PARTIAL PRODUCT ARRAY HEIGHT REDUCTION USING RADIX-16 FOR 64-BIT BOOTH MULTIPLIER:
Reduce the maximum height of the partial product columns to [n/4] for n = 64-bit unsigned
operand. This is in contrast to the conventional maximum height of [(n + 1)/4].
The multiplier algorithm is normally used for higher bit length applications and ordinary multiplier is good for lower order bits.
This document discusses operations on multiple random variables, including:
- The expected value of a function of two random variables X and Y is the sum of the expected values of the functions.
- Joint moments describe the relationship between multiple random variables and can be used to find properties like covariance and correlation.
- Two random variables are jointly Gaussian if their joint density function follows a specific form, and properties of Gaussian random variables include being fully defined by their first and second moments.
- Transformations of multiple random variables, such as applying a linear transformation, preserve properties like expected value and covariance if the original variables were Gaussian.
This document provides an overview of activation functions in deep learning. It discusses the purpose of activation functions, common types of activation functions like sigmoid, tanh, and ReLU, and issues like vanishing gradients that can occur with some activation functions. It explains that activation functions introduce non-linearity, allowing neural networks to learn complex patterns from data. The document also covers concepts like monotonicity, continuity, and differentiation properties that activation functions should have, as well as popular methods for updating weights during training like SGD, Adam, etc.
Seminar on Digital Multiplier(Booth Multiplier) Using VHDLNaseer LoneRider
This is my Mini project. It is very clear and has lots of animation in it. If you like to know about booth algorithm and VHDL this the perfect presentation. Download it and see as SLIDE SHOW. You will enjoy more of my work, Give blessings.
The document discusses digital signal processing and the fast Fourier transform (FFT) algorithm. It explains that the FFT reduces the computational complexity of the discrete Fourier transform (DFT) by exploiting symmetry and periodicity properties. Specifically, the number of multiplications required for an N-point DFT using FFT is Nlog2N, compared to N2 for direct computation of the DFT. The document also describes decimation-in-time and decimation-in-frequency as two common FFT algorithms.
The document discusses latches and flip-flops. It describes how a latch stores a data value and is non-volatile, while a flip-flop is built from two back-to-back latches clocked on opposite phases. Various latch and flip-flop circuit designs are presented, including pass transistor, transmission gate, buffered input/output, and master-slave designs. Features like enables, resets, and sets are also discussed along with their synchronous and asynchronous implementations in flip-flops. Different flip-flop circuit styles are covered such as transmission gate, complementary CMOS, precharge, and self-gating designs.
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The document discusses the z-transform, which is the discrete-time equivalent of the Laplace transform. It defines the z-transform and provides examples of calculating the z-transform for various sequences, including the unit impulse, unit step function, sinusoids, and exponential sequences. It also discusses properties of the z-transform such as the region of convergence and relationship to the discrete-time Fourier transform.
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This document summarizes basic rate-changing components in multirate digital signal processing. It describes upsamplers and downsamplers, which increase or decrease the sampling rate by an integer factor. It also discusses the time-domain and frequency-domain models of upsamplers and downsamplers, and how they respectively introduce zero samples or skip samples while changing the sampling rate and bandwidth.
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Nyquist criterion for distortion less baseband binary channelPriyangaKR1
binary transmission system
From design point of view – frequency response of the channel and transmitted pulse shape are specified; the frequency response of the transmit and receive filters has to be determined so as to reconstruct [bk]
This document discusses Nyquist's criterion for distortionless transmission of binary signals over a baseband channel. It states that intersymbol interference (ISI) can be eliminated by choosing a transmit filter response P(f) that satisfies the Nyquist criterion. An ideal rectangular pulse shape meets the criterion but is physically unrealizable. A more practical raised cosine pulse is proposed, which introduces a rolloff factor to trade off excess bandwidth for slower decay. The full-cosine case provides additional zero-crossings that aid synchronization but doubles the bandwidth.
This document discusses efficient algorithms for computing the discrete Fourier transform (DFT), specifically the fast Fourier transform (FFT). It covers several FFT algorithms including decimation-in-time, decimation-in-frequency, and the Goertzel algorithm. The decimation-in-time algorithm recursively breaks down the DFT computation into smaller DFTs by decomposing the input sequence. This allows the computation to be performed in O(NlogN) time rather than O(N^2) time for a direct DFT computation. The document also discusses optimizations like in-place computation to reduce memory usage.
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This document contains questions from a M.Tech degree examination on digital system design using Verilog. The first question asks to design a night light logic circuit using a 2:1 multiplexer and modify it for multiple lamps. The second question asks to design a color selection logic for an inkjet printer to enable the correct driver based on a multibit color selection signal, including devising a minimal length code and handling an increased number of colors. In general, the document contains questions related to digital logic design, coding, and Verilog implementation.
The document describes the design of an arithmetic logic unit (ALU) for an embedded system as a final project. Key details include:
1. The ALU is designed with a 5-stage pipeline and performs operations like addition, subtraction, logical operations, and multiplication on 16-bit operands from registers.
2. It includes modules for basic logic functions like AND, OR, XOR, and NOT as well as a carry look-ahead adder and multiplier.
3. The project is implemented in Verilog HDL with modules, registers, and always blocks to control the flow through each pipeline stage on each clock cycle.
This document discusses various designs for digital multipliers. It begins by reviewing the basic building blocks used in digital circuits and how binary multiplication works by adding partial products. It then describes approaches for implementing multiplication, including right shift and add serial multipliers and faster parallel array and tree multipliers. Booth encoding is introduced as a technique to reduce the number of stages in a multiplier. Implementation details are provided for array and Wallace tree multipliers, including the use of compression cells like the (4,2) counter. Optimization goals for multipliers differ from adders in emphasizing reducing the critical path.
IRJET- Efficient Design of Radix Booth MultiplierIRJET Journal
The document proposes a method to optimize binary radix-16 Booth multipliers by reducing the maximum height of the partial product columns from (n + 1)/4 to n/4 for n-bit operands. This is achieved by performing a short carry-propagate addition in parallel to the regular partial product generation, which reduces the maximum height by one row. The method allows further optimizations in the partial product array reduction stage in terms of area, delay, and power. It can also allow additional partial products to be included without increasing delay. The method is generally applicable but provides the most benefit for 64-bit radix-16 Booth multipliers.
The document describes a design for a low power 32x32 multiplier that combines Booth and Vedic multiplication architectures. It partitions each 32-bit input into two 16-bit blocks, uses 16x16 Booth multipliers to generate partial products for each block, and employs 16x16 Vedic multipliers and carry select adders to add the partial products. This combined architecture achieves lower power and faster performance than individual Booth or Vedic multipliers. The design is implemented using Xilinx Vivado and evaluated for applications such as floating point multiplication.
Design and testing of systolic array multiplier using fault injecting schemesCSITiaesprime
Nowadays low power design circuits are major important for data transmission and processing the information among various system designs. One of the major multipliers used for synchronizing the data transmission is the systolic array multiplier, low power designs are mostly used for increasing the performance and reducing the hardware complexity. Among all the mathematical operations, multiplier plays a major role where it processes more information and with the high complexity of circuit in the existing irreversible design. We develop a systolic array multiplier using reversible gates for low power appliances, faults and coverage of the reversible logic are calculated in this paper. To improvise more, we introduced a reversible logic gate and tested the reversible systolic array multiplier using the fault injection method of built-in self-test block observer (BILBO) in which all corner cases are covered which shows 97% coverage compared with existing designs. Finally, Xilinx ISE 14.7 was used for synthesis and simulation results and compared parameters with existing designs which prove more efficiency.
Design and Implementation of a Programmable Truncated Multiplierijsrd.com
Truncated multiplication reduces part of the power required by multipliers by only computing the most-significant bits of the product. The most common approach to truncation includes physical reduction of the partial product matrix and a compensation for the reduced bits via different hardware compensation sub circuits. However, this result in fixed systems optimized for a given application at design time. A novel approach to truncation is proposed, where a full precision multiplier is implemented, but the active section of the partial product matrix is selected dynamically at run-time. This allows a power reduction trade off against signal degradation which can be modified at run time. Such architecture brings together the power reduction benefits from truncated multipliers and the flexibility of reconfigurable and general purpose devices. Efficient implementation of such a multiplier is presented in a custom digital signal processor where the concept of software compensation is introduced and analysed for different applications. Experimental results and power measurements are studied, including power measurements from both post-synthesis simulations and a fabricated IC implementation. This is the first system-level DSP core using a fine-grain truncated multiplier. Results demonstrate the effectiveness of the programmable truncated MAC (PTMAC) in achieving power reduction, with minimum impact on functionality for a number of applications. Software compensation also shown to be effective when deploying truncated multipliers in a system.
The proposed system is an efficient processing of 16-bit Multiplier Accumulator using Radix-8 and Radix-16 modified Booth Algorithm and other adders (SPST adder, Carry select adder, Parallel Prefix adder) using VHDL (Very High Speed Integrated Circuit Hardware Description Language). This proposed system provides low power, high speed and fewer delays. In both booth multipliers, comparison between the power consumption (mw) and estimated delay (ns) are calculated. The application of digital signal processing like fast fourier transform, finite impulse response and convolution needs high speed and low power MAC (Multiplier and Accumulator) units to construct an added. By reducing the glitches (from 1 to 0 transition) and spikes (from 0 to 1 transition), the speed of operation is improved and dynamic power is reduced. The adder designed with SPST avoids the unwanted glitches and spikes, reduce the switching power dissipation and the dynamic power. The speed can be improved by reducing the number of partial products to half, by grouping of bits in the multiplier term. The proposed Radix-8 and Radix-16 Modified Booth Algorithm MAC with SPST reduces the delay and obtain low power consumption as compared to array MAC.
This document discusses the implementation of a Radix-4 Booth multiplier using VHDL. It begins with an introduction to multipliers and their importance in digital circuits. It then provides background on Booth multiplication algorithms and related work that has been done to improve multiplier speed and efficiency. The methodology section describes the design of a configurable Booth multiplier that can detect the bit range of the operands and perform the multiplication accordingly in fewer cycles to reduce delay. Simulation results are provided to verify the operation of the Radix-4 Booth multiplier design for different input values.
This document describes a proposed VLSI implementation of a high-speed DCT architecture for H.264 video codec design. It presents a Booth radix-8 multiplier-based multiply-accumulate (MAC) unit to improve throughput and minimize area complexity for 8x8 2D DCT computation. The proposed MAC architecture achieves a maximum operating frequency of 129.18MHz while reducing area by 64% compared to a regular merged MAC unit with a conventional multiplier. FPGA implementation and performance analysis demonstrate the suitability of the proposed DCT architecture for applications in HDTV systems.
Design of a Novel Multiplier and Accumulator using Modified Booth Algorithm w...IRJET Journal
The document describes a novel design for a multiplier and accumulator (MAC) unit using the modified Booth algorithm and parallel self-timed adder (PASTA). The modified Booth algorithm reduces the number of partial products compared to a regular multiplication process, lowering delay. A carry save adder design is also proposed to further improve performance in terms of computation speed, power consumption, and area compared to a conventional design using the modified Booth algorithm. Simulation results show the proposed MAC design with PASTA has better performance and reduced area overhead and critical path delay compared to conventional methods.
Design and Implementation of Low Power DSP Core with Programmable Truncated V...ijsrd.com
The programmable truncated Vedic multiplication is the method which uses Vedic multiplier and programmable truncation control bits and which reduces part of the area and power required by multipliers by only computing the most-significant bits of the product. The basic process of truncation includes physical reduction of the partial product matrix and a compensation for the reduced bits via different hardware compensation sub circuits. These results in fixed systems optimized for a given application at design time. A novel approach to truncation is proposed, where a full precision vedic multiplier is implemented, but the active section of the truncation is selected by truncation control bits dynamically at run-time. Such architecture brings together the power reduction benefits from truncated multipliers and the flexibility of reconfigurable and general purpose devices. Efficient implementation of such a multiplier is presented in a custom digital signal processor where the concept of software compensation is introduced and analyzed for different applications. Experimental results and power measurements are studied, including power measurements from both post-synthesis simulations and a fabricated IC implementation. This is the first system-level DSP core using a high speed Vedic truncated multiplier. Results demonstrate the effectiveness of the programmable truncated MAC (PTMAC) in achieving power reduction, with minimum impact on functionality for a number of applications. On comparison with the previous parallel multipliers Vedic should be much more fast and area should be reduced. Programmable truncated Vedic multiplier (PTVM) should be the basic part implemented for the arithmetic and PTMAC units.
A Review of Different Methods for Booth MultiplierIJERA Editor
In this review paper, different type of implementation of Booth multiplier has been studied. Multipliers has great importance in digital signal processor, so designing a high-speed multiplier is the need of the hour. Advantages of using modified booth multiplier algorithm is that the number of partial product is reduced. Different types of addition algorithms are also discussed which are used for addition operation of multiplier.
Multipliers play an important role in today’s digital signal processing (DSP) and various other
applications. Multiplication is the most time consuming process in various signal processing operations like
convolution, circular convolution, auto-correlation and cross-correlation. With advances in technology, many
researchers have tried and are trying to design multipliers which offer either of the following- high speed, low
power consumption, regularity of layout and hence less area or even combination of them in multiplier. However
area and speed are two conflicting constraints. So improving speed results always in larger areas. So here we try
to find out the best trade off solution among the both of them. To have features like high speed and low power
consumption multipliers several algorithms have been introduced .In this paper, we describes Multipliers by using
various algorithm in VLSI technology.
International Journal of Computational Engineering Research(IJCER)ijceronline
This document summarizes a research paper on designing a high-speed arithmetic architecture for a parallel multiplier-accumulator based on the radix-2 modified Booth algorithm. It presents the design of a modified Booth multiplier using a carry lookahead adder for high accuracy with a fixed width. It also proposes a high-speed MAC architecture that improves performance by eliminating the accumulator and modifying the carry-save adder to add lower bits in advance, reducing the number of inputs to the final adder. The proposed CSA architecture can efficiently implement the operations of the new MAC arithmetic.
International Journal of Computational Engineering Research(IJCER)ijceronline
This document summarizes a research paper on designing a high-speed arithmetic architecture for a parallel multiplier-accumulator based on the radix-2 modified Booth algorithm. It presents the design of a modified Booth multiplier using a carry lookahead adder for high accuracy with a fixed width. It also proposes a high-speed MAC architecture that improves performance by eliminating the accumulator and modifying the carry-save adder to add lower bits in advance, reducing the number of inputs to the final adder. The proposed CSA architecture can efficiently implement the operations of the new MAC arithmetic.
International Journal of Engineering Research and DevelopmentIJERD Editor
This document describes a proposed VLSI architecture for an optimized low power digit serial finite impulse response (FIR) filter using multiple constant multiplications (MCM). It introduces an algorithm to optimize the area of digit serial MCM operations at the gate level by considering implementation costs of digit serial addition, subtraction, and shift operations. The proposed filter architecture aims to reduce area and power compared to designs using generic digit serial multipliers through the use of MCM blocks optimized for area. Experimental results indicate the algorithm leads to lower complexity digit serial MCM designs.
This document summarizes a research paper on designing low power digit serial finite impulse response (FIR) filters using multiple constant multiplication (MCM) techniques. It proposes an architecture that optimizes the area of digit serial MCM operations at the gate level by considering the implementation costs of digit serial addition, subtraction and shift operations. An algorithm is presented to design digit serial FIR filters under a shift-adds architecture to reduce area compared to designs using generic digit serial multipliers. Experimental results show the technique leads to lower complexity digit serial MCM designs.
IRJET - Design of a Low Power Serial- Parallel Multiplier with Low Transition...IRJET Journal
This document describes the design of a low power serial-parallel multiplier that uses a modified radix-4 Booth algorithm. It aims to improve performance over a standard serial-parallel Booth multiplier in terms of area, delay, and power. The proposed multiplier generates partial products conditionally, adding only non-zero Booth encodings and skipping zero operations to reduce transitions and increase throughput. It is implemented using FPGA technology to evaluate its utility for applications like digital signal processing and machine learning that require high performance multiplication.
This document discusses optimizing the gate-level area of digit-serial finite impulse response (FIR) filter designs using multiple constant multiplication (MCM) blocks. It introduces the problem of designing a digit-serial MCM operation with minimal area and presents algorithms to formalize the area optimization problem. Results show the proposed algorithms and digit-serial MCM architectures efficiently design digit-serial MCM operations and FIR filters with reduced area compared to existing approaches.
In present day MAC unit is demanded in most of the Digital signal processing. Function of addition and multiplication is performed by the MAC unit. MAC operates in two stages. Firstly, multiplier computes the given number output and the result is forwarded to second stage i.e. addition/accumulation operates. Speed of multiplier is important in MAC unit which determines critical path as well as area is also of great importance in designing of MAC unit. Multiplier plays an important roles in many digital signal processing (DSP) applications such as in convolution, digital filters and other data processing unit. Many research has been performed on MAC implementation. This paper provides analysis of the research and investigations held till now.
VHDL Implementation of High Speed and Low Power BIST Based Vedic MultiplierIRJET Journal
This document describes a VHDL implementation of a built-in self-test (BIST) based Vedic multiplier circuit that aims to achieve high speed and low power consumption. A linear feedback shift register (LFSR) based test pattern generator (TPG) is used to generate random test vectors for the circuit under test, which is a 4-bit Vedic multiplier. The proposed design is simulated using Xilinx tools and VHDL. Simulation results show the BIST-based Vedic multiplier operating along with the test vectors from the TPG. Power analysis on different FPGAs shows the design has low dynamic power consumption.
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PARTIAL PRODUCT ARRAY HEIGHT REDUCTION USING RADIX-16 FOR 64-BIT BOOTH MULTIPLIER.pptx
1. PARTIAL PRODUCT ARRAY HEIGHT REDUCTION USING
RADIX-16 FOR 64-BIT BOOTH MULTIPLIER
PRESENTED BY(A11 Batch):
G. NIRMALA (18MG1A0404)
M.LAKSHMI (18MG1A0416)
D.SURYA TEJA (18MG1A0432)
SK.MUJEEB (18MG1A0449)
G.TEJESH REDDY (18MG1A0435)
INTERNAL GUIDE:
Mr. K. RAMESH, M. TECH.
ASSISTANT PROFESSOR.
SREE VAHINI INSTITUTE OF SCIENCE AND TECHNOLOGY, TIRUVURU
DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERNG
2. CONTENTS
o ABSTRACT
o INTRODUCTION
o EXISTING SYSTEM
o PROPOSED SYSTEM
o BLOCK DIAGRAM
o SCHEMATIC
o WORKING
o SOFTWARE DEVELOPMENT
o EXPERIMENTAL RESULT
o ADVANTAGES
o APPLICATIONS
o FUTURE SCOPE
o CONCLUSION
o REFERENCES
3. ABSTRACT
Reduce the maximum height of the partial product columns to [n/4] for n = 64-bit unsigned
operand. This is in contrast to the conventional maximum height of [(n + 1)/4].
Therefore, a reduction of one unit in the maximum height is achieved.
It may allow further optimizations of the partial product array reduction stage in terms of
area/delay/power
Additional addends to be included in the partial product array without increasing the delay.
The multiplier algorithm is normally used for higher bit length applications and ordinary
multiplier is good for lower order bits.
These two methods are combined to produce the high speed multiplier for higher bit length.
The main objective of this research paper is to design architecture for radix-4 complex
Multiplier by rectifying the problems in the existing method and to improve the speed by using
the common Boolean logic (cbl).
4. INTRODUCTION
Binary multipliers are widely used building block element in the design of microprocessors
Embedded systems, and therefore, they are an important target for implementation
optimization.
Current implementations of binary multiplication follow the steps of
1) Recoding of the multiplier in digits in a certain number system
2) Digit multiplication of each digit by the multiplicand, resulting in a certain number of
partial products
3) Reduction of the partial product array to two operands using multi operand addition
techniques and
4) Carry propagate addition of the two operands to obtain the final result.
5. EXISTING SYSTEM
The problem of existing architecture is reduced by removing bits from the remainders. To
reduce the maximum height of the partial product bit array we perform a short carry
propagate addition in parallel to the regular partial product generation.
This short addition reduces the maximum height by one row and it is faster than the
regular partial product generation.
The elements of the bit array to be added by the short adder the resulting partial product bit
array after the short addition.
Comparing both figures, we observe that the maximum height is reduced from 17 to 16 for
n = 64.
6. PROPOSED SYSTEM
In this work, we have presented a method to reduce by one the maximum height of the
partial product array for 64-bit radix-16 Booth recoded magnitude multipliers.
This reduction may allow more flexibility in the design of the reduction tree of the
pipelined multiplier.
We have shown that this reduction is achieved with no extra delay for n ≥ 32 for a cell-
based design.
The method can be extended to Booth recoded radix-8 multipliers, signed multipliers
and combined signed/unsigned multipliers
7. BLOCK DIAGRAM
High level view of the recoding and partial product generation stage
including our proposed scheme
8. WORKING
Booth's multiplication algorithm is a multiplication algorithm that multiplies two signed
binary numbers in two’s complement notation
Partial product for multiplying two or three digit numbers in columns that can be easier
by making use of standard algorithm of multiplication.
In a large number multiplication grouping the number to multiply into parts, multiply
the parts separately, and then add.
A product formed by multiplying the multiplicand by one digit of the multiplier when
the multiplier has more than one digit.
Partial products are used as intermediate steps in calculating larger products. The partial
product to solve a multiplication equation can set it up like a traditional long
multiplication equation.
10. SOFTWARE DEVELOPMENT
XILINX ISE SOFTWARE
The Integrated Software Environment (ISE™) is the Xilinx® design software suite that allows you to take your
design from design entry through Xilinx device programming. The ISE Project Navigator manages and
processes your design through the following steps in the ISE design flow.
1.DESIGN ENTRY: Design entry is the first step in the ISE design flow. During design entry, you create your
source files based on your design objective.
2. SYNTHESIS: After design entry and optional simulation you run synthesis
3. IMPLEMENTATION: After synthesis, you run design implementation, which converts the logical design
into a physical file format that can be downloaded to the selected target device.
4.VERIFICATION: You can verify the functionality of your design at several points in the design flow.
You can use simulator software to verify the functionality and timing of your design or a portion of your design
5.DEVICECONFIGURATION: After generating a programming file, you configure your device.
During configuration, you generate configuration files and download the programming files from a host
computer to a Xilinx device.
15. ADVANTAGES
This technique is general, but its impact (reduction of one row without increasing
the critical path of the partial product generation stage) depends on the specific
timing of the different components.
Therefore, it cannot claim a successful result for all practical values of r and n
and different implementation technologies.
Thus, it concentrates on a specific instance: a 64-bit radix-16 Booth recoded
unsigned multiplier implemented with a synthesis tool and a standard-cell library.
Therefore by using radix-16 since it is the most complex case, among the practical
values of the radix, for the design of our scheme.
16. APPLICATIONS
It has the most basic advantage in digital signal processing.
It is used along with multiplier-accumulator (MAC)that reduces the partial
derivatives of multiplication product with ease in circuitry.
It increases the efficiency of the system by enhancing its speed.
Better performance in low cost at low power consumption.
17. FUTURE SCOPE
we will extend an optimization for binary radix-32 (modified) Booth recoded
multipliers to reduce the maximum height of the partial product columns to [n/4]
for n = N-bit unsigned operands.
This is in contrast to the conventional maximum height of [(n + 1)/4]. Therefore, a
reduction of one unit in the maximum height is achieved.
This reduction may add flexibility during the design of the pipelined multiplier to
meet the design goals, it may allow further optimizations of the partial product
array reduction stage in terms of area/delay/power and/or may allow additional
addends to be included in the partial product array without increasing the delay.
The method can be extended to Booth recoded radix-8 multipliers, signed
multipliers, combined signed/unsigned multipliers, and other values of n.
18. CONCLUSION
Pipelined large word length digital multipliers are difficult to design under the constraints of
core cycle time (for nominal voltage), pipeline depth, power and energy consumption and area.
Low level optimizations might be required to meet these constraints.
In this work, we have presented a method to reduce by one the maximum height of the partial
product array for 64-bit radix-16 Booth recoded magnitude multipliers.
This reduction may allow more flexibility in the design of the reduction tree of the pipelined
multiplier. We have shown that this reduction is achieved with no extra delay for n ≥ 32 for a
cell-based design.
The method can be extended to Booth recoded radix-8 multipliers, signed multipliers and
combined signed/unsigned multipliers. Radix-8 and radix-16 Booth recoded multipliers are
attractive for low power designs, mainly to the lower complexity and depth of the reduction tree,
and therefore they might be very popular in this era of power-constrained designs with
increasing overheads due to wiring.
19. REFERENCES
[1] Weiqiang Liu, Liangyu Qian, Chenghua Wang, and Jie Han “Design of Approximate Radix
4 Booth Multipliers for Error-Tolerant Computing ,” IEEE Trans
[2] F. Lamberti et al., “Reducing the computation time in (short bit-width) twos complement
multipliers,” IEEE Trans. Comput., vol. 60, no. 2, pp. 148– 156, Feb. 2011.
[3] N. Petra et al., “Design of fixed-width multipliers with linear compensation function,”
IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 58, no. 5, pp. 947–960, May 2011.
[4] S. Galal et al., “FPU generator for design space exploration,” in Proc. 21st IEEE Symp.
Comput. Arithmetic (ARITH), Apr. 2013, pp. 25–34.
[5] K. Tsoumanis et al., “An optimized modified booth recoder for efficient design of the add-
multiply operator,” IEEETrans.Circuits Syst.I,Reg. Papers, vol. 61, no. 4, pp. 1133–1143, Apr.