We consider the multihop broadcasting problem for @math nodes placed uniformly at random in a disk and investigate the number of hops required to transmit a signal from the central node to all other nodes under three communication models: Unit-Disk-Graph (UDG), Signal-to-Noise-Ratio (SNR), and the wave superposition model of multiple input single output (MISO). In the MISO model, informed nodes cooperate to produce a stronger superposed signal. We do not consider the problem of transmitting a full message nor do we consider interference. In each round, the informed senders try to deliver to other nodes the required signal strength such that the received signal can be distinguished from the noise. We assume sufficiently high node density @math . In the unit-disk graph model, broadcasting needs @math rounds. In the other models, we use an Expanding Disk Broadcasting Algorithm, where in a round only triggered nodes within a certain distance from the initiator node contribute to the broadcasting operation. This algorithm achieves a broadcast in only @math rounds in the SNR-model. Adapted to the MISO model, it broadcasts within @math rounds. All bounds are asymptotically tight and hold with high probability, i.e. @math .
A transmission with cooperative beamforming requires phase synchronization of the collaborating transmitters to produce a beam and sharing the data to transmit. The authors of @cite_39 present therefore a two phase scheme: in phase one, the message is spread among nodes in a disk in the plane around the node holding the original message. The open-loop and closed-loop approach can be used to synchronize nodes to the destination or a known node position and time synchronization. In phase two, the synchronized nodes jointly transmit the message towards the destination.
@cite_39
1997887828
We propose and demonstrate a scalable architecture for distributed receive beamforming. In a receive cluster of N + 1 nodes receiving a message from a distant transmitter, N nodes are designated as amplify-and-forward relays and one node is designated as the receiver. The relay nodes apply a phase shift to their received signal and forward it such that their forwarded signals add up constructively at the receiver, with received SNR scaling linearly with N. This approach transforms a distributed receive beamforming problem on the "long link" from transmitter to receive cluster into a distributed transmit beamforming problem on the "short link" from relays to receiver, so that the number of degrees of freedom on the short link need not scale with N. A key simplification relative to distributed transmit beamforming is that, for stable oscillators, relay frequency synchronization is not required. For oscillators with drift, we provide a simple rule of thumb for when explicit frequency synchronization can be avoided. Explicit timing alignment can also be avoided by exploiting the timing of the message received on the long link. That leaves the problem of phase adjustment at the relays, and we employ an algorithm originally invented for distributed transmit beamforming for this purpose, using one bit (per iteration) feedback from the receiver. Experimental results with software-defined radios (whose oscillators have significant drift) demonstrate that the expected gains in received signal strength are obtained with the proposed architecture.