Digital logic degin, Number system
- 2. 2’s complement numbers › Addition and subtraction Binary coded decimal Gray codes for binary numbers ASCII characters Moving towards hardware › Storing data › Processing data
- 3. 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. Numbers in hexadecimal 0 1 1 0 1 + 1 1 0 1 1 -------------- 1 0 1 0 0 0 carry
- 4. 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. Numbers in hexadecimal 0 0 1 0 1 + 1 0 1 0 0 -------------- 1 1 0 0 1
- 5. 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. Digit BCD Code Digit BCD Code 0 0000 5 0101 1 0001 6 0110 2 0010 7 0111 3 0011 8 1000 4 0100 9 1001 3 2 9
- 6. ° BCD not very efficient ° Used in early computers (40s, 50s) ° Used to encode numbers for seven- segment displays. ° Easier to read?
- 7. 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. Digit Binary Gray Code 0 0000 0000 1 0001 0001 2 0010 0011 3 0011 0010 4 0100 0110 5 0101 0111 6 0110 0101 7 0111 0100 8 1000 1100 9 1001 1101 10 1010 1111 11 1011 1110 12 1100 1010 13 1101 1011 14 1110 1001 15 1111 1000
- 8. 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).Character ASCII (bin) ASCII (hex) Decimal Octal A 1000001 41 65 101 B 1000010 42 66 102 C 1000011 43 67 103 … Z a … 1 ‘
- 9. ° 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?
- 10. 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. Information BitsP 1 1 0 0 0 0 1 1 Added even parity bit 0 1 0 0 0 0 1 1 Added odd parity bit
- 11. Concatenate a parity bit to the ASCII code for the characters 0, X, and = to produce both odd-parity and even-parity codes. Character ASCII Odd-Parity ASCII Even-Parity ASCII 0 0110000 10110000 00110000 X 1011000 01011000 11011000 = 0111100 10111100 00111100
- 12. • Binary cells store individual bits of data • Multiple cells form a register. • Data in registers can indicate different values • Hex (decimal) • BCD • ASCII Binary Cell 0 0 1 0 1 0 1 1
- 13. Data can move from register to register. Digital logic used to process data We will learn to design this logic Register A Register B Register C Digital Logic Circuits
- 14. Data input at keyboard Shifted into place Stored in memory NOTE: Data input in ASCII
- 15. We need processing We need storage We need communication You will learn to use and design these components.
- 16. 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 Next time: Building logic circuits!