Look-ahead Adders (CLA) and CMOS power gating based CLA are used for maximizing the speed of the multiplier and to improve the power dissipation with minimum delay. CMOS logic is based on radix 2(binary) number system. In arithmetic operation, major issue corresponds to carry in binary number system. Higher radix number system like Quaternary Signed Digit (QSD) can be used for performing arithmetic operations without carry. The proposed system designed an array multiplier with Quaternary Signed Digit number system (QSD) based Carry Look-Ahead Adder (CLA) to improve the performance. Generally, the quaternary devices require simpler circuit to process same amount of data than that needed in binary logic devices. Hence the Quaternary logic is applied in the CLA to improve the speed of adder and high throughput. In array multiplier architecture, instead of full adders, carry look-ahead adder based on QSD are used. This facilitates low consumption of power and quick multiplication. Tanner EDA tool is used for simulating the proposed multiplier circuit in 180 nm technology. With respect to area, Power Delay Product (PDP), Average power proposed QSD CLA multiplier is compared with Power gating CLA and CLA multiplier.     

一种用于实现任意基数值时序逻辑的阈值存储电路

本文基于多值时序电路的次态方程和输出方程最小项展开式,提出了一种具有任意值输入、双轨二值输出的阈值存储电路设计方案,它和多值与或门配合,运用Disjoint代数能够设计出任意基数值时序电路.文中通过三值九进制计数器的设计,阐明了任意基数值时序电路的设计方法.     

论域理论在超大规模集成电路逻辑设计上的运用

认为传统的二值布尔不利于大规模集成电路的设计,尤其是在逻辑门电路上.为此引入了三值逻辑.此三值逻辑是基于集成电路的物理性质,且碰巧等同于Kleene的三值逻辑.鉴于Kleene三值逻辑的不完备性,文章将论域理论以及普通不动点算子运用于此,使三值逻辑获得此逻辑系统的单调完备性定理.文章认为这个结果有利于集成电路设计的可靠性,具有广阔的应用前景.     

Design of low power multi-ternary digit multiplier in CNTFET technology

This work introduces the method to implement energy efficient designs of arithmetic units such as a ternary full adder, ripple carry adder, single-trit multiplier and multi-trit multiplier using carbon nanotube field effect transistors (CNTFETs). A CNTFET unique feature of the threshold voltage variation by changing the CNT diameter, make it a suitable alternative for being employed in ternary logic designs. In designing the proposed circuits, decoder circuit functionality is realized by various threshold detector circuits tuned to a specific logical threshold voltage value. The multiplier circuit is designed by combing the capacitive logic and the minority function. In order to test the practicability of proposed circuits in cascaded circuits, multi-digit adder and multiplier circuits are constructed. The proposed multi-digit multiplier structure is based on classical Wallace multiplier and includes various optimized versions of adder and multiplier circuits. Extensive simulation has been done to examine the competency of proposed designs under different test conditions. The design of 3-trit multiplier formed by combing the proposed adder and multiplier circuits shows 16 times reduction in power consumption as well as energy consumption in comparison to previous multiplier design.     

高速RS编译码器的设计及其FPGA实现

在分析RS（Reed-Solomon）码编译码基本原理的基础上,对编码过程中的乘法电路实现进行了深入分析,对译码过程中用于错误位置多项式和错误值多项式计算的BM（Berlekamp-Massey）迭代算法进行改进,并设计了适合于FPGA硬件实现的伴随式计算策略和钱搜索电路。硬件实现结果表明,改进算法能有效节省硬件资源,在Xilinx公司的XC4VSX35 FPGA上仅需要总资源的15%就可以实现（31,15）RS码编译码器电路,且在200 MHz系统时钟频率时达到10 Mb/s的译码速率,实现了高速数据处理。     

下三角矩阵神经网络逻辑电路的设计

由于人工神经网络的卓越优点，为制造超高速，高可靠和可编程的数字集成电路提供了新途径，具有下三角形连接矩阵的Ｈｏｐｆｉｅｌｄ模型在同一输入下仅有唯一的平衡点。本文将讨论基于这种网络模型的组合逻辑电路的逻辑设计方法，以最小化神经元个数为目标的启发式优化算法及权电阻网络参数的计算方法。     

A recursive method for synthesizing quantum/reversible quaternary parallel adder/subtractor with look-ahead carry

Multiple-valued quantum logic circuits are a promising choice for future quantum computing technology since they have several advantages over binary quantum logic circuits. Adder/subtractor is the major component of the ALU of a computer and is also used in quantum oracles. In this paper, we propose a recursive method of hand synthesis of reversible quaternary full-adder circuit using macro-level quaternary controlled gates built on the top of ion-trap realizable 1-qudit quantum gates and 2-qudit Muthukrishnan–Stroud quantum gates. Based on this quaternary full-adder circuit we propose a reversible circuit realizing quaternary parallel adder/subtractor with look-ahead carry. We also show the way of adapting the quaternary parallel adder/subtractor circuit to an encoded binary parallel adder/subtractor circuit by grouping two qubits together into quaternary qudit values.     

EDOA: an efficient delay optimization approach for mixed-polarity Reed-Muller logic circuits under the unit delay model

Delay optimization has recently attracted significant attention. However, few studies have focused on the delay optimization of mixed-polarity Reed-Muller (MPRM) logic circuits. In this paper, we propose an efficient delay optimization approach (EDOA) for MPRM logic circuits under the unit delay model, which can derive an optimal MPRM logic circuit with minimum delay. First, the simplest MPRM expression with the fewest number of product terms is obtained using a novel Reed-Muller expression simplification approach (RMESA) considering don’t-care terms. Second, a minimum delay decomposition approach based on a Huffman tree construction algorithm is utilized on the simplestMPRM expression. Experimental results on MCNC benchmark circuits demonstrate that compared to the Berkeley SIS 1.2 and ABC, the EDOA can significantly reduce delay for most circuits. Furthermore, for a few circuits, while reducing delay, the EDOA incurs an area penalty.     

Reversible binary subtractor design using quantum dot-cellular automata

In the field of nanotechnology, quantum dot-cellular automata (QCA) is the promising archetype that can provide an alternative solution to conventional complementary metal oxide semiconductor (CMOS) circuit. QCA has high device density, high operating speed, and extremely low power consumption. Reversible logic has widespread applications in QCA. Researchers have explored several designs of QCA-based reversible logic circuits, but still not much work has been reported on QCA-based reversible binary subtractors. The low power dissipation and high circuit density of QCA pledge the energy-efficient design of logic circuit at a nano-scale level. However, the necessity of too many logic gates and detrimental garbage outputs may limit the functionality of a QCA-based logic circuit. In this paper we describe the design and implementation of a DG gate in QCA. The universal nature of the DG gate has been established. The QCA building block of the DG gate is used to achieve new reversible binary subtractors. The proposed reversible subtractors have low quantum cost and garbage outputs compared to the existing reversible subtractors. The proposed circuits are designed and simulated using QCA Designer-2.0.3.     

Evolutionary Algorithms and Theirs Use in the Design of Sequential Logic Circuits

In this paper an approach based on an evolutionary algorithm to design synchronous sequential logic circuits with minimum number of logic gates is suggested. The proposed method consists of four main stages. The first stage is concerned with the use of genetic algorithms (GA) for the state assignment problem to compute optimal binary codes for each symbolic state and construct the state transition table of finite state machine (FSM). The second stage defines the subcircuits required to achieve the desired functionality. The third stage evaluates the subcircuits using extrinsic Evolvable Hardware (EHW). During the fourth stage, the final circuit is assembled. The obtained results compare favourably against those produced by manual methods and other methods based on heuristic techniques.     

基于基因表达式的演化硬件进化和优化算法

电路进化设计是可进化硬件研究的重要内容．针对电路进化设计做了如下工作：(1)融合了数据挖掘、基因表达式编程与传统电路进化技术,提出两阶段电路进化方法．该方法包括基于表达式树遗传编程进化算法的电路进化阶段和基于挖掘频繁数字电路算法的电路优化阶段。(2)给出了详尽的实验．实验表明6次多项式函数发现的平均进化代数为442代、乘法器电路的平均进化代数为2292代．比笛卡尔遗传编程和NEHF(Novel Evolvable Hardware Framework)快6倍以上．用MFDC对乘法器电路进化结果进行挖掘后,得到了比传统电路更有效的乘法器电路。     

A signed binary addition circuit based on an alternative class of addition tables

Redundant signed digit number systems have been used as a basis for the construction of fast arithmetic circuits for several years. In particular, addition circuits with no carry-ripple effects have been developed using signed binary arithmetic systems. This paper presents a general class of signed binary addition tables and provides a framework for constructing various tables. The existence of an entire class of tables provides a circuit designer with an additional degree of freedom while developing addition circuitry. The choice of the exact form of the addition table can be based on the dominant desired characteristics of the resultant circuit. An example of a circuit derived for area minimization is presented and compared to another signed binary addition circuit that was previously published. Both circuits were optimized and mapped to 20 different CMOS cell libraries. The experimental results indicate an average decrease in area of 26% and an average decrease in dynamic power consumption of 29% with an average increase in delay of only 4.4%.     

Evolutionary design of complex approximate combinational circuits

Functional approximation is one of the methods allowing designers to approximate circuits at the level of logic behavior. By introducing a suitable functional approximation, power consumption, area or delay of a circuit can be reduced if some errors are acceptable in a particular application. As the error quantification is usually based on an arithmetic error metric in existing approximation methods, these methods are primarily suitable for the approximation of arithmetic and signal processing circuits. This paper deals with the approximation of general logic (such as pattern matching circuits and complex encoders) in which no additional information is usually available to establish a suitable error metric and hence the error of approximation is expressed in terms of Hamming distance between the output values produced by a candidate approximate circuit and the accurate circuit. We propose a circuit approximation method based on Cartesian genetic programming in which gate-level circuits are internally represented using directed acyclic graphs. In order to eliminate the well-known scalability problems of evolutionary circuit design, the error of approximation is determined by binary decision diagrams. The method is analyzed in terms of computational time and quality of approximation. It is able to deliver detailed Pareto fronts showing various compromises between the area, delay and error. Results are presented for 16 circuits (with 27–50 inputs) that are too complex to be approximated by means of existing evolutionary circuit design methods.     

一种改进的基于FPGA的32位对数变换器的设计与实现

对数变换器是对数乘法器的重要组成部分,它们以精度换取更快的速度.设计并实现了一种基于FPGA的32位二进制对数变换器,主要由先导"1"检测电路、移位逻辑和误差校正电路组成,通过有效的误差校正算法提高了计算精度;给出了一种新的4位、16位和32位的基于FPGA的并行先导"1"检测电路PLOD,在保持低延时的同时,减小了先导"1"检测电路的功耗和面积;改进了现有的6-域校正算法,在提高精度的同时保持了硬件电路的规整性,降低了系统复杂度及面积和功耗开销;分两站流水实现校正操作,提高了系统的吞吐率;改进后的校正电路将对数操作的最大误差由30%降低到20%,区域1的平均误差大幅度降低.     

Controlled gates for multi-level quantum computation

Multi-level (ML) quantum logic can potentially reduce the number of inputs/outputs or quantum cells in a quantum circuit which is a limitation in current quantum technology. In this paper we propose theorems about ML-quantum and reversible logic circuits. New efficient implementations for some basic controlled ML-quantum logic gates, such as three-qudit controlled NOT, Cycle, and Self Shift gates are proposed. We also propose lemmas about r-level quantum arrays and the number of required gates for an arbitrary n-qudit ML gate. An equivalent definition of quantum cost (QC) of binary quantum gates for ML-quantum gates is introduced and QC of controlled quantum gates is calculated.     

On figures of merit in reversible and quantum logic designs

Five figures of merit including number of gates, quantum cost, number of constant inputs, number of garbage outputs, and delay are used casually in the literature to compare the performance of different reversible or quantum logic circuits. In this paper we propose new definitions and enhancements, and identify similarities between these figures of merit. We evaluate these measures to show their strength and weakness. Instead of the number of gates, we introduce the weighted number of gates, where a weighting factor is assigned to each quantum or reversible gate, based on its type, size and technology. We compare the quantum cost with weighted number of gates of a circuit and show three major differences between these measures. It is proved that it is not possible to define a universal reversible logic gate without adding constant inputs. We prove that there is an optimum value for number of constant inputs to obtain a circuit with minimum quantum cost. Some reversible logic benchmarks have been synthesized using Toffoli and Fredkin gates to obtain their optimum values of number of constant inputs. We show that the garbage outputs can also be used to decrease the quantum cost of the circuit. A new definition of delay in quantum and reversible logic circuits is proposed for music line style representation. We also propose a procedure to calculate the delay of a circuit, based on the quantum cost and the depth of the circuit. The results of this research show that to achieve a fair comparison among designs, figures of merit should be considered more thoroughly.      

Toward novel designs of reversible ternary 6:2 Compressor using efficient reversible ternary full-adders

Integrated circuits always face with two major challenges including heat caused by energy losses and the area occupied. In recent years, different strategies have been presented to reduce these two major challenges. The implementations of circuits in a reversible manner as well as the use of multiple-valued logic are among the most successful strategies. Reversible circuits reduce energy loss and ultimately eliminate the problem of overheating in circuits. Preferring multiple-valued logic over binary logic can also greatly reduce area occupied of circuits. When switching from binary logic to multiple-valued logic, the dominant thought in binary logic is the basis of designing computational circuits in multiple-valued logic, and disregards the capabilities of multiple-valued logic. This can cause a minimal use of multiple-valued logic capabilities, increase complexity and delay in the multiple-valued computational circuits. In this paper, we first introduce an efficient reversible ternary half-adder. Afterward, using the reversible ternary half-adder, we introduce two reversible versions of traditional and comprehensive reversible ternary full-adders. Finally, using the introduced reversible ternary full-adders, we propose two novel designs of reversible ternary 6:2 Compressor. The results of the comparisons show that although the proposed circuits are similar to or better than previous corresponding designs in terms of criteria number of constant input and number of garbage outputs, they are superior in criterion quantum cost.