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Overflow

Overflow. Signed binary is in fixed range -2 n-1  2 n-1 If the answer for addition/subtraction more than the range, it is overflow Two situation where overflow can happen: Positive + positive = negative (enough n-bit) Negative + negative = positive(more than n-bit). Overflow.

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Overflow

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  1. Overflow • Signed binary is in fixed range • -2n-1 2n-1 • If the answer for addition/subtraction more than the range, it is overflow • Two situation where overflow can happen: • Positive + positive = negative (enough n-bit) • Negative + negative = positive(more than n-bit) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  2. Overflow • Example: Binary number 4-bit (second complement) • Range : -2n-1 2n-1-1 • Range : (1000)2s(0111) 2s • Range : (-8) 10 (+7)10 • Two situation where overflow can happen: • Positive + positive = negative (enough n-bit) • Negative + negative = positive(more than n-bit) 0101 = 5 1001 = -7 0100 = 4 1010 = -6 ----------- ----------- 1001 10011 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  3. Overflow • Example: Binary number 4-bit (second complement) • Range : -2n-1 2n-1-1 • Range : (1000)2s(0111) 2s • Range : (-8) 10 (+7)10 (Overflow exist) (ignore final carry) (Overflow exist) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  4. Fixed Point Number • Signed number and unsigned number representation is given in fixed point number • Binary point is assumed to have fixed location, if it is located at the end of the number • It can represent integer number between –128 to 127 (for 8-bit binary complement) Binary point MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  5. Fixed Point Number fraction • Generally, other locations in binary point position • Example: If two fraction bit is used, we can represent: Binary point MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  6. Floating Point Number • Fixed point number has limited range • To represent extremely large or extremely small number, we use floating point number (like scientific number) • Example: • 0.23X1023(really large number) • 0.1239X10-10(really small number) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  7. Floating Point Number • Floating point number is divided into three parts mantissa, base and exponent • Base always fixed in number system • Therefore, only need mantissa and exponent MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  8. Floating Point Number • Mantissa always in normalize form: (base 10) 23X1021 is normalized to 0.23X1023 (base 10) –0.0017X1021 is normalized to -0.17X1019 (base 10) 0.01101X103 is normalized to 0.1101X102 • 16-bit floating point number might contain 10-bit mantissa and 6-bit exponent • More exponent, the greater its range • More mantissa, the greater its persistence MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  9. Arithmetic with Floating Point Number • Arithmetic with floating point number is much difficult • MULTIPLICATION The steps: • multiply with the mantissa • Add its exponent • normalized MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  10. Arithmetic with Floating Point Number • Example (Normalization) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  11. Arithmetic with Floating Point Number • ADDITION Steps: • Equalize their exponent • Add their mantissa • Normalize them MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  12. Arithmetic with Floating Point Number • Example: (0.12x102)10 + (0.0002x104 ) 10 = (0.12x102) 10 +(0.02x102) 10 = (0.12+0.02) 10 x 102 = (0.14x102 ) 10 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  13. Binary Coded Decimal (BCD) • Decimal number is normally used by human. Binary number is normally used by computer. It is expensive to exchange between each other. • If used only little calculation, we can use coding scheme for decimal number. • One of the scheme is BCD, or also called 8421 code. • Which represent every decimal digit with 4-bit binary code. MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  14. Binary Coded Decimal (BCD) Decimal Digit • There are code which is not used, e.g. (1010)BCD,(1011)BCD,….,(1111)BCD. This code is said to be an error. • Easy to convert but the arithmetic is hard • Suitable as interface such as keyboard input and digital reading Decimal Digit MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  15. Binary Coded Decimal (BCD) Decimal Digit • Example: Notes: BCD is not similar to binary Example: (243)10=(11101010)2 Decimal Digit MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  16. Gray Code • No weight • Only one bit change from one code number to the others • Suitable for error detection Decimal Binary Gray Code Decimal Binary Gray Code MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  17. Gray Code MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  18. Gray Code MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  19. Convert Binary Code to Gray Code • Fixed MSB • From left to right, add each coupled binary code bit next to each other to get Gray code bit, ignore carry • Example: convert binary 10110 to Gray code MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  20. Convert Gray Code to Binary Code • Fixed MSB • From left to right, add each coupled binary code executed with Gray code bit at the next position, ignore carry • Example: convert Gray 10110 to binary code MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  21. Other Decimal Code • Self compliment code: excess-3 code, 84-2-1, 2*421 • Error detection code: Biquinary code (bi=two, quinary=five) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  22. Self Compliment Code • Example: Excess-3, 84-2-1, 2*421 • Code represented by coupled compliment-digit which compliment each other MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  23. Alphanumeric Code • Part of numbers, computer also handle textual data • Set which always used includes: Letters : ‘A’,…..,‘Z’ and ‘a’,…..,‘z’ Digits : ‘0’,…..,‘9’ Special Characters: ‘$’, ‘’, ‘!’, ‘,’, ‘.’,…. Not Printable: SOH, NULL, BELL,…. • Most of the time, it is represented by 7 or 8-bit MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  24. Alphanumeric Code • Two standard that are frequently used ASCII (American Standard Code for Information Interchange) EBCDIC (Extended BCD Interchange Code) • ASCII: 7-bit, add with parity bit for error detection (odd,even parity) • EBCDIC: 8-bit MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  25. Alphanumeric Code • ASCII Table MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  26. Error Detection Code • Error can exist in transmission. It must be detected so that retransmission can be requested • With binary number, mostly exist 1-bit error. Example: 0010 is transmitted incorrectly as 0011, or 0000, or 0110, or 1010 • Biquinary using additional 3-bit to detect error. For one error detection, only one extra bit is needed MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  27. Error Detection Code • Parity Bit • Even parity: number of bit 1 is even • Odd parity: number of bit 1 is odd • Example: Odd parity MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  28. Error Detection Code • Parity Bit can detect odd error and not even error (if odd is set) Example: For odd parity number 10011=>10001 (detected) 10011=>10101 (not detected) • Parity bit can also be used on data block MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  29. Error Detection Code • Sometimes, it is not enough to detect code, we need to correct it • Error correction is expensive in practical, we only need to use one bit error correction • Popular technique: Hamming Code • Add k-bit to n-bit number to produce n+k bit • Number the bit 1 on bit n+k • Every parity bit is on the number range MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  30. Error Detection Code • E.g: For 8-bit number, we need 4 parity bit 12 bit number are 0001,0011,…,1100. Every 4 bit parity is used to detect group of bit. Every parity bit is for themselves and has bit ‘1’ on certain position bit MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  31. Error Detection Code • Therefore: P1= parity for bit {3,5,7,9,11} P2= parity for bit {3,6,7,10} P4= parity for bit {5,6,7,12} P8= parity for bit {9,10,11,12} MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  32. Error Detection Code • Given 8-bit number: 1100 0100 • Assume even parity is P1= parity for bit {3,5,7,9,11} = 0 P2= parity for bit {3,6,7,10} = 0 P4= parity for bit {5,6,7,12} = 1 P8= parity for bit {9,10,11,12} = 1 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  33. Error Detection Code • To check error, execute checking code C1= XOR {1,3,5,7,9,11} C2= XOR {2,3,6,7,10} C4= XOR {4,5,6,7,12} C8= XOR {8,9,10,11,12} If C8 C4 C2 C1=0000 therefore no error, if otherwise C8 C4 C2 C1 show position, there is an error for only one bit • Example MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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