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(solution) Exercise 9.16 Consider the bit-reversal scheme for the parallel channel described in Section 9.2.2.


Exercise 9.16 Consider the bit-reversal scheme for the parallel channel described in Section 9.2.2. Strictly speaking, the condition in (9.57) is not true for every integer between 0 and 2 R − 1. However, the set of integers for which this is not true is small (i.e., expurgating them will not change the multiplexing rate of the scheme). Thus the bit-reversal scheme with an appropriate expurgation of codewords is approximately universal for the 2-parallel channel. A reading exercise is to study [118] where the expurgated bit-reversal scheme is described in detail. Q7 Exercise 9.17 Consider the bit-reversal scheme described in Section 9.2.2 but with every alternate bit flipped after the reversal. Then for every pair of normalized code- word differences, it can be shown that l d 1 d 2 l 2 1 > , 64 · 2 2 R (9.104) where the data rate is R bits / s / Hz per sub-channel. Argue now that the bit-reversal scheme with alternate bit flipping is approximately universal for the 2-parallel channel. A reading exercise is to study the proof of (9.104) in [118]. Hint : Compare (9.104) with (9.53) and use the result derived in Exercise 9.14.

 


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