Contents Communications Theory Parallel vs. serial transmission Transmission Capacity (Shannon) Error detection and correction Communications Media Optical fibers Coaxial cables Twisted pairs Wireless

Error Detection Example Belgian Bank Account Numbers Bank account number structure Bank identification : 3 digits Account number : 7 digits Error detection : 2 digits The ten first digits modulo 97 are appended for error detection purposes. This algorithm allows detection of all single digit errors Example : 140-0571659-08. 1400571659 MOD 97 = 08 140-0671659-08. 1400671659 MOD 97 = 01

Error detection and correction Length of messages : Informative message: Redundancy: # Messages send: # Messages received: Hamming Distance (X-Y): k + r <= LMax k bits r bits, f(inf.mess.) 2 k 2 k+r i=1 |Xi-Yi| k+r

Error detecting codes 00 11 k = 1; r = 1; red.bit = inf.bit. 00 01 11 10 Hd = 2 00 11 Single bit errors are detected if hamming distance between legitimate messages > 1. No guessing is possible as erroneous messages are at equal distances from several correct ones.

Error correcting codes k = 1; r = 2; red.bits = inf.bit. 001 011 111 101 000 010 110 100 Hd = 3 000 111 Hamming distance between legitimate messages > 2. This implies that each erroneous message is closer to one correct message than to any other.

Error correcting codes for single bit error correction Required Overhead for single bit error correction k+r < 2r information redundancy Overhead 1 <= 4 <= 11 <= 26 <= 57 <= 120 <= 247 2 3 4 5 6 7 8 200 % 75 % 36 % 19 % 11 % 6 % 3 %

Error Correction Error detecting codes Correction by retransmission of erroneous blocks If few errors, very low overhead Most common approach to error correction in data communications Error correcting codes Very high overhead with short data blocks Longer data blocks can have multiple errors Used when retransmission impossible or impractical Also used when error rate rather high. Error correcting codes for long blocks, with multiple errors exist and are used (trellis encoding)