To Use the Codebook Information or Not: A Study of the Compress-and-Forward Relay Strategy
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The motivation of this thesis is to understand how nodes can cooperate in a particular relay channel, say a relay channel with orthogonal link between relay and destination. We are especially interested in the scenario where relay cannot decode the message sent because the coding rate is too large vis-a-vis what it can handle, and try to investigate the optimality of compress-and-forward strategy for this scenario. Specifically, noting that relay's compression is based on the unconditional distribution of its observation, it is natural to ask if relay can do better if it utilizes the source's codebook information, such that the performance of the relay network is improved? To answer this key question, we need to find the posterior Conditional Distribution of channel Output given Codebook Information (namely CDOCI) for the channel between source and relay. Firstly, we model it as Binary Symmetric Channel (BSC) and show that under the now standard random coding framework, if the input distribution is uniform, then with high probability, the CDOCI is asymptotically uniform when coding rate is greater than the channel capacity and the block length is sufficiently large. Then it is shown that under the discrete memoryless channel (DMC), for those strongly typical output sequences, with high probability, the CDOCI is also asymptotically uniform and close to the unconditional distribution, for rate above capacity and sufficiently large block length. These results implicate that relay can hardly do better with codebook information used. To confirm this implication, we show that for sufficiently large block length, the rate needed for relay to forward its observation when the codebook information is utilized approaches the rate needed when the relay simply ignore the codebook information, if the coding rate at source is larger than channel capacity. Now the answer to the above key question is apparent: in the cases of BSC and DMC, even if the relay tries to utilize the information obtained by knowing the codebook used at the source, it can hardly do better than simply ignore the codebook information. Therefore, the compress-and-forward strategy is kind of optimal in this sense, under the random coding framework.