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Example 2-input 4-output DMC

Example 2-input 4-output DMC. Example. Example BSC. Transition probability 0.3’ Recall: MLD decodes to closest codeword in Hamming distance Note: extremely poor channel! Channel capacity = 0.12. Example. 1. 3. 2. 4. 4. 4. 5. 4. 6. 5. 7. 4. 5. 2. 5. 3. 4. 2.

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Example 2-input 4-output DMC

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  1. Example 2-input 4-output DMC

  2. Example

  3. Example BSC • Transition probability 0.3’ • Recall: MLD decodes to closest codeword in Hamming distance • Note: extremely poor channel! • Channel capacity = 0.12

  4. Example 1 3 2 4 4 4 5 4 6 5 7 4 5 2 5 3 4 2 3 6 6

  5. The Viterbi algorithm for the binary-input AWGN channel • Recall from Chapter 10: • Metrics: Correlation • Decode to the codeword v whose correlation rv with r is maximum • Symbol metrics: rv

  6. Performance bounds • Linear code, and symmetric DMC: • We can w. l. o. g. assume that the all zero codeword is transmitted • ..and that if we make a mistake, the wrong codeword is something else • Thus the decoding mistake will take place at the time instant in the trellis when the wrong codeword and the zero codeword merge • Therefore, interested in the elementary codewords (first event errors)

  7. More about first event errors

  8. Performance bounds: BSC • Then the first-event error probability at time t is • Assume that the wrong codeword has weight d Pd: Channel specific

  9. Simplifications of Pd B Pb (E)

  10. Performance bounds: Binary-input AWGN • Asymptotic coding gain  = 10 log 10 (Rdfree/2) • BSC derived from AWGN:

  11. Pd for binary-input AWGN v correct: (-1,...,-1) v’ incorrect Sum of d ind.Gaussian random variables (-1,N0/2Es) Count only positions where v’ and v differ

  12. Pd for binary-input AWGN • Asymptotic coding gain  = 10 log 10 (Rdfree)

  13. Rate 1/3 convolutional code 3.3 dB • = 10 log10 (1/3*7) = 3.68dB

  14. Some further comments Replace this by a bound • Union bound poor for small SNR • Contributions from higher order terms becomes significant • When to stop the summation? • Other bounds: Divide summation in two parts near/far

  15. Performance: Optimum R=1/2 codes on AWGN

  16. Performance: Optimum R=1/2 codes on Hard quantized AWGN

  17. R=1/2, =4 code, varying quantization

  18. Rate 1/3 codes

  19. Rate 2/3 codes

  20. Code design: Computer search • Criteria • High dfree • Low Adfree (Low Bdfree)? • High slope/low truncation length* • Use Viterbi algorithm to determine the weight properties • Example:

  21. Example: Viterbi algorithm for distance properties calculation 3 4 6 5 7 6 8 7 5 7 6 7 7 7 7

  22. Suggested exercises • 12.6-12.21

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