Partial Product Reduction Tree Research Articles

A high‐efficient imprecise discrete cosine transform block based on a novel full adder and Wallace multiplier for bioimages compression

SummarySophisticated systems take advantage of approximate circuits for energy management. A gate‐level structure is introduced to make a novel imprecise full adder (FA) cell by the gate diffusion input (GDI) strategy. The carbon nanotube field‐effect transistor (CNTFET) technology lowers the FA power, and the swing issue is resolved by the dynamic threshold (DT) technique. The FA specifications are 10 transistors, two internal nodes, 0.191‐μm2 area, three errors, and an error rate (ER) of 37.5%. Some inputs are directly passed to the final stage of the FA to reduce the delay and dynamic power. The FA is used in three 4:2 compressors with 12, 16, and 20 transistors, respectively. The compressors are based on the modified stacking concept to attain low power, small area, and high speed. The compressors are used up in the partial product reduction tree (PPRT) of a new 8‐bit inexact multiplier. Regarding the multiplier, the power–delay product (PDP) is decreased by 25.21%, and the normalized mean error distance (NMED) shows high accuracy. An imprecise discrete cosine transform (DCT) is implemented by the multiplier for image compression, and compared with the literature, the peak signal‐to‐noise ratio (PSNR) and structural similarity index metric (SSIM) are amended by 8.26% and 7.29%, respectively.

A 0.35 V 376 Mb/s Configurable Long Integer Multiplier for Subthreshold Encryption

A configurable long integer multiplier tailored to subthreshold operation is presented for ultra-low-power asymmetric en/decryption in semi-passive or passive systems. The multiplier is composed of radix-4 booth de/encoders and two ${544\times 16}$ -bit partial product reduction tree arrays and reconfigurable data paths. The architecture can be configured for multiple multiplications with different bit size. Novel 6–2 compressors are applied to the reduction tree with 4–2 compressors. The tree can compresses 16 bit products and carry-ins into 1 bit sum and carry within the delay of 6 XOR gates. Innovatively, a novel custom ratioed logic style is adopted in critical logic paths to fundamentally speed up the subthreshold operation. Fabricated in 65-nm CMOS, the proposed design indicates competent subthreshold operation. It can operate at 0.35-V supply with throughput of 376 Mb/s, with power consumption of 20.8 ${\mu }\text{W}$ in 1024-bit (modular) multiplication.

High Performance Parallel Decimal Multipliers Using Hybrid BCD Codes

A parallel decimal multiplier with improved performance is proposed in this paper by exploiting the properties of three different binary coded decimal (BCD) codes, namely the redundant BCD excess-3 code (XS-3), the overloaded decimal digit set (ODDS) code and the BCD-4221/5211 code. The signed-digit radix-10 recoding is used to recode the BCD multiplier to the digit set [-5, 5] from [0, 9]. The redundant BCD XS-3 code is adopted to generate the multiplicand multiples in a carry-free manner. The XS-3 coded partial products (PPs) are converted to ODDS PPs to fit binary partial product reduction (PPR). In this paper, a regular decimal PPR tree using ODDS and BCD-4221/5211 codes is proposed; it consists of a binary PPR tree block, a non-fixed size BCD-4221 counter block and a BCD-4221/5211 PPR tree block. The decimal carry-save algorithm based on BCD-4221/5211 is used in the PPR tree to obtain high performance multipliers. Moreover, an improved PPG circuit and an improved parallel prefix/carry-select decimal adder are proposed to further improve the performance of the proposed multipliers. Analysis and comparison using the 45 nm technology show that the proposed decimal multipliers are faster and require less hardware area than previous designs found in the technical literature.

Fast Radix-10 Multiplication Using Redundant BCD Codes

We present the algorithm and architecture of a BCD parallel multiplier that exploits some properties of two different redundant BCD codes to speedup its computation: the redundant BCD excess-3 code (XS-3), and the overloaded BCD representation (ODDS). In addition, new techniques are developed to reduce significantly the latency and area of previous representative high-performance implementations. Partial products are generated in parallel using a signed-digit radix-10 recoding of the BCD multiplier with the digit set [-5, 5], and a set of positive multiplicand multiples (0X, 1X, 2X, 3X, 4X, 5X) coded in XS-3. This encoding has several advantages. First, it is a self-complementing code, so that a negative multiplicand multiple can be obtained by just inverting the bits of the corresponding positive one. Also, the available redundancy allows a fast and simple generation of multiplicand multiples in a carry-free way. Finally, the partial products can be recoded to the ODDS representation by just adding a constant factor into the partial product reduction tree. Since the ODDS uses a similar 4-bit binary encoding as non-redundant BCD, conventional binary VLSI circuit techniques, such as binary carry-save adders and compressor trees, can be adapted efficiently to perform decimal operations. To show the advantages of our architecture, we have synthesized a RTL model for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math>$16\times 16$</tex-math></inline-formula> -digit and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math>$34\times 34$</tex-math></inline-formula> -digit multiplications and performed a comparative survey of the previous most representative designs. We show that the proposed decimal multiplier has an area improvement roughly in the range 20-35 percent for similar target delays with respect to the fastest implementation.

Design and Implementation of Modified Partial Product Reduction Tree for High Speed Multiplication

This paper presents the details of hardware implementation of modified partial product reduction tree using 4:2 and 5:2 compressors. Speed of multiplication operation is improved by using higher compressors .In order to improve the speed of the multiplication process within the computational unit; there is a major bottleneck that is needed to be considered that is the partial products reduction network which is used in the multiplication block. For implementation of this stage require addition of large operands that involve long paths for carry propagation. The proposed architecture is based on binary tree constructed using modified 4:2 and 5:2 compressor circuits. Increasing the speed of operation is achieved by using higher modified compressors in critical path. Our objective of work is, to increase the speed of multiplication operation by minimizing the number of combinational gates using higher n:2 compressors. The experimental test of the proposed modified compressor is done using Spartan-3FPGA device (XC3S400 PQ-208). Using tree architectures for the partial products reduction network represent an attractive solution that is frequently applied to speed up the multiplication process. The simulation result shows 4:2 and 5:2 compressor output which is done using Questa Sim 6.4c Mentor Graphics tool.

Design and Implementation of Modified Partial Product Reduction Tree for High Speed Multiplication

Design and Implementation of Modified Partial Product Reduction Tree for High Speed Multiplication

Compressor tree synthesis on commercial high-performance FPGAs

Compressor trees are a class of circuits that generalizes multioperand addition and the partial product reduction trees of parallel multipliers using carry-save arithmetic. Compressor trees naturally occur in many DSP applications, such as FIR filters, and, in the more general case, their use can be maximized through the application of high-level transformations to arithmetically intensive data flow graphs. Due to the presence of carry-chains, it has long been thought that trees of 2- or 3-input carry-propagate adders are more efficient than compressor trees for FPGA synthesis; however, this is not the case. This article presents a heuristic for FPGA synthesis of compressor trees that outperforms adder trees and exploits carry-chains when possible. The experimental results show that, on average, the use of compressor trees can reduce critical path delay by 33% and 45% respectively, compared to adder trees synthesized on the Xilinx Virtex-5 and Altera Stratix III FPGAs.

A High-Speed Multiplication Algorithm Using Modified Partial Product Reduction Tree

A High-Speed Multiplication Algorithm Using Modified Partial Product Reduction Tree

A fully redundant decimal adder and its application in parallel decimal multipliers

Decimal hardware arithmetic units have recently regained popularity, as there is now a high demand for high performance decimal arithmetic. We propose a novel method for carry-free addition of decimal numbers, where each equally weighted decimal digit pair of the two operands is partitioned into two weighted bit-sets. The arithmetic values of these bit-sets are evaluated, in parallel, for fast computation of the transfer digit and interim sum. In the proposed fully redundant adder (VS semi-redundant ones such as decimal carry-save adders) both operands and sum are redundant decimal numbers with overloaded decimal digit set [0, 15]. This adder is shown to improve upon the latest high performance similar works and outperform all the previous alike adders. However, there is a drawback that the adder logic cannot be efficiently adapted for subtraction. Nevertheless, this adder and its restricted-input varieties are shown to efficiently fit in the design of a parallel decimal multiplier. The two-to-one partial product reduction ratio that is attained via the proposed adder has lead to a VLSI-friendly recursive partial product reduction tree. Two alternative architectures for decimal multipliers are presented; one is slower, but area-improved, and the other one consumes more area, but is delay-improved. However, both are faster in comparison with previously reported parallel decimal multipliers. The area and latency comparisons are based on logical effort analysis under the same assumptions for all the evaluated adders and multipliers. Moreover, performance correctness of all the adders is checked via running exhaustive tests on the corresponding VHDL codes. For more reliable evaluation, we report the result of synthesizing these adders by Synopsys Design Compiler using TSMC 0.13 μm standard CMOS process under various time constrains.

New method for high performance multiply-accumulator design

This study presents a new method of 4-pipelined high-performance split multiply-accumulator (MAC) architecture, which is capable of supporting multiple precisions developed for media processors. To speed up the design further, a novel partial product compression circuit based on interleaved adders and a modified hybrid partial product reduction tree (PPRT) scheme are proposed. The MAC can perform 1-way 32-bit, 4-way 16-bit signed/unsigned multiply or multiply-accumulate operations and 2-way parallel multiply add (PMADD) operations at a high frequency of 1.25 GHz under worst-case conditions and 1.67 GHz under typical-case conditions, respectively. Compared with the MAC in 32-bit microprocessor without interlocked piped stages (MIPS), the proposed design shows a great advantage in speed. Moreover, an improvement of up to 32% in throughput is achieved. The MAC design has been fabricated with Taiwan Semiconductor Manufacturing Company (TSMC) 90-nm CMOS standard cell technology and has passed a functional test.

Toward architecture-based test-vector generation for timing verification of fast parallel multipliers

Fast parallel multipliers that contain logarithmic partial-product reduction trees pose a challenge to simulation-based high-accuracy timing verification, since the reduction tree has many reconvergent signal branches. However, such a multiplier architecture also offers a clue as how to attack the test-vector generation problem. The timing-critical paths are intimately associated with long carry propagation. We introduce a multiplier test-vector generation method that has the ability to exercise such long carry propagation paths. Through extensive circuit simulation and static timing analysis, we evaluate the quality of the test vectors that result from the new method. Especially for fast multipliers with a pronounced carry propagation, the timing-critical vectors manage to stimulate a path, which has a delay that comes close to the true worst case delay. We investigate the complexity and run-time for the test-vector generation, and derive timing-critical vectors up to a factor word length of 54 bits.

Synthesis of arithmetic circuits considering layout effects

In deep submicron technology, wires are equally or more important than logic components since wire-related problems such as crosstalk noise is much critical in system-on-chip design. Recently, a method for generating a partial product reduction tree with optimal-timing using bit-level adders to implement arithmetic circuits has been proposed, which outperforms the current best designs. However, in the conventional approaches, interconnects are not primary components to be optimized in the synthesis of arithmetic circuits, mainly due to its integration complexity or unpredictable wire effects, thereby resulting in unsatisfactory layout results with long and messy wire connections. To overcome the limitation, we propose a new module generation/synthesis algorithm for arithmetic circuits utilizing carry-save-adder (CSA) modules, which not only optimizes the circuit timing but also generates a much more regular interconnect topology of the final circuits. Specifically, we propose a two-step algorithm: (Phase 1: CSA module generation) we propose an optimal-timing CSA module generation algorithm for an arithmetic expression under a general CSA timing model; then (Phase 2: Bit-level interconnect refinements), we optimally refine the interconnects between the CSA modules while retaining the global CSA-tree structure produced by Phase 1. We show that the timing of the circuits produced by our approach is equal or almost close to that in most test cases (even without including the interconnect delay), and at the same time, the interconnects in layout are short and regular.

Optimal circuits for parallel multipliers

We present new design and analysis techniques for the synthesis of parallel multiplier circuits that have smaller predicted delay than the best current multipliers. V.G. Oklobdzija et al. (1996) suggested a new approach, the Three-Dimensional Method (TDM), for Partial Product Reduction Tree (PPRT) design that produces multipliers that outperform the current best designs. The goal of TDM is to produce a minimum delay PPRT using full adders. This is done by carefully modeling the relationship of the output delays to the input delays in an adder and, then, interconnecting the adders in a globally optimal way. Oklobdzija et al. suggested a good heuristic for finding the optimal PPRT, but no proofs about the performance of this heuristic were given. We provide a formal characterization of optimal PPRT circuits and prove a number of properties about them. For the problem of summing a set of input bits within the minimum delay, we present an algorithm that produces a minimum delay circuit in time linear in the size of the inputs. Our techniques allow us to prove tight lower bounds on multiplier circuit delays. These results are combined to create a program that finds optimal TDM multiplier designs. Using this program, we can show that, while the heuristic used by Oklobdzija et al. does not always find the optimal TDM circuit, it performs very well in terms of overall PPRT circuit delay. However, our search algorithms find better PPRT circuits for reducing the delay of the entire multiplier.

Design strategies for optimal hybrid final adders in a parallel multiplier

In this paper we address the problem of adding twon-bit numbers when the bit arrival times are arbitrary (but known in advance). In particular we address a simplified version of the problem where the input arrival times for theith significant bits of both addends are the same, and the arrival timesti have a profile of the form: $$t_0 \leqslant t_1 \leqslant \cdot \cdot \cdot t_{p + 1} \geqslant \cdot \cdot \cdot \geqslant t_{n - 1} $$ This profile is important because it matches the signal arrival time profile of the reduced partial products in a parallel multiplier before they are summed in the final adder. In this paper we present a design strategy specific to arrival time profiles generated by partial product reduction trees constructed by optimal application of the Three Dimensional Method presented by Oklobdzija, Villeger, and Liu and subsequently analyzed by Martel, Oklobdzija, Ravi, and Stelling. This strategy can be used to obtain adders for any arrival time profile that matches the above form, as well as a broad class of arrival time profiles where even greater variation in the input times is allowed. Finally, we show that our designs significantly out-perform the standard adder designs for the uniform signal arrival profile, yielding faster adders that (for these profiles) are also simpler and use fewer gates.

A method for speed optimized partial product reduction and generation of fast parallel multipliers using an algorithmic approach

This paper presents a method and an algorithm for generation of a parallel multiplier, which is optimized for speed. This method is applicable to any multiplier size and adaptable to any technology for which speed parameters are known. Most importantly, it is easy to incorporate this method in silicon compilation or logic synthesis tools. The parallel multiplier produced by the proposed method outperforms other schemes used for comparison in our experiment. It uses the minimal number of cells in the partial product reduction tree. These findings are tested on design examples simulated in 1 /spl mu/ CMOS ASIC technology.