1 / 16

Riyadh Philanthropic Society For Science Prince Sultan College For Woman

Riyadh Philanthropic Society For Science Prince Sultan College For Woman Dept. of Computer & Information Sciences CS 251 Introduction to Computer Organization & Assembly Language Lecture 7 (Computer System Organization) Error Detection & Correction. Outline. From Text Book: 2.2.4

neona
Télécharger la présentation

Riyadh Philanthropic Society For Science Prince Sultan College For Woman

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Riyadh Philanthropic Society For Science Prince Sultan College For Woman Dept. of Computer & Information Sciences CS 251 Introduction to Computer Organization & Assembly Language Lecture 7 (Computer System Organization) Error Detection & Correction

  2. Outline • From Text Book: 2.2.4 • Error Detection & Error Correction • Hamming Distance • Hamming Algorithm Error Detection & Error Correction

  3. Error Detection & Correction Error Detection & Error Correction • Computer memories could make errors due to electric spikes leading to incorrect behavior • To protect against such behavior, two ways are adopted • Error-detection codes • Error-Correction codes • Extra bits are added to the memory word in a special way so that when the word is read out from memory these extra bits are checked for errors

  4. m r n (codeword) Memory Errors – Code words Error Detection & Error Correction • Consider a memory word of size m bits to which r redundant check bits are added • The total length of the unit is n = m+ r • This n bit unit is called a code word

  5. Hamming Distance Error Detection & Error Correction • Exclusive OR (XOR) is used to know the number of bits that differ between two code words • Example 10001001 10110001 00111000 • Hamming Distance is the number of bit positions in which two valid code words differ • In the above example it is 3

  6. Code Hamming Distance Error Detection & Error Correction • With m-bit memory word, 2m words are valid (i.e. all word combinations) • With n-bit code words, only 2m out of 2n will be valid. • If an invalid code word is read out of memory, the computer knows that an error occurred • There are algorithms for computing the check bits • Using this algorithm, the set or list of valid code words could be set • Compute the hamming distance between the code words • The minimum hamming distance computed will set the

  7. Code Hamming Distance (Cont.) Error Detection & Error Correction • With m-bit memory word, 2m words are valid (i.e. all word combinations) • With n-bit code words, only 2m out of 2n will be valid. • If an invalid code word is read out of memory, the computer knows that an error occurred • There are algorithms for computing the check bits and setting the list of valid code words • Compute the hamming distance (HD) between the code words ( two at a time) • The minimum hamming distance computed will set the Code Hamming Distance

  8. Code Hamming Distance Properties Error Detection & Error Correction • Error-detecting and error-correcting properties of a code depends on its hamming distance • To detect d single bit errors in a code word read A distance d + 1 code is needed (minimum HD between 2 code words) • With such a code there is no way that d single-bit errors can change a valid codeword into another valid code word • To correct d single bit errors in a code word read A distance 2d + 1 code is needed (minimum HD between 2 code words) • the valid code words are so far apart that even with d changes, the original codeword is still closer than any other codeword, so it can be uniquely determined

  9. Parity Bit (Error Detection Example) Error Detection & Error Correction • Consider the word 1011011, adding a parity bit • Even parity, to have even number of 1s  1011011 1 • Odd parity, to have odd number of 1s  1011011 0 • The code with such a parity bit will have a hamming distance of 2, as it will take 2 single bit error to move from one code word to another • With this, a single error could be detected • No errors could be corrected • When the error is detected, an error signal is set and the program will stop running

  10. Error Correction Example Error Detection & Error Correction • Consider the following code that has only 4 valid code words 00000 00000, 00000 11111, 11111 00000, 11111 11111 • The hamming distance of the code is 5 • Single-bit errors that could be detected is 4 • Single-bit errors that could be corrected is 2 • If the code word 00000 00111 is received • If there is only a double error, then the computer knows that the value should be 00000 11111 • If there is a triple error, (the three lower ones are wrong) then the error wouldn’t be corrected

  11. A A A error 1 0 1 1 1 1 0 0 0 0 0 1 1 B B C C 0 1 0 1 0 B C Even parity added Error in AC Error Detection for 4-bit Word Encoding of 1100 Error Detection & Error Correction

  12. Hamming Algorithm - Idea Error Detection & Error Correction • Hamming algorithm is used to build error correcting codes for any memory word size • In a hamming code, r parity bits are added to m bit data word forming a new word of length n = m + r • The bits are numbered starting at 1 (not 0) with bit 1 the leftmost (high order) bit. • All bits whose bit numbers are a power of 2 are parity bits • All other bits are data bits

  13. Hamming Algorithm (16-bits Example) Error Detection & Error Correction • Ex. 16-bit word • Each of the parity bits checks specific bit positions • The parity bit is set so that the number of 1s in the bit positions checked is even • Bit 1 checks bits: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21 • Bit 2 checks bits: 2, 3, 6, 7, 10, 11, 14, 15, 18, 19 • Bit 4 checks bits: 4, 5, 6, 7, 12, 13, 14, 15, 20, 21 • Bit 8 checks bits: 8, 9, 10, 11, 12, 13, 14, 15 • Bit 16 checks bits: 16, 17, 18, 19, 20, 21

  14. Hamming Algorithm (16-bits Example) Error Detection & Error Correction • Ex. 16-bit word • Each of the parity bits checks specific bit positions • The parity bit is set so that the number of 1s in the bit positions checked is even • Bit 1 checks bits: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21 • Bit 2 checks bits: 2, 3, 6, 7, 10, 11, 14, 15, 18, 19 • Bit 4 checks bits: 4, 5, 6, 7, 12, 13, 14, 15, 20, 21 • Bit 8 checks bits: 8, 9, 10, 11, 12, 13, 14, 15 • Bit 16 checks bits: 16, 17, 18, 19, 20, 21

  15. Hamming Algorithm (16-bits Example) Error Detection & Error Correction • Ex. 16-bit word • In general, bit b is checked by those bits b1, b2, ... bj such that b1 + b2 + ... + bj = b. • Ex. Bit 5 is checked by bits 1 and 4  1 + 4 = 5. • Ex. Bit 7 is checked by bits 1, 2 and 4  1 + 2 + 4 = 7

  16. Hamming’s Algorithm (1 Error Example) Error Detection & Error Correction • Ex. 16-bit word = 1111 0000 1010 1110 • If bit 5 was inverted (0 instead of 1): • The 5 parity bits will be checked, with the following results: • Parity bit 1 incorrect (1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21 contain five 1’s). • Parity bit 2 correct (2, 3, 6, 7, 10, 11, 14, 15, 18, 19 contain six 1’s). • Parity bit 4 incorrect (4, 5, 6, 7, 12, 13, 14, 15, 20, 21 contain five 1’s). • Parity bit 8 correct (8, 9, 10, 11, 12, 13, 14, 15 contain two 1’s). • Parity bit 16 correct (16, 17, 18, 19, 20, 21 contain four 1’s). • Add up all incorrect parity bits 1 + 4 = 5 5 was inverted.

More Related