Cooperative Relaying at Finite SNR–-Role of Quantize-Map-and-Forward
Quantize-Map-and-Forward (QMF) relaying has been shown to achieve the optimal diversity-multiplexing trade-off (DMT) for arbitrary slow fading full-duplex networks as well as for the single-relay half-duplex network. A key reason for the DMT-optimality of QMF is that quantizing at the noise level suffices to achieve the cut-set bound approximately to within an additive gap, without the requirement of any instantaneous channel state information (CSI). However, DMT only captures the high SNR performance and potentially, limited CSI at the relay can help improve the performance in moderate SNR regimes. In this work we propose an optimization framework for QMF relaying over slow fading channels. Focusing on vector Gaussian quantizers, we optimize the outage probability for the full-duplex single relay by finding the best quantization level according to the available CSI at the relays. For the half-duplex relay channel, we find jointly optimal quantizer distortions and relay schedules using the same framework. Also, for the $N$-relay diamond network, we derive an universal quantizer that uses only the information of network topology. The universal quantizer sharpens the additive approximation gap of QMF from the conventional $\Theta(N)$ bits/s/Hz to $\Theta(\log(N))$ bits/s/Hz. Analytical solutions to channel-aware optimal quantizers for two-relay and symmetric $N$-relay diamond networks are also derived. In addition, we prove that suitable hybridizations of our optimized QMF schemes with Decode-Forward (DF) or Dynamic DF protocols provides significant finite SNR gains over the individual schemes.