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

3. 3  Computer memories could make errors due to electric spikes leading to incorrect behavior  To protect against such behavior, two ways are adopted 1.Error-detection codes 2.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. 4Error 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 m r n (codeword)

5. 5Error 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. 6Error Detection & Error Correction  With m-bit memory word, 2 m words are valid (i.e. all word combinations)  With n-bit code words, only 2 m out of 2 n 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. 7Error Detection & Error Correction  With m-bit memory word, 2 m words are valid (i.e. all word combinations)  With n-bit code words, only 2 m out of 2 n 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. 8Error 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. 9Error 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. 10Error 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. 11Error Detection & Error Correction A BC 1 1 0 0 Encoding of 1100 A BC 1 1 0 0 Even parity added 0 01 A BC 1 1 0 1 Error in AC 0 01 error

12. 12Error 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. 13Error 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 212019181716151413121110987654321

14. 14Error 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 212019181716151413121110987654321

15. 15Error 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 212019181716151413121110987654321

16. 16Error 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. 011101101000001110100 212019181716151413121110987654321