 Open Access
 Authors : Sulbha Bhongale, Rajendra Patel
 Paper ID : IJERTV12IS060150
 Volume & Issue : Volume 12, Issue 06 (June 2023)
 Published (First Online): 29062023
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Implementation of Radix4 (32 bit) Booth Multiplier using VHDL
Sulbha Bhongale *1, Rajendra Patel2,
Department of electronics and communication
1, 2 Infinity Management & Engineering College, Sagar
Abstract –
Area, power, delay is the key design factor in VLSI circuit. In the design of microprocessor, graphical systems, multimedia systems, and DSP system Multiplier is a very essential part.
Multiplication alone uses about 15% of the IC's total power. It is crucial to have a design that is efficient in terms of speed, area, and performance of the multiplier, and for the same, Booth's multiplication algorithm offers a relatively simple strategy with a noticeable improvement in performance.
In this article, we offer a signed radix4 Booth multiplier implementation and design. There have been discussions about a number of collation parameters, including the Wallace tree multiplier, the Booth radix2, and the Modified radix4 Booth multiplier. A different multiplier is compared to the radix4 Booth 32bit implementation.
Any multiplier has limitations, one of which is the quantity of partial products. This limitation is overcome by the multiplication Booth algorithm, which allows 32bit multiplication to be calculated in 16 latches. In this study, we present a multiplier that, with reduced latency, minimises the partial product row via improved booth technique. We created a radix4 booth multiplier, and the software Xilinx ISE 14.4 was used to synthesise it. By multiplying two 32bit signed numbers, the design has been confirmed. The results are compared to published literature. Utilising the most recent version of the Xilinx ISE Design Suite tool, the simulation and synthesis results are provided.
Keywords –
Radix4 Booth multiplier, radix2, Ripple carry adder, Xilinx 14.4.
INTRODUCTION
The multiplication is one of the most important arithmetic operations and it is an important factor in signal processing algorithm, in digital signal processor, multimedia and microprocessor application. The highspeed processing is the key requirement of modern days applications. In many Digital Signal Processing (DSP) applications, including convolution, fast Fourier transform (FFT), filtering, in the arithmetic and logic units of microprocessors, and in graphics processors, multiplicationbased operations like multiply and accumulate (MAC) and inner product are currently implemented [1]. The most commonly used elements in any digital circuit design are digital multipliers. They are used to carry out any operation since they are quick, dependable, and effective components. The addandshift method is used for multiplication.
The multiplicand is a number which is to be added and number of times it is called as multiplier. The repetitive addition is considered as slow multiplication. The speed of multiplier is a key factor in digital signal processing and generalpurpose processor. The design of multiplier inconsistent with delay, time, area, size to minimum power dissipation. Therefore, continue high performance, the speed and area of multiplier are major design issues and they need to be improved for enhancing performance [2]
The architecture of multiplier can be divided into three stages namely, partial product generation stage, partial product reduction stage and final addition of reduced partial products. Partial products are generated by multiplying the multiplicand with each bit of the multiplier. Generally, multiplication is done by generating partial products and then added. The speed of the multiplier depends on fast the generation of partial products and addition of these partial products. The reduction of no of partial product and use of efficient adder circuit for speeding up the process is the desired approach. Booth multiplier offers an advantage and accomplish both the steps and hence performance can be improved. In addition, the Radix4 Booth multiplier can be performed both signed and unsigned number [3]. The aim of booth algorithm scheme to increase the speed of multiplication process, as compared to conventional multiplier such as array multiplier, Wallace tree multiplier, radix2 multiplier. Booth multiplication helps to reduce the number of iteration step and result is faster computation. In this paper we present 32bit multiplication by using Modified Booth (radix4) algorithm and its implementation using VHDL platform [4].
LITERATURE REVIEW
Various multiplier is contrast by using commercial variable like delay, power, and area etc to interpret which multiplier was applicable for situation. After contrast all, it completes that radix4 booth multiplier was applicable for situation where speed is the critical concern [5]. Implementation of radix4 booth multiplier is done by using ripple carry adder and came to concluded that parallel multiplier is much better than serial multiplier due to less area occupied and low power consumption.
The Booth algorithm was invented by Andrew D. Booth [6]. The radix2 algorithm treats both signed and unsigned number. Partial product is generated by using this algorithm, which firstly encode the multiplier bits. The radix2 and radix4 are two algorithms to obtained partial product generation and reduce no of partial product and final addition of reduced partial product.
In various DSP application, all of multiplier output bits were not used, but only upper bits of output were used. Kidmahi proposed prune unsigned multiplier for this idea. This prune scheme can be applied for booth multiplier which can be used in real DSP system more efficiently. Also prune booth multiplier guaranteed 0 input to 0 output that was not provided before. Prune booth multiplier reduced about 37 to 48 % of area and about power consumption.
Several designs and methods are suggested to begin a modular point of view for enhancing power efficiency [7]. It initiates that algorithmbased structure decrease gate switching activity considerably. As a result, power consumption in multiplier is reduced [8]. It is initiate that data complication various distribution of gate level digital circuit has significant effect on power dissipation. Regarding this actual design of the chip can be employed by using historical algorithm by analysing placement option subject to optimum space allocation similarly selection booth algorithm and modified booth algorithm may reduce power consumption as outcome of data complexity. It is initiate that in multiplier circuit, Modified booth algorithm reduces power consumption as compared to other method of multiplication on making implementation of radix4 booth multiplier has power reduction than the conventional multiplier.
Booth multiplier can be composed based on dynamic span observation of multiplier and which can be compose either 32bit multiplication.
In reported literature, several approaches and designs have been reported [912]. 16 bits Booth multipliers with radix2, radix4 methods using VHDL have been demonstrated and their performance have been analysed. Niharika and Dilip Kumar, Analysis of Booth multiplier using radix2 and radix4 using VHDL. They have demonstrated 16bit Booth multiplier using Radix2 and radix 4 method. The designed has been proposed which can raise the performance in term of the area, and the speed of the multiplier.
The optimization of digital circuit can lower the number of transistors resulting into reduced area on the chip. Thus, the low power consumption can also be achieved [6].
Sagar Pradhan et al have proposed the design and implementation of low power and area efficient 32bit multiplir. They have demonstrated two high speed multipliers, namely, Modified booth multiplier (MBM) and Wallace tree multiplier are combined with carry select adder and then they formed a hybridized that gives high speed computation and power consumption is low. Wallace tree multiplier is maintained for fast addition and then last CSA is used for final accumulation. By considering factor such as speed, power, the hybrid multiplier gives better result than conventional design. The proposed design considering the term such as delay, area, power, and their product by hand with logical effort and through custom design. It has been proposed that the Booth multiplier with Wallace tree implementation energy efficient relative to the other proposed design.
Manpreet Manna & Sukhmeet kaur, have also carried out the implementation of Modified Booth algorithm (radix4) and compare with Booth algorithm (radix2). They proposed the method of implementing the parallel MAC with the smallest possible delay. A new architecture of multiplier and accumulator (MAC) for highspeed arithmetic by combining multiplication with accumulation and devising a carry look ahead adder (CLA). Modified Booth multiplication algorithm is designed using high speed adder. High speed adder is used to speed up the operation of multiplication. The radix4 booth multiplier has higher computation speed than radix2 booth multiplier, coding of both the multiplier is done in VHDL and simulated using Xilinx ISE 9.1i software has been used and implemented on FPGA Xc35505pq208 [13].
A demonstrative radix2 Booth Multiplier is described below. The faster and smaller multiplication of 2 complement of binary integer are employ by Booth technique. Therefore, Multiplication completed by Booth recoding algorithm. So first we have to recode the multiplier first. Every bit of recoded multiplier can take any value from 0, 1,1. In order to obtain, 2 bits of multiplier are conduct at a time by conjoining technique. Thus, in radix2, clustering of multiplier bits start from LSB for which the first block utilizes only single bit of multiplier and one more bit supposed to be zero [5]. By using following steps recoded multiplier for radix 2 are obtained.

Add a zero to the LSB side of given multiplier.

By using conjoining technique, clustering of two bits of multiplier and recode the number using following table.
Table 1: Truth table for Booth Multiplier
Figure 1: Flow chart of Booth Multiplier
Procedure;
M is the multiplicand Q is the multiplier
Consider 1 bit register Q1 and initialize to 0 Consider register A and initialize to 0 Condition;

If Q0, and Q1 are same 11 or 00 then perform arithmetic right shift by 1 bit

If Q0 and Q1 = 01 then perform A = A + M and then perform arithmetic right shift

If Q0 and Q1 = 10 then perform A = A – M then perform arithmetic right shift

Example 5 x 4 = 20

Multiplicand = 5

Multiplier = 4
Table 2 presents step by step implementation of multiplication (5 x 4) of the two numbers by Booth approach Table 2: Multiplication of two number (5*4) in step by step using Booth algorithm

The main version of Booth algorithm (radix2) has certain drawbacks. It becomes inconvenient while designing parallel multiplier and there is a string of acute 1s, the algorithm becomes ineffective. These drawbacks are overcome by using Modified Booth algorithm
Modified Radix4 Booth Multiplier
One more advancement in the multiplier is by reducing the number of partial products. To attain high speed multiplication, algorithm using parallel counter like modified booth algorithm has been exhibit and accomplished. It decreases the number of partial products and reduces number of multiplicands [9]. It groups three consecutive bits at a time, in that threebit 2 bits on left hand side are present bit and third bit is the carry bit from previous pair of bits. The multiplier performs much faster than array multiplier for extended operands because its time to ended the operation is proportional to the log of the word length of operand. The major advantage of this multiplier is that if the successive bits in multiplicand are same, then addition can be skipped [14]. The number of partial products is downed by a factor of half by using the technique of booth recoding [15]. The grouping of bits and booth recording determine the number of partial products.
Figure 2: Grouping of bits from the multiplier term
In a given multiplier, the group of three bits are considered, it initiates from the LSB bit and first group is formed by two bits, onward consider a group of three bits in which one bit will be overlapped on the previous group as shown in figure 4. Thus, groups multiplier will result in the production of bits between these five bits are as follows 2, 1, 0, +1, and +2 which are tabulated in table 3.
Table 3: Recoding table of radix4 Booth multiplier
Block 
Recoded digits 
Operation 
000 
0 
0 
001 
+1 
+1 
010 
+1 
+1 
011 
+2 
+2 
100 
2 
2 
101 
1 
1 
110 
1 
1 
111 
0 
0 
Algorithm:

Check in case the multiplier and multiplicand are having negative sign if so convert 2s complement

Add implied zero to the multiplier and then convert the multiplier into other form using Booth recoding table shown in above table

Now multiply the multiplier with multiplicand

If the multiplier was zero then the multiplicand multiplied with zero will result in the output is zero

If the multiplier was +1 then, the result of the multiplicand with the multiplier was same as the multiplicand

If the multiplier was 1 then, the result was calculated by 2s complement of the multiplicand


Finally add all generated partial products in ripple carry adder.
Methodology
From the principle of Booth Multiplication, it can be manifested that the addition/subtraction operation can be omitted if the consecutive bits in the multiplicand are same. If 3 consecutive bits are same then addition/subtraction operation can be omitted. therefore, normally the delay related with Booth Multiplication are smaller than that with Array Multiplier. Although, the execution of Booth Multiplier for delay is input data dependant. In the worst case the delay with booth multiplier is on par with Array Multiplier [16]. The method of Booth recording reduces the numbers of adders and hence the delay required to produce the partial sums by examining three bits at a time. The high performance of booth multiplier comes with the drawback of power consumption. The reason is large number of adder cells required that consumes large power [3].
Implementation and Result
Multiplication based on Radix4 Booth algorithm has been simulated on ISIM simulator of Xilinx 14.4 software and it is shown in Fig. 3 and implemented on FPGA platform for which the results are obtained.
Figure 3: Properties of Xilinx 14.4i Module
In the design of Booth multiplier for Radix4, there are two inputs namely Multiplier [32:0] and Multiplicand [32:0] and single output product [64:0] and the carry output. The schematic is presented in Fig. 4.
Figure 4: shows RTL of Booth Multiplier using Radix4
Output is verified for input of different ranges. In this case the simulation results are shown for input value in binary for multipier 0000000000000000000000000000011 and a multiplicand 0000000000000000000000000010000. The result of above multiplication is given in binary form and the product is 000000000000000000000000000000000000000000000000000000000110000. The simulation result is presented.
Synthesis result of radix4
The multiplier has been synthesized using Xilinx ISE 14.4. The RTL schematic of Booth multiplier using Radix4 has been shown in Fig.5
Figure 5: shows simulation result using Radix4
In the RTL schematic of Booth multiplier multiplicand [32:0] and multiplier [32:0] represent the 32 bit input operands and product [64:0] represent 64 bit output product. Figure shows the internal RTL of the Booth multiplier and figure the device utilization of Booth multiplier using Radix4.
Figure 6: shows Internal RTL of Booth Multiplier using Radix4
The delay analysis is performed using various device family. The details are tabulated in table 46 as shown below.
Table 4: Device utilization of Booth Multiplier using Radix4
Table 5: Shows Device utilization proposed Radix4 Booth multiplier
Device & Family 
Vertex 5 
Vertex 6 
Vertex 7 
Spartan 3 
Spartan 6 
Number of slices 
2522 
2514 
2514 
1881 
2514 
Number of fully used LUTFFF pair 
0 
0 
0 
3542 
0 
Number of Bonded IOB 
129 
129 
129 
129 
129 
Table 6: Comparison table of Radix4 Booth Multiplier
Design 
Proposed Radix4 Booth Multiplier 
Reference paper number 2 

Device and Family 
Vertex 5 
Vertex 7 
Spartan 3 
Spartan 3 
Delay(ns) 
35.80ns 
26.32ns 
111.29ns 
500ns 
Conclusion and future work:
After doing some work and observed a lot of difficulty, we maintain to accomplish the main aim of the work is to implement Booth algorithm for the design of radix4 Booth multiplier using ripple carry adder architecture and performed delay analysis using various device family that shown in table 46. We evaluate the time delay in design with Ripple carry adder. Design and implementation of radix4 Booth multiplier using VHDL and synthesized results were presented. The Radix4 Booth multiplier have been simulated using Xilinx platform and implemented on vertex XC7z010Zcg400. The proposed Modified Booth Multiplier has taken less computation time to multiply the two 32 bits binary number and analysis shows that the delay obtained for Modified Booth multiplier using RCA is 26.32 ns which is quite low compared with reported literature [1]. Ramteke et al reported delay of 500ns to complete 8bit multiplication. It has proved that that it can be useful to apply a Radix4 Booth Multiplier in highspeed multiplier because gain in time obtained due to reduction in partial product to N/2, by using efficient encoding method, which save multiplier area and reduced delay at the same time. The highspeed implementation of such multiplier has wide range of application in image processing, arithmetic logic unit and VLSI signal processing. In future the algorithm can be extended to Radix8 for which complexity is somewhat high, but the generated partial product will reduce to N/3.
REFERENCES:
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