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Digital logic degin, Number system

1.  2’s complement numbers › Addition and subtraction  Binary coded decimal  Gray codes for binary numbers  ASCII characters  Moving towards hardware…
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  • 1.  2’s complement numbers › Addition and subtraction  Binary coded decimal  Gray codes for binary numbers  ASCII characters  Moving towards hardware › Storing data › Processing data
  • 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.  Numbers in hexadecimal 0 1 1 0 1 + 1 1 0 1 1 -------------- 1 0 1 0 0 0 carry
  • 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.  Numbers in hexadecimal 0 0 1 0 1 + 1 0 1 0 0 -------------- 1 1 0 0 1
  • 4.  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
  • 5. ° BCD not very efficient ° Used in early computers (40s, 50s) ° Used to encode numbers for seven- segment displays. ° Easier to read?
  • 6.  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
  • 7.  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 ‘
  • 8. ° 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?
  • 9.  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
  • 10.  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
  • 11. • 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
  • 12.  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
  • 13.  Data input at keyboard  Shifted into place  Stored in memory NOTE: Data input in ASCII
  • 14.  We need processing  We need storage  We need communication  You will learn to use and design these components.
  • 15.  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!
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