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154 Part 2 Lecture 7

# 154 Part 2 Lecture 7 - Miscellaneous Topics in Coding 1...

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Misc 1 5/15//2007 Information Theory and Coding © K. Sam Shanmugan, 2007 Miscellaneous Topics in Coding 1. Erasure Correction 2. Non-binary (Reed-Solomon) Codes 3. Concatenated Codes 4. Turbo Codes

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Misc 2 5/15//2007 Information Theory and Coding © K. Sam Shanmugan, 2007 Erasure Correction In many instances, the receiver may not be sure of its output and may declare a symbol to be an “erasure”. For example, Matched filter Output : 1.2, -0.9, 0.05 , -1.2, 0.8,. . Output bits: 1, 0, E , 0, 1 E: Erasure flag With erasure indication , the correct value of the erased bit in not known, but the location of the erased bit is known The error control decoder can use this information to correct transmission errors as well as erasures If a block code has a minimum distance of d min , it can correct any combination of α errors and γ errors per block simultaneously , where 2 α + γ +1 ≤ d min There are many different algorithms for simultaneously correcting erasures and errors; A simple algorithm and an example is given in the next chart
Misc 3 5/15//2007 Information Theory and Coding © K. Sam Shanmugan, 2007 Erasure and Error Correction 1. Replace all the erasure symbols with 0’s and decode 2. Then replace all erasure symbols with ‘1’ s and decode Of the two code words found in 1 and 2, the one corresponding to the smallest number of errors corrected outside of the γ erased positions is chosen as the final decoded output Example : Consider a (6,3) code with a minimum distance of 3 and the following code words 000000 110100 011010 101110 101001 011101 110011 000111 Suppose the transmitted code word was 110011 and received bits are EE 0011 Decoded output with 00 0011 000 1 11 one corrected bit (One correction) Decoded output with 11 0011 110011 zero corrected bit ( zero correction) Output = 110011

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Misc 4 5/15//2007 Information Theory and Coding © K. Sam Shanmugan, 2007 Non-binary Codes Reed-Solomon Codes Reed Solomon codes use nonbinary code symbols from an alphabet of size 2 m ( The code symbols however are converted to binary bits prior to transmission) Encoding and decoding are done using finite filed arithmetic The values of n,k,m and d min satisfy the following relationships (n,k) = ( 2 m - 1 , 2 m - 1 -2t) ; d min = (n-k) + 1 This code can correct any combination of α M-ary symbols errors and γ M-ary symbol erasures per block simultaneously, 2 α + γ+1 ≤ d min < n-k Input Binary (n,k) R-S Encoder M-ary symbols M=2 m M-ary to Binary Channel bits Input block size = k*m binary bits Or k M-ary symbols Output block size = n*m binary bits Or n M-ary symbols
Misc 5 5/15//2007 Information Theory and Coding © K. Sam Shanmugan, 2007 Primary Advantage of R-S Codes R-S codes are particularly useful for burst-error correction For example, consider a (255, 247), m = 8, R-S code ; Each R-S code symbol in this case is a byte n – k = 8 , and hence the code can correct 3 symbol errors per block Now suppose during transmission there is a burst error of 25 consecutive

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154 Part 2 Lecture 7 - Miscellaneous Topics in Coding 1...

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