Synthesis And Simulation Of 8×8-Bit Modified Booth’s Multiplier

Synthesis And Simulation Of 8×8-Bit Modified Booth’s Multiplier

Lakhvinder Kaur

M.Tech Student(ECE), YCOE, Punjabi University Guru kashi Campus , Talwandi Sabo 151302 , Punjab(India)

Parminder Singh Jassal

Assistant Prof(ECE), YCOE, Punjabi University Guru kashi Campus , Talwandi Sabo 151302 , Punjab (India)

Abstract

The booths algorithm is the most frequently used method for multiplication. This algorithm allows the reduction in the number of partial products to be compressed in the carry adder. Thus the compression speed can be enhanced. In this paper a multiplier, using Modified Booths multiplication algorithm is developed. After applying modified Booths algorithm to the inputs, simple operations like twos complement or shifting plus twos complement by proceeding simple arithmetic shift are done. The number of partial products are reduced by N/2 by using this algorithm.

Key words: Xilnx 9.2i, Model-sim XE-II 5.7g, VHDL.

1. INTRODUCTION

   A binarymultiplier is an electronic

   circuit used in digital electronics, such as a computer, to multiply two binary numbers. The multiplier is a device which is used to perform the multiplication operation. The adders and multipliers are the basic components of designing of communication circuits. The multiplier is the basic key component of any digital signal processing system[1],[2]. Multiplication includes two basic steps, generation of partial products and their accumulation[2]. Multipliers are key components of many high performance systems such as FIR filters, microprocessors, digital signal processors and multimedia applications[3]. The type of the multiplier used for an application is based upon the requirements of the application. There are different types of

   multiplier available according to the requirements. The difference is in the way in which data is processed for the multiplication, the examples are serial multipliers, parallel multipliers and serial- parallel multipliers[18]. Using VHDL language a single component can be described using all three styles of modeling[22].

2. MODIFIED BOOTH MULTIPLIER

   The Modified Booth multiplier is an extension of Booths multiplier. In Modified Booth, the number of partial products reduced by N/2, that is half of total partial products as compare to simple multiplication process[4]. So, clearly if the number of partial products become reduced, the area of the multiplier also will reduced and automatically as the result of it, the speed will increased. So, this multiplier is more efficient.

3. MODIFIED BOOTH ALGORITHM

   The Modified Booth algorithm is the most frequently used method to generate partial products. The partial products are reduced by N/2 by using this algorithm. So as the result of this, the multiplier can be implemented using less hardware components as compare to conventional multiplier. This algorithm can save multiplier layout area and reduces delay at the same time which are the important design advantages[1]. One of the method for high speed multiplier is to enhance the

   parallelism by reducing the number of calculating stages .Booths encoding reduces partial products to N/2. It converts the multiplier from radix-2 to radix-4 using redundant digit set {-2, -1, 0, 1, 2}. So in new multiplier with radix-4 there are only N/2 digits. New multiplier's P digits are defined by following formula:

   Pj = Y2j + Y2j-1 -2Y2j+1 with Y-1 = 0

   The Modified Booth Algorithm takes following steps for multiplication:

   • Modified Booth Encoder.

   • Partial Product Generator.

   • Sign Extension

     1. MODIFIED BOOTH ENCODER (MBE)

        Modified Booth encoding is most often used to avoid variable size partial product arrays. Before designing a MBE, the multiplier B has to be converted into a Radix-4 number by dividing them into three digits respectively according to Booth Encoder Table given afterwards. Prior to convert the multiplier, a zero is appended into the Least Significant Bit (LSB) of the multiplier. approach Instead of eight partial products being generated using conventional multiplier. Table 1. shows the truth table for a Booth encoder. The encoder takes inputs YN+1, YN and YN-1 from the multiplier bus and produces a 1 or a 0 for each operation: single, double, negative(X, 2X and Neg).

        Figure 1. Grouping of bits for MBE scheme[6]

        This Booth multiplier technique is to increase speed by reducing the number of partial products by half. The operand that is

        booth encoded is called multiplier, and the other operand is called multiplicand.

        Table.1. Truth Table For Booth Encoder

        YN +1	YN	YN-1	Operati on	X	2 X	Ne g
        0	0	0	+0 x X	0	0	0
        0	0	1	+1 x X	1	0	0
        0	1	0	+1 x X	1	0	0
        0	1	1	+2 x X	0	1	0
        1	0	0	-2 x X	0	1	1
        1	0	1	-1 x X	1	0	1
        1	1	0	-1 x X	1	0	1
        1	1	1	-0 x X	0	0	1

        In most of the cases MBE scheme is used for generating PP, because of its ability to reduce the number of PP by half[7]. The truth table shows the function of booth encoder. If a 3-bit binary input sequence is given at the input, and perform the operation as mentioned infront of it, the partial products will be generated.

        Figure 2. Booth Encoder[2]

        Figure 3. Booth Decoder[2]

        As shown in the figure 2 and 3, the booth encoder and decoder cirrcuit is shown. As mentioned in figure 1, there are total 3

        combinations for generating a partial product i.e, the obtained partial product is dependent upon the combination of three binary bits of the multiplier.

     2. PARTIAL PRODUCT GENERATOR (PPG)

        Partial product generator is the combination circuit of the product generator. Product generator is designed to produce the product by multiplying the multiplicand X by 0, 1, -1, 2 or -2. For product generator, multiply by zero means the multiplicand is multiplied by 0. Multiply by 1 means the product still remains the same as the multiplicand value. Multiply by -1 means that the product is the twos complement form of the number. Multiply by -2 is to shift left one bit the twos complement of the multiplicand value and multiply by 2 means just shift left the multiplicand by one place.

     3. SIGN EXTENSION CORRECTOR

     Sign Extension Corrector is designed to enhance the ability of the booth multiplier to multiply not only the unsigned number but as well as the signed number. As shown in Table 3.2 when bit 7 of the multiplicand X(X7) is zero(unsigned number) and Yn+1 is equal to one, then sign E will have one vale (become signed number for resulted partial product).

     Table 2. Truth Table for Sign Extension when X7 is zero.

     X7	YN+1	YN	YN-1	E
     0	0	0	0	0
     0	0	0	1	0
     0	0	1	0	0
     0	0	1	1	0
     0	1	0	0	1
     0	1	0	1	1
     0	1	1	0	1
     0	1	1	1	0

     It is the same when the bit 7 of the multiplicand X(X7) is one (signed number) and Yn+1 is equal to zero, the sign E will have a new value.

     Table 3. Truth Table for Sign Extension when X7 is one.

     X7	YN+1	YN	YN-1	E
     1	0	0	0	0
     1	0	0	1	1
     1	0	1	0	1
     1	0	1	1	1
     1	1	0	0	0
     1	1	0	1	0
     1	1	1	0	0
     1	1	1	1	0

     However when both the value of A7 and Yn+1 are equal either to zero or one, the sign E will have a value zero(unsigned number). For the case when all three bits of the multiplier value Yn+1, Yn and Yn-1 are equal to zero or one, the sign E will direct have a zero value independent to the X7 value[8].

4. RESULTS

1. RTL view

   Figure 4. Top module of 8 x 8 multiplier

   Figure 5 RTL schematic of 8 x 8 bit multiplier

   As shown, a and b are the 8-bit inputs for the multiplier. The signals clk and

   reset are also input signals.

2. Simulations results

In this work, the Xilinx ISE and Model- Sim simulator are used. Here shows the outputs taken after the simulation on both of the simulators.

For the given inputs as folows: Input a(7 to 0) : 10111011(binary) Input b(7 to 0) : 01111011(binary) output: 0101100111011001(binary).

The output is verified on both the simulators and that is same.

Figure 5. Simulation results of 8×8 bit multiplier for given input.

Figure 6. Simulation results of 8×8 bit multiplier for given input.

The following results are also verified. Input a(7 to 0) : 11110001(binary) Input b(7 to 0) : 01100111(binary) Output : 0110011110001110(binary)

The power consumption for the design is 121 mW, with a thermal temperature of 28 0 C. The table 4 shows the results for the 8- bit multiplier compared with the reference work[9].

Selected device – 3s500ecp132

Speed grade – -4

Table 4. Comparison of results

REFERENCES

1. C.-Y. Lee And C.W. Chiou, Low-Complexity Bit- Parallel Multipliers For A Class Of Gf(2m) Based On Modified Booths Algorithm, International Journal Of Computers And Applications, Vol. 29, No. 4, 2007.

2. Shiann-Rong Kuang, Modified Booth Multipliers With A Regular Partial Product Array, IEEE Transactions On Circuits And SystemsIi: Express Briefs, Vol. 56, No. 5, May 2009.

3. Soojin Kim And Kyeongsoon Cho, Design Of High- Speed Modified Booth Multipliers Operating At Ghz Ranges, World Academy Of Science, Engineering And Technology, 2010.

4. J. Bhasker, A VHDL Primer,third edition, 2008.

5. Li-Rong Wang, Shyh-Jye Jou And Chung-Len Lee, Awell-Structuredmodified Booth Multiplier Design, 2008 IEEE.

6. Ki-Seon Cho, Jong-On Park, Jin-Seok Hong, Goang- Seog Choi, 54×54-Bit Radix-4 Multiplierbased On Modified Booth Algorithm, GLSVLSI03, April 28-29, 2003, Washington, Dc, Usa.

7. Jung-Yup Kang, A Simple High-Speed Multiplier Design, IEEE Transactions On Computers, Vol. 55, No. 10, October 2006..

8. P. Assady, A New Multiplication Algorithm Using High-Speed Counters, European Journal Of Scientific Research, ISSN 1450-216x Vol.26 No.3 (2009), Pp.362- 368.

9. G. Jaya Prada1, N.C. Pant, Design And Verification Of Faster Multiplier, International Journal Of Engineering Research And Applications (IJERA), Vol. 1, Issue 3, Pp.683-686, 2009.