AKT211 – CAO 08 – Computer Memory (2) Ghifar Parahyangan Catholic University Okt 31, 2011

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

Outline Error Correction Single error correction Double error correction

THE BASIC OF ERROR CORRECTION

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

Single-bit error Only 1 bit in data unit has changed

Error Correcting Code (ECC) Function

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

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

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

Hamming (7,4) Example

HAMMING ALGORITHM GENERALIZATION FOR SINGLE ERROR CORRECTION

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

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

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

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

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

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

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

Hamming SEC-DED Code (2)

Any Question ?

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

Exercises Dengan penggunaan algoritma Hamming, berapakah jumlah check bit yang dibutuhkan jika data bit berukuran 1024-bit ? Terdapat data bit sebanyak 8-bit tersimpan di dalam memori yang isinya 11000010. Dengan menggunakan algoritma Hamming, tentukan nilai check bit yang akan tersimpan pada memori. Untuk data word 8-bit 00111001, check bit yang tersimpan adalah 0111. Anggap terjadi error pada pembacaan memori. Ketika data bit tersebut di baca ulang dari memori, nilai check bit yang terhitung adalah 1101. Berapakah sebenarnya nilai data bit yang error?

Week 8 Assignment Bentuklah persamaan XOR untuk menentukan SEC code (check bit) dengan menggunakan algoritma Hamming untuk data bit berukuran 16-bit. Bagaimana hasil check bit apabila menerima masukan data bit 0101000000111001 ? Simulasikan bagaimana algoritma Hamming dapat mengoreksi error apabila terjadi error di data bit posisi ke-5 (0101000000101001). Jelaskan jawaban Anda selengkap-lengkapnya.

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