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_18: 
@cite_14: The efficiency of routing algorithms in ad-hoc networks is measured by delay, throughput, and energy. Here, we focus on routing algorithms optimizing energy consumption while providing small routing delay. For this, we exploit the sender beamforming gain in the line-of-sight path-loss model, where multiple nodes (each with a single antenna) cooperate for beamforming and the routing algorithms provide distributed self-synchronization. While direct point-to-point communication over distance d in the line-of-sight model needs transmission power Θ(d 2 ), and multi-hop power needs power Θ(d) and delay Θ(d), we can reduce the power to Θ(√d) or Θ(log d) depending on the geometry. We present three algorithms with different trade-offs. The first algorithm is designed for grid nodes in the plane and has a point-to-point delay of Θ(log d) and overall power consumption of Θ(√d). The second algorithm for the same geometry decreases the delay to Θ(1 over e log log d) with power Θ((√d) 1+e ) for e > 0. The third algorithm requires a three-dimensional grid network and achieves a delay of Θ(log d) and reduces the energy needed by all nodes to Θ(log d).
@cite_4: We consider a wireless network with n users distributed over a square area A = n. Under line-of-sight propagation, this network has Θ(√n) degrees of freedom. At high SNR, these degrees of freedom can be readily achieved by multi-hop relaying between nodes. At low SNR however, the performance is determined by the power transfer in the network. We show that none of the existing architectures can achieve optimal capacity scaling. We develop a beamforming architecture where signals are relayed by coherent combining over multiple clusters. The key ingredient is an analysis of the beamforming gain achievable between two clusters of nodes under the line-of-sight propagation model. This result reveals a new regime for large two-dimensional wireless networks, where beamforming techniques are needed to achieve capacity.
@cite_9: 
@cite_0: We consider a dense fading multi-user network with multiple active multi-antenna source-destination pair terminals communicating simultaneously through a large common set of K multi-antenna relay terminals in the full spatial multiplexing mode. We use Shannon-theoretic tools to analyze the tradeoff between energy efficiency and spectral efficiency (known as the power-bandwidth tradeoff) in meaningful asymptotic regimes of signal-to-noise ratio (SNR) and network size. We design linear distributed multi-antenna relay beamforming (LDMRB) schemes that exploit the spatial signature of multi-user interference and characterize their power-bandwidth tradeoff under a system-wide power constraint on source and relay transmissions. The impact of multiple users, multiple relays and multiple antennas on the key performance measures of the high and low SNR regimes is investigated in order to shed new light on the possible reduction in power and bandwidth requirements through the usage of such practical relay cooperation techniques. Our results indicate that point-to-point coded multi-user networks supported by distributed relay beamforming techniques yield enhanced energy efficiency and spectral efficiency, and with appropriate signaling and sufficient antenna degrees of freedom, can achieve asymptotically optimal power-bandwidth tradeoff with the best possible (i.e., as in the cutset bound) energy scaling of K-1 and the best possible spectral efficiency slope at any SNR for large number of relay terminals. Furthermore, our results help to identify the role of interference cancellation capability at the relay terminals on realizing the optimal power- bandwidth tradeoff; and show how relaying schemes that do not attempt to mitigate multi-user interference, despite their optimal capacity scaling performance, could yield a poor power- bandwidth tradeoff.
@cite_5: We consider n wireless ad hoc network nodes with one antenna each and equidistantly placed on a line. The transmission power of each node is just large enough to reach its next neighbor. For this setting we show that a message can be broadcasted to all nodes in time O(log n) without increasing each node's transmission power. Our algorithm needs O(log n) messages and consumes a total energy which is only a constant factor larger than the standard approach where nodes sequentially transmit the broadcast message to their next neighbors. We obtain this by synchronizing the nodes on the fly and using MIMO (multiple input multiple output) techniques. To achieve this goal we analyze the communication capacity of multiple antennas positioned on a line and use a communication model which is based on electromagnetic fields in free space. We extend existing communication models which either reflect only the sender power or neglect the locations by concentrating only on the channel matrix. Here, we compute the scalar channel matrix from the locations of the antennas and thereby only consider line-of-sight-communication without obstacles, reflections, diffractions or scattering. First, we show that this communication model reduces to the SINR power model if the antennas are uncoordinated. We show that n coordinated antennas can send a signal which is n times more powerful than the sum of their transmission powers. Alternatively, the power can be reduced to an arbitrarily small polynomial with respect to the distance. For coordinated antennas we show how the well-known power gain for MISO (multiple input single output) and SIMO (single input multiple output) can be described in this model. Furthermore, we analyze the channel matrix and prove that in the free space model no diversity gain can be expected for MIMO. Finally, we present the logarithmic time broadcast algorithm which takes advantage of the MISO power gain by self-coordinating wireless nodes.
@cite_10: This paper explores the usage of cooperative multiple input multiple output (MIMO) technique to minimize energy consumption used to establish communications among distant nodes in a wireless sensor network (WSN). As energy depletion is an outstanding problem in WSN research field, a number of techniques aim to preserve such resource, especially by means of savings during communication among sensor nodes. One such wide used technique is multi-hop communication to diminish the energy required by a single node to transmit a given message, providing a homogeneous consumption of the energy resources among the nodes in the network. However, it is not the case that multi-hop is always more efficient than single-hop, even that it may represent a great depletion of a single node’s energy. In this paper a cooperative MIMO transmission technique for WSN is presented, which is compared to single-hop and multi-hop transmission ones, highlighting its advantages in relation to both. Simulation results support the statement about the utility in applying the proposed technique for energy saving purposes.
While this research is largely concerned with the diversity gain, we study the physical limitations of the energy gain. In @cite_0 @cite_18 , a method is presented to amplify the signal by using spatially distributed nodes. They explore the trade-off between energy efficiency and spectral efficiency with respect to network size. In @cite_4 , a distributed algorithm is presented in which rectangular collaborative clusters of increasing size are used to produce stronger signal beams. In @cite_9 , the asymptotic behavior of the rounds for a unicast has been analyzed in great detail and an upper and lower bound of @math rounds has been proven. If the nodes are placed on the line it takes an exponential number of rounds @cite_5 . The generalization of these observations for different path loss models can be found in @cite_10 . In @cite_14 it is shown that the sum of all cooperating sender power can be reduced to the order of one sender, while maintaining a logarithmic number of rounds to send a message over an @math hop distance.