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Understanding Error Correction in Computer Memory Systems

Explore the basics of error correction in computer memory systems, covering topics such as semiconductor system errors, hard and soft failures, ECC functions, Hamming error-correcting codes, and Hamming SEC-DED codes. Learn about error detection and correction mechanisms and how they enhance memory reliability.

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Understanding Error Correction in Computer Memory Systems

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  1. AKT211 – CAO08 – Computer Memory (2) Ghifar Parahyangan Catholic University Okt 31, 2011

  2. Last Course Review • Computer Memory System • Memory Characteristics • Memory Hierarchy • RAM Basic Technology • Semiconductor • SRAM vs DRAM • Advanced RAM Organization • SDRAM vs DDR-RAM

  3. Outline • Error Correction • Single error correction • Double error correction

  4. THE BASIC OF ERROR CORRECTION

  5. Semiconductor System Error • Hard Failures • Permanent physical defect so that it can’t reliably store data • Stuck at 0 or 1 or switch erratically between 0 and 1 • Soft Error • Random, nondestructive event that alters the contents of one or more cell without damaging the memory • Caused by power supply problems or alpha particles • Most modern main memory systems include logic for both detecting and correcting errors

  6. Single-bit error • Only 1 bit in data unit has changed

  7. Error Correcting Code (ECC) Function

  8. Simples form of Error Detection • Using a parity bit • A bit that is added to ensure that the number of bits with the value ‘1’ in a set of bits is even or odd • Only for detecting 1-bit error, not more, nor correcting ! • E.g.: no error • E.g.: 1 bit error A wants to transmit: 1001 A computes parity bit value : 1^0^0^1 = 0 A adds parity bit and sends : 10010 B receives : 10010 B computes parity : 1^0^0^1 = 0 B reports correct transmission after observing expected result A wants to transmit: 1001 A computes parity bit value : 1^0^0^1 = 0 A adds parity bit and sends : 10010 *** TRANSMISION ERROR *** B receives : 11010 B computes parity : 1^1^0^1 = 1 B reports incorrect transmission after observing unexpected result

  9. Hamming Error-Correcting Code • linear error-correcting code • can detect up to d-1 bit errors • can correct (d-1)/2 • d is the minimum hamming distance between all pairs in the code words

  10. Hamming (7, 4) Code • encodes 4 data bits (d1, d2, d3, d4) into 7 bits by adding 3 parity bits (p1, p2, p3) • single error correction

  11. Hamming (7,4) Example

  12. HAMMING ALGORITHM GENERALIZATION FOR SINGLE ERROR CORRECTION

  13. Generalization of the Hamming Single Error Correction • The comparison logic receives as input two K-bit values • A bit-by-bit comparison is done by taking the XOR • The result is called the syndrome word • The value 0 indicates that no error was detected and otherwise • We can determine the position from that syndrome word

  14. Required criteria for Hamming Error Correction • If the syndrome contains all 0s, no error has been detected • If the syndrome contains one and only one bit set to 1, then an error has occurred in one of the n check bits. No correction is needed • If the syndrome contains more than one bit set to 1, then numerical value of the syndrome indicates the position of the data bit in error

  15. SEC Step-by-Step • Determine how long the code (check bits) must be • Determine the stored position for each bit in M data bits and K check bits • Construct the appropriate XOR function that match with the required criteria

  16. 1. Determine how long the code must be • M : number of bits in data bits • K : number of bits in code bits • Because an error could occur on any of the M data bits or K check bits, we must have : • e.g.: for a word of 8 data bits (M=8), we have 2K – 1 ‹ M + K K=3 : 23 – 1 < 8 + 3 K=4: 24– 1 >8 + 4

  17. 2. Determine the stored position Let’s see the explanation ! 

  18. 3. Construct the XOR Function Again, let’s see the explanation ! 

  19. Hamming SEC-DED Code • Nowadays, more commonly, semiconductor memory is equipped with a single-error-correcting, double-error-detecting (SEC-DED) code • Needs 1 extra parity bit that indicates whether the total number of 1s is even or odd • Enhances the reliability of the memory, but adds the cost of complexity • E.g. : • The IBM 30xx implementations used an 8-bit SEC-DED code for each 64 bits of data in main memory • The size is actually about 12% larger than is apparent to the user

  20. Hamming SEC-DED Code (2)

  21. Any Question ?

  22. Reference • Chapter 5.2: Error Correction (Stallings, William. Computer Organization and Architecture, 8th ed. Prentice Hall. 2010)

  23. Exercises • Denganpenggunaanalgoritma Hamming, berapakahjumlah check bit yang dibutuhkanjika data bit berukuran 1024-bit ? • Terdapat data bit sebanyak 8-bit tersimpandidalammemori yang isinya 11000010. Denganmenggunakanalgoritma Hamming, tentukannilai check bit yang akantersimpanpadamemori. • Untuk data word 8-bit 00111001, check bit yang tersimpanadalah 0111. Anggapterjadierrorpadapembacaanmemori. Ketika data bit tersebutdibacaulangdarimemori, nilai check bit yang terhitungadalah 1101. Berapakahsebenarnyanilai data bit yang error?

  24. Week 8 Assignment • Bentuklahpersamaan XOR untukmenentukan SEC code (check bit) denganmenggunakanalgoritma Hamming untuk data bit berukuran 16-bit. Bagaimanahasil check bit apabilamenerimamasukan data bit 0101000000111001 ? Simulasikanbagaimanaalgoritma Hamming dapatmengoreksi error apabilaterjadi error di data bit posisi ke-5 (0101000000101001). JelaskanjawabanAndaselengkap-lengkapnya.

  25. THANK YOU

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