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 .
@cite_39: Via collaborative beamforming, nodes in a wireless network are able to transmit a common message over long distances in an energy efficient fashion. However, the process of making available the same message to all collaborating nodes introduces delays. In this paper, a medium access control-physical (MAC-PHY) cross-layer scheme is proposed that enables collaborative beamforming at significantly reduced collaboration overhead. It consists of two phases. In the first phase, nodes transmit locally in a random access time-slotted fashion. Simultaneous transmissions from multiple source nodes are viewed as linear mixtures of all transmitted packets. In the second phase, a set of collaborating nodes, acting as a distributed antenna system, beamform the received analog waveform to one or more faraway destinations. This step requires multiplication of the received analog waveform by a complex weight, which is independently computed by each collaborating node, and which allows packets bound to the same destination to add coherently at the destination node. Assuming that each node has access to location information, the proposed scheme can achieve high throughput, which in certain cases exceeds one. Analyses of the average beampattern, networking performance, and symbol error probability corresponding to the proposed scheme are provided.
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.