Principles of Digital Communication And Coding

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4. 15) and (3. 4. 17)] = V4p(l - P) and (Z) A WGN (4. 5. 1) (4. 5. 2) We note also, as was already observed in Sec. 2. 1 1, that if the AWGN channel is converted to the BSC by hard quantization for &JN, < 1, then CONVOLUTIONAL CODES 247 in which case -In Z * -In L/l - 4S s /nN ] ^ -In for a loss of 2/n, or approximately 2 dB, in energy-to-noise ratio.
However, for these two special channels, tighter bounds can be found by obtaining the exact pairwise error probabilities rather than their Bhatta
...charyya bounds. For the BSC, we recall from (2. 10. 14) that, for unmerged segments at distance d from the correct path 7 I tf)p k (l- dodd d k=d!2 + (4. 5. 3) This can be used in the middle expressions of inequalities (4. 4. 5) and (4. 4. 8) to obtain tighter results than (4. 4. 7) and (4. 4. 12) (see also Prob. 4. 10).
Similarly, for the binary-input AWGN channel, we have from (2. 3. 10) that the pairwise error probability for code vectors at distance d is P = (4. 5. 4) While we may substitute this in the above expressions in place of Z d = e d * * 9 a more elegant and useful expression results from noting that (Prob.


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