INTERNATIONAL JOURNAL OF ELECTRONICS AND COMMUNICATION ENGINEERING & TECHNOLOGY (IJECET) International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 ISSN 0976 6464(Print) ISSN 0976 6472(Online) Volume 4, Issue 5, September October, 2013, pp. 132-142 IAEME: www.iaeme.com/ijecet.asp Journal Impact Factor (2013): 5.8896 (Calculated by GISI) www.jifactor.com IJECET I A E M E STUDY ON COMPARISON OF VARIOUS MULTIPLIERS V.Priyanka Brahmaiah 1, L.Dharma Teja 2, Dr.Y.Padma Sai 3 1 ECE Department, VNR Vignana Jyothi Institute of Engg. & Technology, Hyderabad, A.P) 2 (ECE Department, VNR Vignana Jyothi Institute of Engg. & Technology, Hyderabad, A.P) 3 (ECE Department, VNR Vignana Jyothi Institute of Engg. & Technology, Hyderabad, A.P) ABSTRACT Low power consumption and smaller area are some of the most important criteria for the fabrication of DSP systems and high performance systems. Optimizing the speed and area of the multiplier is a major design issue. This paper aims at an efficient implementation of high speed multiplier using the shift and add method, Radix_2, Radix_4 modified Booth multiplier algorithm. The low power consumption quality of booth multiplier makes it a preferred choice in designing different circuits. The result of this paper helps to choose a better option between serial and parallel multiplier in fabricating different systems. The proposed architecture is implemented in mentor graphics tools and targeted to Spartan III FPGA kit. Finally it has been proved that booth multiplier is more efficient than other multipliers in terms of speed and power consumption. Keywords: Multipliers, Radix-2, Radix-4 Booth Multiplier. INTRODUCTION Sizes of devices and the number of transistors are scaling as per Moore s Law. The sources of power consumption on a CMOS chip can be classified as static and dynamic. The actual effort of the circuit to switch is being the dominant component dynamic. A first order approximation of the dynamic power consumption of CMOS circuitry is given by the formula: Where P is the power, C is the effective switch capacitance, V is the supply voltage, and f is the frequency of switching operation. P = C*V2*f 132
The power dissipation arises from the charging and discharging of the circuit node capacitances found on the output of every logic gate. Power management is the careful planning of power budget for every subsystem of a VLSI chip. This is especially important issue for today s complex systems. The most important and successful use of power management is to deactivate a portion of circuit when its computation is not required. Every low-to-high logic transition in a digital circuit incurs a change of voltage, drawing energy from the power supply. A designer at the technological and architectural level can try to minimize the variables in these equations to minimize the overall energy consumption. However, power minimization is often a complex process of trade-offs between speed, area, and power consumption. The current work proposes reduction of dynamic switching power, to reduce switching activity and reduction of gate counts in 16*16 complex multiplier by using higher order compressors. Multipliers require high amount of power and delay during the partial products formation and addition of the numbers. Time to market, rapid technology advancement, increase in chip size, and complexity of digital systems are some of the factors that led to the development of hardware description languages (VHDL and Verilog) along with the automated logic synthesis tools. The VHDL program describing the digital system is technology independent, but the synthesis tool that generates the circuit requires a technology file attached to it. The absence of such file causes the problem of non-possibility of time delay extraction. In this work, the coding of different multipliers is written in VHDL. After the coding they are targeted on the FPGA and by using Xilinx Synthesis Report, the performance of these multipliers has been observed and power consumption is observed using power analyzer in Xilinx 10.1 package. SIMULATION IN TERMS OF TIME AND POWER CONSTRAINTS Multiplication is an important fundamental function in arithmetic operations. In fact, multiplication based operations such as Multiply and Accumulate (MAC) and inner product are among some of the frequently used computations-intensive arithmetic functions currently implemented in many Digital Signal Processors (DSP) applications such as convolution, Fast Fourier Transform (FFT), filtering and others. There are two ways to speed up either by reducing number of partial products and/or by accelerating accumulation. The three types of high-speed multipliers are Array multiplier, Booth multiplier Radix-2 and Radix-4. In this work, analysis is done on these multipliers and they are compared in terms of number of gates, LUT s used and power consumption. Among these multipliers, Booth Multiplier is optimized in terms of power consumption. MULTIPLIER Multipliers are key components of many high performance systems such as FIR filters, microprocessors, digital signal processors, etc. A system s performance is generally determined by the performance of the multiplier because the multiplier is generally the slowest clement in the system. Furthermore, it is generally the most area consuming. Hence, optimizing the speed and area of the multiplier is a major design issue. However, area and speed are usually conflicting constraints so that improving speed results mostly in larger areas. As a result, a whole spectrum of multipliers with different area-speed constraints has been designed with fully parallel. Multipliers at one end of the spectrum and fully serial multipliers at the other end. In between are digit serial multipliers where single digits consisting of several bits are operated on. These multipliers have moderate performance in both speed and area. Multiplication is achieved by adding a list of shifted multiplicands according to the digits of the multiplier. 133
ARRAY MULTIPLIER A Binary multiplier is an electronic hardware device used in digital electronics or a computer or other electronic device to perform rapid multiplication of two numbers in binary representation. It is built using binary adders. The rules for binary multiplication can be stated as follows 1. If the multiplier digit is a 1, the multiplicand is simply copied down and represents the product. 2. If the multiplier digit is a 0 the product is also 0. For designing a multiplier circuit we should have circuitry to provide or do the following three things: 1. It should be capable identifying whether a bit is 0 or 1. 2. It should be capable of shifting left partial products. 3. It should be able to add all the partial products to give the products as sum of partial products. 4. It should examine the sign bits. If they are alike, the sign of the product will be a positive, if the sign bits are opposite product will be negative. The sign bit of the product stored with above criteria should be displayed along with the product. From the above discussion the observation has been done that it is not necessary to wait until all the partial products have been formed before summing them. In fact the addition of partial product can be carried out as soon as the partial product is formed. Notations: a multiplicand b multiplier p Product Binary multiplication (eg n=4) p=a b an 1 an 2_a1a0 bn 1bn 2_b1b0 p2 n 1 p2 n 2_p1 p0 x x x x a x x x x b --------- x x x x b0a20 x x x x b1a21 x x x x b2a22 x x x x b3a23 x x x x x x x x p 134
BOOTH MULTIPLICATION ALGORITHM Figure 1: Array Multiplier Architecture Booth algorithm gives a procedure for multiplying binary integers in signed 2 s complement representation. The booth algorithm is illustrated with the following example: Example, 2 ten x (- 4) ten 0010 two * 1100 two MAKING THE BOOTH TABLE I. From the two numbers, pick the number with the smallest difference between a series of consecutive numbers, and make it a multiplier. i. i.e., 0010 -- From 0 to 0 no change, 0 to 1 one change, 1 to 0 another change, so there are two changes on this one. ii. 1100 -- From 1 to 1 no change, 1 to 0 one change, 0 to 0 no change, so there is only one change on this one. iii. Therefore, multiplication of 2 x ( 4), where 2 ten (0010 two) is the multiplicand and ( 4) ten (1100two) is the multiplier. II. Let X = 1100 (multiplier) Let Y = 0010 (multiplicand) Take the 2 s complement of Y and call it Y Y = 1110 III. Load the X value in the table. IV. Load 0 for X-1 value it should be the previous first least significant bit of X. V. Load 0 in U and V rows which will have the product of X and Y at the end of operation. VI.Make four rows for each cycle; this is because we are multiplying four bits numbers. 135
U V X X-1 0000 0000 1100 0 Load the value 1 st cycle 2 nd cycle 3 rd cycle 4 th cycle Table 1: Step1 in Radix-2 Booth Multiplication BOOTH ALGORITHM Booth algorithm requires examination of the multiplier bits, and shifting of the partial product. Prior to the shifting, the multiplicand may be added to partial product, subtracted from the partial product, or left unchanged according to the following rules: Look at the first least significant bits of the multiplier X, and the previous least significant bits of the multiplier X - 1. I. 0 0 Shift only II. 1 1 Shift only. III. 0 1 Add Y to U, and shift IV. 1 0 Subtract Y from U, and shift or add (-Y) to U and shift V. Take U & V together and shift arithmetic right shift which preserves the sign bit of 2 s complement number. Thus a positive number remains positive, and a negative number remains negative. VI. Shift X circular right shift because this will prevent from using two registers for the X value. VII. Repeat the steps until four cycles are completed. U V X X-1 0000 0000 1100 0 0000 0000 0110 0 0000 0000 0011 0 1110 0000 0011 0 Add Y(0000 + 1110)=1110 1111 0000 1001 1 Shift Table 2: Step3 in Radix-2 Booth Multiplication Process U V X X-1 0000 0000 1100 0 0000 0000 0110 0 0000 0000 0011 0 1110 0000 0011 0 1111 0000 1001 1 1111 1000 1100 1 Shift only Table 3: Step4 in Radix-2 Booth Multiplication Process and Final Result in the Final Cycle 136
After finishing four cycles, so the answer is shown, in the last rows of U and V which is: 11111000 two NOTE: By the fourth cycle, the two algorithms have the same values in the product register. Group of consecutive 0 s in multiplier - no new partial product - only shift partial product right one bit position for every 0 Group of m consecutive 1's in multiplier - less than m partial products generated...01 110... =...10...000... -...00...010... Using SD (signed-digit) notation =...100...010... Xi Xi-1 Operation Comments Yi 0 0 Shift only String of zeros 0 1 1 Shift only String of ones 0 1 0 Subtract & shift Beginning of String of ones 1 0 1 Add & Shift End of a String of ones 1 Table 4: Radix-2 Booth Multiplier Logic Recoding multiplier xn-1 xn- 2...x1 x0 in SD code. Recoded multiplier yn-1 yn-2... y1 y0. xi, xi-1 of multiplier examined to generate yi. Simple recoding is yi = xi-1 xi. DRAWBACKS OF RADIX-2 BOOTH MULTIPLIER There is a problem of isolated one s. To overcome this drawback, instead of comparing two bits at a time, the comparison of three bits at a time should be done. MODIFIED BOOTH ENCODER 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. In this approach instead of eight partial products being generated using conventional multiplier only 4 partial products are generated. Table 4.2. 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). 137
Table 5: Truth table for Modified Booth Encoder 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. 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 in front of it, the partial products will be generated. PARTIAL PRODUCT GENERATOR (PPG) Figure 2: Booth Encoder 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 two s complement form of the number. Multiply by -2 is to shift left one bit the two s complement of the multiplicand value and multiply by 2 means just shift left the multiplicand by one place. 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 value (become signed number for resulted partial product). 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. 138
Table 6: Truth Table for Sign Extension when X7 is zero Table 7: Truth Table for Sign Extension when X7 is one 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]. SIMULATION RESULTS The different multipliers are simulated using Xilinx 10.1 ISE simulator after writing the code. Figure 3: Simulation of Array Multiplier 139
Figure 4: Simulation Of Radix-2 Booth Multiplier Figure 5: Simulation Of Radix-4 Booth Multiplier TIMING REPORT & DESIGN SUMMARY OF MULTIPLIERS The timing report of the multiplier is generated by doing synthesis of code and is targeted on to the Field Programmable Gate logic Array (FPGA).Initially, the package pins are assigned to each port and after that the device have to be configured in Slave/Serial mode and finally the code is dumped into the FPGA and verify the outputs. Figure 6: Design Summary of Array Multiplier 140
Figure 7: Design Summary of Radix-2 Booth Multiplier Figure 8: Design Summary of Radix-4 Booth Multiplier COMPARISON TABLES OF DIFFERENT MULTIPLIERS S.No Multiplier Time Delay 1 Array Multiplier 15.529 ns 2 Radix-2 Booth Multiplier 14.338 ns 3 Radix-4 Booth Multiplier 30.078 ns Table 8: Time Delay Analysis of Various Multipliers SPECIFICATIONS ARRAY MULTIPLER RADIX-2 BOOTH MULTIPLIER RADIX-4 BOOTH MULTIPLIER No. of slices 4% 8% 6% No. of LUTs 4% 8% 6% No. of bonded IOBs 22% 22% 23% Table 9: Performance Comparison of Various Multipliers 141
CONCLUSION This paper gives a clear concept of different multipliers and their comparision. It can be concluded that the parallel multipliers are better than the serial multiplier, the result of power consumption and the total area. In case of parallel multipliers, the total area is much less than that of serial multipliers. Hence the power consumption is also less. This speeds up the calculation and makes the system faster. While comparing the radix 2 and the radix 4 booth multipliers it has been found that radix 4 consumes less power than that of radix 2. This is because it uses almost half number of iterations and adders when compared to radix 2.When all the three multipliers were compared it has been found that array multipliers are most power consuming. This is because it uses a large number of adders. As a result it slows down the system because the system has to perform more number of calculations in case of array multiplier. Multipliers are one the most important component of many systems and it is necessary to find a better solution in case of multipliers. Preferably multipliers should always consume less power and cover less area. REFERENCES [1] Modeling of Time Delay in VHDL-based Design of a Multiplier Z. A bid Electrical Engineering Department Pobox 800, King Saud University, Riyadh 11421. [2] A Two s Complement Parallel Array multiplication Algorithm By Charles.R.Baugh and Bruce.A.Wooley. [3] M. Miyamotoetal, High -Speed and Low-Power Interconnect Technology for Sub-Quarter- Micron ASIC s, IEEE Transaction on Electron Devices, V. 44, No 2, 250, 1997. [4] A. Bellaouar and M. I. Elmasry, Low-Power Digital VLSI Design, Kluwer Academic Publishers, 1995. [5] Low voltage, low power VLSI subsystems By Kiat Seng Yeo, Kaushik Roy. [6] Digital Design By John.F.Wakerly. [7] VHDL Premier and VHDL Synthesis By V.J.Prasad. [8] Computer System Architecture by Marris Mano. [9] B.K.V.Prasad, P.Satishkumar, B.Stephencharles and T.Prasad, Low Power Design of Wallance Tree Multiplier, International Journal of Electronics and Communication Engineering & Technology (IJECET), Volume 3, Issue 3, 2012, pp. 258-264, ISSN Print: 0976-6464, ISSN Online: 0976 6472. [10] Kavita and Umesh Goyal, FPGA Implementation of Vedic Multiplier, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 4, 2013, pp. 150-158, ISSN Print: 0976-6480, ISSN Online: 0976-6499. [11] Dr. Syed Abdul Sattar, Dr. Mohammed Yousuf Khan and Shaik Qadeer, A New Radix-4 FFT Algorithm, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 3, 2013, pp. 251-256, ISSN Print: 0976-6480, ISSN Online: 0976-6499. 142