We explore the SNR-optimal relay functionality in a \emph{memoryless} relaynetwork, i.e. a network where, during each channel use, the signal transmittedby a relay depends only on the last received symbol at that relay. We develop ageneralized notion of SNR for the class of memoryless relay functions. Thesolution to the generalized SNR optimization problem leads to the novel conceptof minimum mean square uncorrelated error estimation(MMSUEE). For the elementalcase of a single relay, we show that MMSUEE is the SNR-optimal memoryless relayfunction regardless of the source and relay transmit power, and the modulationscheme. This scheme, that we call estimate and forward (EF), is also shown tobe SNR-optimal with PSK modulation in a parallel relay network. We demonstratethat EF performs better than the best of amplify and forward (AF) anddemodulate and forward (DF), in both parallel and serial relay networks. Wealso determine that AF is near-optimal at low transmit power in a parallelnetwork, while DF is near-optimal at high transmit power in a serial network.For hybrid networks that contain both serial and parallel elements, and whenrobust performance is desired, the advantage of EF over the best of AF and DFis found to be significant. Error probabilities are provided to substantiatethe performance gain obtained through SNR optimality. We also show that, for\emph{Gaussian} inputs, AF, DF and EF become identical.
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