# Correcting Single-bit Errors with CRC8 in ATM Cell Headers

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1 Freescale Semiconductor Application Note Document Number: AN2918 Rev. 0, 06/2005 Correcting Single-bit Errors with CRC8 in ATM Cell Headers by Bo Lin Computing Platform Division Freescale Semiconductor, Inc. East Kilbride, Scotland 1 Introduction It is well known that CRC is a class of codes providing strong error detection. However, a CRC code can also be used to correct up to two error bits in a data block (Blahut, 1983). In many cases, such as ITU-T I (1999), a CRC code is required to correct a single-bit error in a data block. This application note gives a generic introduction on forward error correction code and describes a very efficient table lookup method to correct a single-bit error in an ATM cell header defined by ITU-T I Contents 1. Introduction Correcting Single-Bit Errors in ATM Cell Headers A Pseudo-Procedure and Numerical Examples Summary References Revision History Freescale Semiconductor, Inc., All rights reserved.

2 Introduction Let d N-1 d N-2 d 1 d 0 denote an N-bit data block D where D s polynomial representation is D(x) = d N-1 x N- 1 + d N-2 x N d 1 x + d 0. A single-bit error occurs in data block D means that there exists a j with 0 j < N such that d j is inverted to d j by the single-bit error. This is equivalent to represent a corrupted data block associated with D as D ( x) = D(x) + x j where + denotes bit-wise modulo 2 addition or simply. On the other hand, given a binary string B = b N-1 b N-2 b 1 b 0, B s CRC8 can be calculated as CRC8(B) = B(x)x 8 mod G(x) where G(x) is the generator polynomial. It is defined as G(x) = x 8 + x 2 + x + 1 in ITU-T I As a result, D s CRC8 can be calculated as CRC8(D) = D(x)x 8 mod G(x). The data block D CRC8 is transmitted. In other words, D(x)x 8 + CRC8(x) is transmitted. On the receiving side, D CRC8 is received where there might be a single-bit error in the received data string D CRC8. This means that the data block D (x)x 8 + CRC8 = D(x)x 8 + CRC8(D) + e x j is received, where 0 j (N 8) and e is either 0 or 1. By verifying CRC8 over the received data block (data part only), we have V(D ) = ((D(x) + f x i ) x 8 mod G(x) (there might be single-bit error at location i) = CRC8(D) + (f x i ) x 8 mod G(x), where 0 i N and f is either 0 (no error) or 1 (single-bit error in data). On the other hand, the received CRC checksum CRC8 = CRC8(D) + h x j mod G(x), where (N 1) < j < (N + 8) and h is either 0 (no error) or 1 which indicates the possible single-bit error in CRC checksum field. It is noticed that only one of f or h can be 1 since the code is designed to correct at most one error over D CRC8. By adding the re-calculated V(D ) and the received CRC8 together, we have S = V(D ) + CRC8 = (CRC8(D) + (f x j ) x 8 + CRC8(D) + (h x j ) x 8 ) mod G(x) = ((f x i ) x 8 + h x j ) mod G(x) (note: + =, N 0 i N and N j < (N + 8)) = e x j mod G(x) with 0 j < (N + 8) since only one of f or h can be 1 and the e is used to replace the two with a wider range. If there is no error, then the S is 0, otherwise, S = x j mod G(x) = S(j) indicating an error at logical location j. To correct the single-bit error, the index j needs to be determined from the calculated S(j) on the receive side. For a CRC code, it is proven that 2 Freescale Semiconductor

4 Correcting Single-Bit Errors in ATM Cell Headers Table 1. The Error-Location and Syndrome-Search Table (continued) For example, if syndrome = 91 is calculated, by searching Table 1, errorloc = 22 is obtained. By comparing with Figure 1, it is found that B2 s second bit is corrupted and can be corrected by B2 = B On the other hand, if syndrome = 123 is calculated, by searching Table 1, no errorloc will be found. This means an uncorrected error or multiple-bit error has occurred in the ATM cell header. The above approach can be described in the following procedure: Procedure errorlocationsearch() 1. S = syndromecalculate() 2. return errorloc = Search Table 1 with S One drawback of the above single-bit error approach is that it requires a search operation on an unsorted (unordered) table which is not very efficient. However, by taking advantage of the small size of the table, sorting the table with errloc, and padding the illegal entries with a special value, such as 1, Table 2 can be obtained for efficiently locating a single-bit error. Table 2. Syndrome-Indexed Error-Location Index Table syndrome errorloc syndrome errorloc syndrome errorloc syndrome errorloc Freescale Semiconductor

5 Correcting Single-Bit Errors in ATM Cell Headers Table 2. Syndrome-Indexed Error-Location Index Table (continued) Freescale Semiconductor 5

6 Correcting Single-Bit Errors in ATM Cell Headers Table 2. Syndrome-Indexed Error-Location Index Table (continued) Freescale Semiconductor

7 A Pseudo-Procedure and Numerical Examples Table 2. Syndrome-Indexed Error-Location Index Table (continued) For example, if syndrome = 91 is calculated, by looking up Table 2, errorloc = 22 is obtained. On the other hand, if syndrome = 123 is calculated, by looking up Table 2, errorloc = 1 is returned and this indicates that an uncorrected error or multiple-bit error has occurred in the ATM cell header. Table 2 only occupies 256 bytes, and its lookup procedure is extremely efficient and can be easily implemented in various Freescale processors. 3 A Pseudo-Procedure and Numerical Examples The following pseudo-procedure summarizes the table lookup method of correcting single-bit errors in ATM cell headers: Procedure errorcorrectionindex() 1. S = syndromecalculate(); 2. if (S == 0) 3. return ( no error ); 4. else 5. errloc = lookup Table 2 with S; 6. if (errloc < 0) 7. return ( uncorrectable error ); 8. else 9. if (0 errloc 7) // i.e. a single-bit error in HEC 10. return ( no error in rxdata ); 11. else 12. errorpattern = (0x )<<(errorLoc 8); 13. rxdata = rxdata errorpattern 14. return ( error corrected ) 15. endif 16. endif 17. endif Example 1: Single-bit error case txcell = txdata txhec = 0x x69 rxcell = rxdata rxhec = 0x x69 verhec = CRC8(0x ) = 0x7F Freescale Semiconductor 7

8 Summary S = verhec rxhec = 0x7F 0x69 = 0x16 = 22 errorloc = 32 after looking up Table 2 with index 22. errorpattern = 0x <<(32 8) = 0x rxdata = 0x x = 0x return ( error corrected ); Example 2: 2 bit error case txcell = txdata txhec = 0x x69 rxcell = rxdata rxhec = 0x x69 verhec = CRC8(0x ) = 0x7F S = syndromecalulate() = verhec rxhec = 0xA9 0x69 = 0xC0 = 192 errorloc = -1 after looking up Table 2 with index 192. Return ( uncorrectable error ) 4 Summary A single-bit error correction can be achieved by calculating a CRC value on the received data field, XORing the calculated CRC with the received CRC checksum to obtain an error syndrome. By looking up Table 2 with the error syndrome, the logical location of a single-bit error can be found. The single-bit error can be corrected by reversing the error bit. 5 References Blahut, R.E, Theory and Error Control Codes. Reading: Addison-Wesley Publishing Company, Inc., ITU-T I (International Telecommunication Union), B-ISDN user-network interface Physical layer specification: General characteristics, 02/ Revision History Table 3 provides a revision history for this application note. Table 3. Document Revision History Revision Number Date Substantive Change(s) 0 06/08/2005 Initial public release. 8 Freescale Semiconductor

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