Error Detection and Correction

1. Error Detection and Correction Martin Weiss

2. Objectives of this Meeting • Describe the major error detection techniques • Describe forward error correction

3. Review of the Last Meeting • Multiplexing • Topologies • Telephone network

4. Data Link Layer Issues • Error control • Flow control • Local addressing

5. Error Detection • Parity • Cyclical redundancy check

6. Vertical and Longitudinal Redundancy Checks Bit 1 Bit 2 Bit n Parity Character 1 b11 b21 bn1 R1 Character 2 b12 b22 bn2 R2 Character m b1m b2m bnm Rm Parity C1 C2 Cn Cn+1 LRC VRC

7. Error Correction • Automatic Response Request (ARQ) • Transmitter sends a message • Receiver acknowledges • In case of NAK, the transmitter retransmits the message

8. Forward Error Correction • Redundancy is built into the message • Detects and corrects errors • Example: Hamming Code

9. Hamming Codes • Developed by R.W. Hamming • Uses multiple parity bits to locate errors in a word • Theory • If n parity bits are used, there are 2n possible combinations • We need to associate each bit position with a combination of parity bits • Thus, we must choose n so that 2n > m (the number of bits in the message)

10. Hamming Code • Assume an 8 bit message • n > log2m • Choose n = 4 m1 m2 m3 m4 m5 m6 m7 m8 p1 x x x x x x x x x x p2 x x x x p3 x x x x p4

11. m1 m2 m3 m4 m5 m6 m7 m8 p1 x x x x x x x x x x p2 x x x x p3 x x x x p4 Hamming Code Even Parity Computation Table Transmitted Sequence p1 p2 m1 p3 m2 m3 m4 p4 m5 m6 m7 m8

12. Hamming Code Error (bit position) Parity Errors <none> <none> p1 p2 p1 & p2 p3 p1 & p3 p2 & p3 p1, p2, & p3 p4 p1 & p4 p2 & p4 p1, p2 & p4 p3 & p4 1 2 3 4 5 6 7 8 9 10 11 12

13. Hamming Code • Consider the bit stream 10010011 • Parity bits are: • p1=1 • p2=1 • p3=0 • p4=0 • Transmitted bit stream is 111000100011

14. Hamming Code • If the received bit stream is • 111010100011 • p1 and p3 fail • 101000100011 • p2 fails • 111000100001 • p1, p2, and p4 fail