Digital Logic Design Lecture 3 Complements , Number Codes and Registers

1. Digital Logic DesignLecture 3Complements,Number Codes and Registers

2. Overview • Complement of numbers • Addition and subtraction • Binary coded decimal • Gray codes for binary numbers • ASCII characters • Moving towards hardware • Storing data • Processing data

3. Complements • In general, we (human beings) express negative numbers by placing a minus (-) sign at the left end of the number. Similarly while representing the integers in binary format, we can leave the left-most bit be the sign bit. If the left-most bit is a zero, the integer is positive; if it is a one, it is negative. • Zero is positive and -0 = 0 • The top-most bit should tell us the sign of the integer. • The negative of a negative integer is the original integer ie., -(-55) is 55. • x - y should give the same result as x + (-y). That is, 8 - 3 should give us the same result as 8 + (-3). • Negative and positive numbers shouldn't be treated in different ways when we do multiplication and division with them.

4. Complement • If we consider only positive numbers, this would allow for all numbers from 0 (naturally represented by 00000000) to and inclusively 255 (represented by 11111111) • The most obvious solution is to spare the first bit as a sign indicator, thus leaving the 7 last bits to represent the numbers. • This simple way to represent negative numbers has for it the equally simple way to compute the negative of a given number: just invert the first bit!. Hence, the number 3, for example, represented as (00000011) will give (10000011) for -3 (we have just toggled the first bit). • This simple way has a drawback: the negative of 0 (00000000) is now (10000000), known as -0

5. 000011002 = 1210 111101002 = -1210 Sign bit Sign bit Magnitude Magnitude Two’s Complement Representation • The two’s complement of a binary number involves inverting all bits and adding 1. • 2’s comp of 00110011 is 11001101 • 2’s comp of 10101010 is 01010110 • For an n bit number N the 2’s complement is (2n-1) – N + 1. • Called radix complement by Mano since 2’s complement for base (radix 2). • To find negative of 2’s complement number take the 2’s complement.

6. Two’s Complement Shortcuts • Algorithm 1 – Simply complement each bit and then add 1 to the result. • Finding the 2’s complement of (01100101)2 and of its 2’s complement… N = 01100101 [N] = 10011011 10011010 01100100 + 1 + 1 --------------- --------------- 10011011 01100101 • Algorithm 2 – Starting with the least significant bit, copy all of the bits up to and including the first 1 bit and then complementing the remaining bits. • N = 0 1 1 0 0 1 0 1 [N] = 1 0 0 1 1 0 1 1

7. Finite Number Representation • Machines that use 2’s complement arithmetic can represent integers in the range -2n-1 <= N <= 2n-1-1 where n is the number of bits available for representing N. Note that 2n-1-1 = (011..11)2 and –2n-1 = (100..00)2 • For 2’s complement more negative numbers than positive. • For 1’s complement two representations for zero. • For an n bit number in base (radix) z there are zn different unsigned values. (0, 1, …zn-1)

8. 0 1 1 0 0 + 0 0 0 0 1 -------------- 0 0 1 1 0 1 0 -------------- 0 1 1 0 1 Add carry Final Result 1’s Complement Addition • Using 1’s complement numbers, adding numbers is easy. • For example, suppose we wish to add +(1100)2 and +(0001)2. • Let’s compute (12)10 + (1)10. • (12)10 = +(1100)2 = 011002 in 1’s comp. • (1)10 = +(0001)2 = 000012 in 1’s comp. Add Step 1: Add binary numbers Step 2: Add carry to low-order bit

9. 1’s Complement Subtraction • Using 1’s complement numbers, subtracting numbers is also easy. • For example, suppose we wish to subtract +(0001)2 from +(1100)2. • Let’s compute (12)10 - (1)10. • (12)10 = +(1100)2 = 011002 in 1’s comp. • (-1)10 = -(0001)2 = 111102 in 1’s comp. 0 1 1 0 0 - 0 0 0 0 1 -------------- 0 1 1 0 0 + 1 1 1 1 0 -------------- 1 0 1 0 1 0 1 -------------- 0 1 0 1 1 1’s comp Step 1: Take 1’s complement of 2nd operand Step 2: Add binary numbers Step 3: Add carry to low order bit Add Add carry Final Result

10. 2’s Complement Addition • Using 2’s complement numbers, adding numbers is easy. • For example, suppose we wish to add +(1100)2 and +(0001)2. • Let’s compute (12)10 + (1)10. • (12)10 = +(1100)2 = 011002 in 2’s comp. • (1)10 = +(0001)2 = 000012 in 2’s comp. 0 1 1 0 0 + 0 0 0 0 1 -------------- 0 0 1 1 0 1 Add Step 1: Add binary numbers Step 2: Ignore carry bit Final Result Ignore

11. 2’s Complement Subtraction • Using 2’s complement numbers, follow steps for subtraction • For example, suppose we wish to subtract +(0001)2 from +(1100)2. • Let’s compute (12)10 - (1)10. • (12)10 = +(1100)2 = 011002 in 2’s comp. • (-1)10 = -(0001)2 = 111112 in 2’s comp. 0 1 1 0 0 - 0 0 0 0 1 -------------- 0 1 1 0 0 + 1 1 1 1 1 -------------- 1 0 1 0 1 1 2’s comp Step 1: Take 2’s complement of 2nd operand Step 2: Add binary numbers Step 3: Ignore carry bit Add Final Result Ignore Carry

12. 0 1 1 0 1 + 1 1 0 1 1 -------------- 1 0 1 0 0 0 carry 2’s Complement Subtraction: Example #2 • Let’s compute (13)10 – (5)10. • (13)10 = +(1101)2 = (01101)2 • (-5)10 = -(0101)2 = (11011)2 • Adding these two 5-bit codes… • Discarding the carry bit, the sign bit is seen to be zero, indicating a correct result. Indeed, (01000)2 = +(1000)2 = +(8)10.

13. 2’s Complement Subtraction: Example #3 • Let’s compute (5)10 – (12)10. • (-12)10 = -(1100)2 = (10100)2 • (5)10 = +(0101)2 = (00101)2 • Adding these two 5-bit codes… • Here, there is no carry bit and the sign bit is 1. This indicates a negative result, which is what we expect. (11001)2 = -(7)10. 0 0 1 0 1 + 1 0 1 0 0 -------------- 1 1 0 0 1

14. 0 1 1 0 1 + 1 1 0 1 1 -------------- 1 0 1 0 0 0 carry 2’s Complement Subtraction • Let’s compute (13)10 - (5)10. • (13)10 = +(1101)2 = (01101)2 • (-5)10 = -(0101)2 = (11011)2 • Adding these two 5-bit codes… • Discarding the carry bit, the sign bit is seen to be zero, indicating a correct result.

15. Data Representation and Communication • Human communication • Includes language, images and sounds • Computers • Process and store all forms of data in binary format • Conversion to computer-usable representation using data formats • Define the different ways human data may be represented, stored and processed by a computer

16. Data Representation and Communication • Numbers are important to computers as they • Represent information precisely • Can be processed • Decimal numbers are favored by humans. Binary numbers are natural to computers. Hence, conversion is required. • If little calculation is required, we can use some coding schemes to store decimal numbers, for data transmission purposes. • Examples: BCD (or 8421), Excess-3, 8-4-2-1, 2421, etc. • Each decimal digit is represented as a 4-bit code. • The number of digits in a code is also called the length of the code.

17. BCD • Binary Coded Decimal (BCD) represents each decimal digit with four bits • Ex. 0011 0010 1001 = 32910 • This is NOT the same as 0011001010012 • Why do this? Because people think in decimal. Note: the following 6 bit patterns are not used: 1010 1011 1100 1101 1110 1111

18. Putting It All Together • BCD not very efficient • Used in early computers (40s, 50s) • Used to encode numbers for seven-segment displays. • Easier to read?

19. Gray Code • Gray code is not a number system. • It is an alternate way to represent four bit data • Only one bit changes from one decimal digit to the next • Useful for reducing errors in communication. • Can be scaled to larger numbers.

20. ASCII Code • American Standard Code for Information Interchange • ASCII is a 7-bit code, frequently used with an 8th bit for error detection (more about that in a bit).

21. ASCII Code Most significant bit Least significant bit

22. ASCII Code e.g., ‘a’ = 1100001

23. ASCII Code Alphabetic codes

24. ASCII Code Numeric codes

25. ASCII Code Punctuation, etc.

26. ASCII Codes and Data Transmission • ASCII Codes • A – Z (26 codes), a – z (26 codes) • 0-9 (10 codes), others (@#$%^&*….) • Complete listing in Mano text • Transmission susceptible to noise • Typical transmission rates (1500 Kbps, 56.6 Kbps) • How to keep data transmission accurate?

27. P Information Bits Parity Codes • Parity codes are formed by concatenating a parity bit, P to each code word of C. • In an odd-parity code, the parity bit is specified so that the total number of ones is odd. • In an even-parity code, the parity bit is specified so that the total number of ones is even. • 1 1 0 0 0 0 1 1 •  • Added even parity bit • 0 1 0 0 0 0 1 1 •  • Added odd parity bit

28. Parity Code Example • Concatenate a parity bit to the ASCII code for the characters 0, X, and = to produce both odd-parity and even-parity codes.

29. Binary Data Storage • Binary cells store individual bits of data • Multiple cells form a register. • Data in registers can indicate different values • Hex (decimal) • BCD • ASCII 0 0 1 0 1 0 1 1 Binary Cell

30. Register A Register B Register C Digital Logic Circuits Register Transfer • Data can move from register to register. • Digital logic used to process data • We will learn to design this logic

31. Transfer of Information • Data input at keyboard • Shifted into place • Stored in memory NOTE: Data input in ASCII

32. Building a Computer • We need processing • We need storage • We need communication • You will learn to use and design these components.

33. Summary • 2’s complement most important (only 1 representation for zero). • Important to understand treatment of sign bit for 1’s and 2’s complement. • Although 2’s complement most important, other number codes exist • ASCII code used to represent characters (including those on the keyboard) • Registers store binary data