Generalized Degrees of Freedom of Noncoherent Diamond Networks

We study the generalized degrees of freedom (gDoF) of the noncoherent diamond (parallel relay) wireless network with asymmetric distributions of link strengths. We use the noncoherent block-fading model introduced by Marzetta and Hochwald, where no channel state information is available at the transmitters or at the receivers and the channels remain constant for a coherence time of T symbol durations. We first derive an upper bound for the capacity of this channel and then derive the optimal structure for the solution of the upper bound optimization problem. Using the optimal structure, we solve the upper bound optimization problem in terms of its gDoF. Using insights from our upper bound signaling solution, we devise an achievability strategy based on a novel scheme that we call train-scale quantize-map-forward (TS-QMF). This scheme uses training in the links from the source to the relays, scaling and quantizing at the relays combined with nontraining-based schemes. We show the optimality of this scheme by comparing it to the upper bound in terms of the gDoF. In noncoherent point-to-point multiple-input-multiple-output (MIMO) channels, where the fading realization is unknown to the transmitter and the receiver, an important tradeoff between communication and channel learning was revealed by Zheng and Tse, by demonstrating that not all the available antennas might be used, as it is suboptimal to learn all their channel parameters. Our results in this paper for the diamond network demonstrate that in certain regimes of relative channel strengths, the gDoF-optimal scheme uses a subnetwork, demonstrating a similar tradeoff between channel learning and communication. In some regimes, it is gDoF-optimal to do relay selection, i.e., use a part of the network. In the other regimes, even when it is essential to use the entire network, it is suboptimal to learn the channel states for all the links in the network, i.e., traditional training-based schemes are suboptimal in these regimes.