The capacity of the symbol-asynchronous single-relay channel is investigated. Symbol asynchronism implies that the codewords transmitted from the relay and the source do not coincide in time at the destination. Due to propagation or implementation effects (e.g. mis-synchronized clocks), symbol asynchronism occurs in many practical situations. In ad hoc scenarios, such as sensor networks, achieving perfect synchronism is challenging (due to the distributed nature of the network). Herein it is shown that the symbol-asynchronous single-relay channel can be modeled as a MIMO relay channel with memory. First, lower and upper bounds on the capacity are derived under the assumption that the receivers know the symbol epochs and the transmitters know their mutual offset. Next, the case where the transmitters only know that the mutual offsets that parameterize the channel belong to an uncertainty set and receivers have no knowledge of the symbol epochs is examined. Conditions for characterizing capacity (when the upper and lower bounds match) are also derived. Simple asynchronous coding schemes for the relay channel are also explored. The proposed coding scheme can be extended to the relay with memory. Bilayer LDPC codes are used to replicate the binning strategy of the relay and MLC coding with MSD decoding is used to counter the effect of memory.
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