DESIGN OF 4 BIT BINARY ARITHMETIC CIRCUIT USING 1 S COMPLEMENT METHOD

e-issn 2455 1392 Volume 2 Issue 4, April 2016 pp. 176-187 Scientific Journal Impact Factor : 3.468 http://www.ijcter.com DESIGN OF 4 BIT BINARY ARITHMETIC CIRCUIT USING 1 S COMPLEMENT METHOD Dhrubojyoti Ghosh 1, Koustuva Kanti Mandal 2, Suchandana Roy Saha 3 1,2,3 ECE Dept., Abacus Institute of Engineering & Management Abstract This paper presents a technique to design a 4 bit binary arithmetic circuit capable of doing addition and subtraction using 1 s complement method.the complement method is essentially used for representation of negative numbers. The explored method of the circuit of 1 s complement method along with the conventional 4 bit adder-subtractor composite unit achieves the design to get perfect result. With the help of this circuit we can add or subtract any two numbers with any sign in an efficient way without employing the human brain. Keywords Control Inverter, Control Sign Input, Composite Unit, Sign Magnitude Bit Output, Full Adders. I. INTRODUCTION In digital circuits, an adder-subtractor is a circuit that is capable of adding and subtracting numbers. The adder-subtractor composite unit find a wide application in binary arithmetic operations in digital technology. Addition of multiple bit numbers can be accomplished using several full adders. Adders are a part of the core of an arithmetic logic unit (ALU).The control unit decides which operation an ALU should perform and sets the ALU operation. Composite unit simplifies the complexity by adding two circuits in a single one[1]. The 4-bit binary Arithmetic circuit using 1 s complement is capable of adding or subtracting two 4 bit numbers resulting in a 5th bit which represents the sign of the output. The complement method are generally performed while performing subtraction. 1 s complement theory is easier to implement by the digital device just by changing 0 s to 1 s and vice versa. The theory says to invert each bit with a NOT gate. The subtraction of two binary numbers may be accomplished by taking 1 s complement of the subtrahend and adding it to the minuend. Control Sign Input bit controls the sign of the input as per requirement and thus can control the addition and subtraction using 1 s complement method in parallel binary full adder circuit[2]. This circuit have inputs two 4 bit binary inputs i.e. A(A0,A1,A2,A3) and B(B0,B1,B2,B3) which is capable of performing arithmetic operation like A+B, -A+B, A-B, -A-B. Digital electronics, or digital (electronic) circuits, are electronics that represent signals by discrete bands of analog levels, rather than by continuous ranges (as used in analogue electronics). All levels within a band represent the same signal state. Because of this discretization, relatively small changes to the analog signal levels due to manufacturing tolerance, signal attenuation or parasitic noise do not leave the discrete envelope, and as a result are ignored by signal state sensing circuitry[3]. In most cases the number of these states is two, and they are represented by two voltage bands: one near a reference value (typically termed as "ground" or zero volts), and the other a value near the supply voltage. These correspond to the "false" ("0") and "true" ("1") values of the Boolean domain, respectively, yielding binary code. @IJCTER-2016, All rights Reserved 176

Digital techniques are useful because it is easier to get an electronic device to switch into one of a number of known states than to accurately reproduce a continuous range of values[4]. II. BASIC TERMINOLOGY BINARY SYSTEM:- Number system that uses only two values (0,1) to represent codes and data are called binary system. Since zeros and ones can be easily represented by two voltages, the binary system is the foundation on which digital technology is built. Every digital computer whether a pocket calculator or a mainframe uses the same binary notation. BIT:-A bit is the smallest unit of data in a computer. A bit has a single binary value, either 0 or 1. Although computers usually provide instructions that can test and manipulate bits, they generally are designed to store data and execute instructions in bit multiples called bytes. Half a byte (four bits) is called a nibble. In some systems, the term octet is used for an eight-bit unit instead of byte. In many systems, four eight-bit bytes or octets form a 32-bit word. In such systems, instruction lengths are sometimes expressed as full-word (32 bits in length) or half-word (16 bits in length). Binary Numbers:- In electronics, binary numbers is the flow of information in the form of zeros and ones used by digital computers and systems. Unlike a linear, or analogue circuits, such as AC amplifiers, which process signals that are constantly changing from one value to another, for example amplitude or frequency, digital circuits process signals that contain just two voltage levels or states, labelled, Logic 0 and Logic 1 [5]. Generally, a logic 1 represents a higher voltage, such as 5 volts, which is commonly referred to as a HIGH value, while a logic 0 represents a low voltage, such as 0 volts or ground, and is commonly referred to as a LOW value. These two discrete voltage levels representing the digital values of 1 s (one s) and 0 s (zero s) are commonly called: BInary digits, and in digital and computational circuits and applications they are normally referred to as binary BITS[1]. Binary Bits of Zeros and Ones Because there are only two valid Boolean values for representing either a logic 1 or a logic 0, makes the system of using Binary Numbers ideal for use in digital or electronic circuits and systems. @IJCTER-2016, All rights Reserved 177

The BINARY NUMBER SYSTEM is a Base-2 numbering system which follows the same set of rules in mathematics as the commonly used decimal or base-10 number system. So instead of powers of ten, ( 10n ) for example: 1, 10, 100, 1000 etc, binary numbers use powers of two, ( 2n ) effectively doubling the value of each successive bit as it goes, for example: 1, 2, 4, 8, 16, 32 etc[3]. The voltages used to represent a digital circuit can be of any value, but generally in digital and computer systems they are kept well below 10 volts. In digital systems theses voltages are called logic levels and ideally one voltage level represents a HIGH state, while another different and lower voltage level represents a LOW state. Digital waveforms or signals consist of discrete or distinctive voltage levels that are changing back and forth between these two HIGH and LOW states. But what makes a signal or voltage Digital and how can we represent these HIGH and LOW voltage levels[4]. Electronic circuits and systems can be divided into two main categories :---- Analogue Circuits Analogue or Linear circuits amplify or respond to continuously varying voltage levels that can alternate between a positive and negative value over a period of time. Digital Circuits Digital circuits produce or respond too two distinct positive or negative voltage levels representing either a logic level 1 or a logic level 0. Analogue Voltage Output Representation : This is an analogue circuit. The output from the potentiometer varies as the wiper terminal is rotated producing an infinite number of output voltage points between 0 volts and Vmax. The output voltage can vary either slowly or rapidly from one value to the next so there is no sudden change between two voltage levels giving a continuous output voltage. Examples of analogue signals include temperature, pressure, liquid levels and light intensity[4]. @IJCTER-2016, All rights Reserved 178

Digital Voltage Output :- International Journal of Current Trends in Engineering & Research (IJCTER) In this digital circuit example, the potentiometer wiper has been replaced by a single rotary switch which is connected in turn to each junction of the series resistor chain, forming a basic potential divider network. As the switch is rotated from one position (or node) to the next the output voltage,vout changes quickly in discrete and distinctive voltage levels representing multiples of 1.0 volts on each switching action, or step, as shown in the output graph. So for example, the output voltage will be 2 volts, 3 volts, 5 volts, etc. but NOT 2.5V, 3.1V or 4.6V. Finer output voltage levels could easily be produced by using a multi-positional switch and increasing the number of resistors within the potential divider chain, therefore increasing the number of discrete steps[5]. Digital Voltage Output Representation :- Then the major difference between an analogue signal or quantity and a digital quantity is that an Analogue quantity is continuously changing over time while a Digital quantity has discrete (step by step) values. LOW to HIGH or HIGH to LOW. Digital Logic Levels :- In all electronic and computer circuits, only two logic levels are allowed to represent a single state. These levels are referred to as a logic 1 or a logic 0, HIGH or LOW, True or False. Most logic systems use positive logic, in which case a logic 0 is represented by zero volts and a logic 1 is represented by a higher voltage, for example, +5 volts as shown[1]. Digital Value Representation:- @IJCTER-2016, All rights Reserved 179

First State Second State Logic 0 Logic 1 LOW HIGH FALSE TRUE Low Level Voltage Output High Level Voltage Output 0V or Ground +5 Volts Generally the switching from one voltage level, 0 to 1 or 1 to 0 is made as quickly as possible to prevent miss switching of the logic circuit. In standard TTL (transistor-transistor-logic) IC s there is a pre-defined range of input and output voltage limits for defining what exactly is a logic 1 value and what is a logic 0 value. TTL Input & Output Voltage Levels:- @IJCTER-2016, All rights Reserved 180

Then, when using a +5 volt supply any voltage input between 2.0v and 5v is recognised as a logic 1 value and any voltage input of below 0.8v is recognised as a logic 0 value. While the output of a logic gate between 2.7v and 5v represents a logic 1 value and a voltage output below 0.4v represents a logic 0 value. This is called positive logic. Then binary numbers are commonly used in digital and computer circuits and are represented by either a logic 0 or a logic 1. Binary numbering systems are best suited to the digital signal coding of binary, as it uses only two digits, one and zero, to form different figures[4]. BINARY CODES :-We can categorize binary codes in following types: Weighted codes: These codes have fixed weights for different binary positionse.g. BCD codes, 8 4-2 -1, 2 4 2 1 etc Non-weighted codes: These codes don t have fixed weights for different binary positionse.g. excess- 3 codes, gray code Sequential codes: In this coding system, we have consecutive codes whose decimal equivalents differ only by 1 e.g. excess-3 codes Random codes: These codes don t have consecutive codes differing by 1 in their decimal equivalentse.g. gray codes Alphanumeric codes: These codes are used to represent different alphabets. We can have codes for 26 alphabets both in lowercase and higher case, numeric 0-9 and some special symbols like @,#,$,%,& etc. ASCII codes are an example of alphanumeric codes[2]. A. One s Complement of a Signed Binary Number:- One s Complement or 1 s Complement as it is also termed, is another method which we can use to represent negative binary numbers in a signed binary number system. In one s complement, positive numbers (also known as non-complements) remain unchanged as before with the sign-magnitude numbers. Negative numbers however, are represented by taking the one s complement (inversion, negation) of the unsigned positive number. Since positive numbers always start with a 0, the complement will always start with a 1 to indicate a negative number. The one s complement of a negative binary number is the complement of its positive counterpart, so to take the one s complement of a binary number, all we need to do is change each bit in turn. Thus the one s complement of 1 is 0 and vice versa, then the one s complement of 10010100 2 is simply 01101011 2 as all the 1 s are changed to 0 s and the 0 s to 1 s. The easiest way to find the one s complement of a signed binary number when building digital arithmetic or logic decoder circuits is to use Inverters. The inverter is naturally a complement generator and can be used in parallel to find the 1 s complement of any binary number as shown[3]. @IJCTER-2016, All rights Reserved 181

1) 1 s Complement Using Inverters Then we can see that it is very easy to find the one s complement of a binary number N as all we need do is simply change the 1 s to 0 s and the 0 s to 1 s to give us a -N equivalent. Also just like the previous sign-magnitude representation, one s complement can also have n-bit notation to represent numbers in the range from: -2 (n-1) - 1 and +2 (n-1) - 1. For example, a 4-bit representation in the one s complement format can be used to represent decimal numbers in the range from -7 to +7 with two representations of zero: 0000 (+0) and 1111 (-0) the same as before. B. Addition and Subtraction Using One s Complement :- One of the main advantages of One s Complement is in the addition and subtraction of two binary numbers. In mathematics, subtraction can be implemented in a variety of different ways as A B, is the same as saying A + (-B) or -B + A etc. Therefore, the complication of subtracting two binary numbers can be performed by simply using addition. We saw in the Binary Adder tutorial that binary addition follows the same rules as for the normal addition except that in binary there are only two bits (digits) and the largest digit is a 1, (just as 9 is the largest decimal digit) thus the possible combinations for binary addition are as follows: 0 0 1 1 + 0 + 1 + 0 + 1 0 1 1 1 0 ( 0 plus a carry 1 ) When the two numbers to be added are both positive, the sum A + B, they can be added together by means of the direct sum (including the number and bit sign), because when single bits are added together, 0 + 0, 0 + 1, or 1 + 0 results in a sum of 0 or 1. This is because when the two bits to be added together are odd ( 0 + 1 or 1 + 0 ), the result is 1. Likewise when the two bits to be added together are even ( 0 + 0 or 1 + 1 ) the result is 0 until you get to 1 + 1 then the sum is equal to 0 plus a carry 1. Let s look at a simple example[2,3]. Subtraction of Two Binary Numbers :- An 8-bit digital system is required to subtract the following two numbers 115 and 27 from each other using one s complement. So in decimal this would be: 115-27 = 88. @IJCTER-2016, All rights Reserved 182

First we need to convert the two decimal numbers into binary and make sure that each number has the same number of bits by adding leading zero s to produce an 8-bit number (byte). Therefore: 115 10 in binary is: 01110011 2 27 10 in binary is: 00011011 2 Now we need to find the complement of the second binary number, (00011011) while leaving the first number (01110011) unchanged. By changing all the 1 s to 0 s and 0 s to 1 s, the one s complement of 00011011 is equal to 11100100. Adding the first number and the complement of the second number gives: 01110011 + 11100100 O v e r f l o w 1 01010111 Since the digital system is to work with 8-bits, only the first eight digits are used to provide the answer to the sum, and we simply ignore the last bit (bit 9). This bit is call an overflow bit. Overflow occurs when the sum of the most significant (left-most) column produces a carry forward. This overflow or carry bit can be ignored completely or passed to the next digital section for use in its calculations. Overflow indicates that the answer is positive. If there is no overflow then the answer is negative[5]. The 8-bit result from above is: 01010111 (the overflow 1 cancels out) and to convert it back from a one s complement answer to the real answer we now have to add 1 to the one s complement result. 01010111 + 1 01011000 So the result of subtracting 27 ( 00011011 2 ) from 115 ( 01110011 2 ) using 1 s complement in binary gives the answer of: 01011000 2 or (64 + 16 + 8) = 88 10 in decimal. CIRCUIT DETAILS The circuit has ten inputs. A3, A2,A1,A0 represent the first 4 bit input A and Ca is the control sign input of A. B3,B2,B1,B0 represent the second 4 bit input B and Cb is the control sign input of B. When the number is positive control sign input is 0 and when the number is negative control sing input is 1. We use some logic gates for implementing this circuit like X-OR, OR etc and 4 bit parallel adder. We obtain Cab after performing the X-OR operation between the two control sign inputs Ca and Cb. Here 4 stage X-OR gates are used to complement the input values A and B according to the requirements. Then we apply this outputs in the 4 bit parallel adder as input and it produces the sum bits represented as S 3,S 2,S 1,S 0 and a carry out represented as Cout1. T is connected with 1st parallel adder s carry input to implement 1 s complement method. K is the 2nd parallel adder s complement controller factor. This is mainly an indicator of the two operations when othe 5th output shows the extra output which represents the high ranges of data which is the final carry out Cout. The 6th,7th and 8th bits are grounded respectively. The 9th output denoted as Sm is the sign magnitude bit which represents the sign of the output. @IJCTER-2016, All rights Reserved 183

Figure-1.Logic Diagram @IJCTER-2016, All rights Reserved 184

THE TRUTH TABLE FOR THE ABOVE CIRCUIT : TABLE:-1 Ca A3 A2 A1 A0 Cb B3 B2 B1 B0 Sm Cout S3 S2 S1 S0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 TABLE:-2 Ca A3 A2 A1 A0 Cb B3 B2 B1 B0 Sm Cout S3 S2 S1 S0 0 1 0 0 0 1 0 0 1 1 0 0 0 1 0 1 TABLE:-3 Ca A3 A2 A1 A0 Cb B3 B2 B1 B0 Sm Cout S3 S2 S1 S0 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 0 TABLE:-4 Ca A3 A2 A1 A0 Cb B3 B2 B1 B0 Sm Cout S3 S2 S1 S0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 0 0 TABLE:-5 Ca A3 A2 A1 A0 Cb B3 B2 B1 B0 Sm Cout S3 S2 S1 S0 1 0 1 1 1 1 0 1 0 1 1 0 1 1 0 0 @IJCTER-2016, All rights Reserved 185

TABLE:-6 Ca A3 A2 A1 A0 Cb B3 B2 B1 B0 Sm Cout S3 S2 S1 S0 1 1 0 0 0 0 0 0 1 1 1 0 0 1 0 1 III. RESULT From the circuit of 4 bit binary adder- subtractor composite unit by using 1 s complement we construct the tables shown above. The results are explained below. For table 1, A=+8(A3=1,A2=0,A1=0,A0=0) and Ca=0 and B=+8(B3=1,B2=0,B1=0,B0=0) and Cb=0. At the output port Sm=0 that means the answer is positive and Co=1,S3=0,S2=0,S1=0,S0=0 so the final result is +16. For table 2, A=+8(A3=1,A2=0,A1=0,A0=0) and Ca=0 and B=-3(B3=0,B2=0,B1=1,B0=1) and Cb=1. At the output port Sm=0 that means the answer is positive and Co=0,S3=0,S2=1,S1=0,S0=1 so the final result is +5. For table 3, A=-8(A3=1,A2=0,A1=0,A0=0) and Ca=1 and B=-8(B3=1,B2=0,B1=0,B0=0) and Cb=1. At the output port Sm=1 that means the answer is negative and Co=1,S3=0,S2=0,S1=0,S0=0 so the final result is -16. For table 4, A=+7(A3=0,A2=1,A1=1,A0=1) and Ca=0 and B=+5(B3=0,B2=1,B1=0,B0=1) and Cb=0. At the output port Sm=0 that means the answer is positive and Co=0,S3=1,S2=1,S1=0,S0=0 so the final result is +12. For table 5, A=-7(A3=0,A2=1,A1=1,A0=1) and Ca=1 and B=-5(B3=0,B2=1,B1=0,B0=1) and Cb=1. At the output port Sm=1 that means the answer is negative and Co=0,S3=1,S2=1,S1=0,S0=0 so the final result is -12. For table 6, A=-8(A3=1,A2=0,A1=0,A0=0) and Ca=1 and B=+3(B3=0,B2=0,B1=1,B0=1) and Cb=0. At the output port Sm=1 that means the answer is negative and Co=0,S3=0,S2=1,S1=0,S0=1 so the final result is -5. IV. CONCLUSION This circuit adds or subtracts two 4 bits numbers and gives us an accurate result with accurate sign. Besides getting an accurate number as result, it is also essential to check the number which is obtained as result is positive or negative that can be done by looking at the result itself i.e. When we are adding two 4-bits numbers, the result is of 5 bits and the 6th bit (MSB) denotes the sign of the number or the output obtained. We can also apply this circuit for 5-bits addition and subtraction in that case the result will be of 6 bits and the 6th bit will denote the sign of the number and the remaining 5 bits are considered as our output. This circuit finds a wide application in digital world of binary operation. This circuit is also very easy to implement as it is using 1 s complement method which can be easily obtained by using a NOT gate. @IJCTER-2016, All rights Reserved 186

REFERENCES [1] D. Ghosh and S. R. Saha, Design of 4bit binary arithmetic circuit using 2 s complement method, IOSR Journal of Electronics and Communication Engineering (IOSR-JECE), 8(4), 2013, pp.39-42. [2] D. Ghosh, K. K. Mandal and S. R. Saha, Design of digital arithmetic circuit using BCD code & 9 s complement method, International Journal of Electronics & Communication Technology (IJECT),4(4),2013,pp.115-117. [3] D. Ghosh, Kazi M. Qabir, I. Bhattacharya and G.Banerjee, Design of Digital Arithmetic Circuit using BCD/Excess-3 Logic & 9 s Complement Method, International journal of Advance Research(IJOAR), 2(5), 2014, pp.1-7. [4] D. Ghosh, Design of 4-Bit Adder Subtractor Composite Unit Using 2 s Complement Method http://www.academia.edu/4020322/design_of_4bit_adder_subtractor_composite_unit_using_ 2S_COMPLEMENT_METHOD [5] D. Ghosh and N. Bhanja, Design of digital arithmetic circuit using Exess-3 Code & 9 s Complement Method, IOSR Journal of Electronics and Communication Engineering (IOSR-JECE), 9(3), 2014, pp.15-18. BIOGRAPHIES DHRUBOJYOTI GHOSH received the B.Tech & M.Tech Degree in Electronics and Communication Engineering from the University of W.B.U.T,Kolkata,West Bengal, in 2007 & 2009 respectively. Currently, He is an Assistant Professor of Electronics and Communication Engineering of Abacus Institute of Engineering & Management, West Bengal, India. His teaching and research areas include Digital Electronic circuits, Digital Communication Engineering. Dhrubojyoti Ghosh (Assistant Professor) may be reached at dhrubo_ghosh@rediffmail.com SUCHANDANA ROY SAHA received the B.Tech & M.Tech Degree in Electronics and Communication Engineering from the University of W.B.U.T,Kolkata,West Bengal, in 2007 & 2011 respectively. Currently, She is an Assistant Professor of Electronics and Communication Engineering of Abacus Institute of Engineering & Management, West Bengal, India. Her teaching and research areas include Digital Electronic circuits, Digital Communication Engineering. Suchandana Roy Saha (Assistant Professor) may be reached at suchandana.roysaha@gmail.com KOUSTUVA KANTI MANDAL received the B.Tech & M.Tech Degree in Electronics and Communication Engineering from the University of W.B.U.T,Kolkata,West Bengal, in 2007 & 2009 respectively. Currently, He is an Assistant Professor of Electronics and Communication Engineering of Abacus Institute of Engineering & Management, West Bengal, India. His teaching and research areas include Digital Electronic circuits, Digital Signal Processing. Koustuva Kanti Mandal (Assistant Professor) may be reached at koustuva@gmail.com @IJCTER-2016, All rights Reserved 187