Channel Coding: An Introduction to Error Detection and Correction, Slides of Digital Communication Systems

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Channel Coding

Binit Mohanty Ketan Rajawat

Recap…

 Information is transmitted through channels (eg. Wires, optical fibres and even air)  Channels are noisy and we do not receive what was transmitted

Repetition Coding

 Assume 1/3 repetition  What is the probability of error?  (^) If crossover probability p = 0.01, Pe ≈

 (^) Here coding rate R = 1/3. Can we do better? How much better?

2 3 2 3 P C p ( 1 p ) p e

Shannon’s Theorem

 Given,  (^) A noisy channel (some fixed p)  (^) A value of Pe which we want to achieve “We can transmit through the channel and achieve this probability of error at a maximum coding rate of C(p)”  (^) Is it counterintuitive?  (^) Do such good codes exist?

Parity Check Codes

 #information bits transmitted = k  #bits actually transmitted = n = k+  Code Rate R = k/n = k/(k+1)  Error detecting capability = 1  Error correcting capability = 0

2-D Parity Check

 Rate?  Error detecting capability?  Error correcting capability? 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 0 0 1 1 1 Bottom row consists of check bit for each column Last column consists of check bits for each row

Linear Block Codes

 Linearity  Example : 4/7 Hamming Code  (^) k = 4, n = 7  (^) 4 message bits at (3,5,6,7)  (^) 3 parity bits at (1,2,4)  (^) Error correcting capability =  (^) Error detecting capability = 2  (^) What is G? , 1 1 c  m G

c m G

2 2

c c ( m m ) G 1 2 1 2   

Cyclic codes

 Special case of Linear Block Codes  Cyclic shift of a codeword is also a codeword  (^) Easy to encode and decode,  (^) Can correct continuous bursts of errors  (^) CRC (used in Wireless LANs), BCH codes, Hamming Codes, Reed Solomon Codes (used in CDs)

Convolutional Codes

 Encoder consists of shift registers forming a finite state machine  Decoding is also simple – Viterbi Decoder which works by tracking these states  First used by NASA in the voyager space programme  Extensively used in coding speech data in mobile phones

Achieving Capacity

 Do Block codes and Convolutional codes achieve Shannon Capacity? Actually they are far away  Achieving Capacity requires large k (block lengths)  Decoder complexity for both codes increases exponentially with k – not feasible to implement

Summary

 There is a limit on the how good codes can be  Linear Block Codes and Convolutional Codes have traditionally been used for error detection and correction  Turbo codes in 1993 introduced a new way of designing very good codes with feasible decoding complexity