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 .
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.
@cite_18 @cite_14 @cite_4 @cite_9 @cite_0 @cite_5 @cite_10
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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. Distributed transmit beamforming with N cooperating nodes, each with fixed transmit power, provides a received power scaling with N2, corresponding to a “power pooling” gain of N and a beamforming gain of N. Prior work has shown that the optimal beamforming solution can be attained using a decentralized, iterative algorithm based on one bit (per iteration) feedback broadcast from the receiver to the transmitters. The algorithm is provably convergent in a noiseless setting, and is the basis for several successful prototypes. In this paper, we develop a framework for providing analytical insight into the effect of receiver noise, with the following key question in mind: can we bootstrap the algorithm from the incoherent power-pooled solution to operate in a regime in which the received SNR per node can be made arbitrarily small as we scale up the number of nodes N? Our analytical computations, validated by simulations, yield a somewhat negative answer: while the power-pooling gain guarantees a linear increase in received power with N, the per-node SNR cannot be scaled down with N if we wish to attain a quadratic increase in received power. Specifically, the fraction of the ideal beamforming gain attained using the one-bit algorithm is asymptotically independent of N, and depends only on the per-node SNR. However, the one-bit algorithm provides significant performance gains in practical regimes with a moderate number of cooperating nodes: the per-node SNR required for attaining a substantial fraction of the beamforming gain is low enough (e.g., − 5dB for 65 of the beamforming gain) to provide significant extension in operation regimes, while providing aggregate SNRs which permit reliable communication at high spectral efficiency: for example, starting from −5 dB per-node SNR, we obtain about 11 dB aggregate SNR with 10 cooperating nodes, and 17 dB SNR with 20 cooperating nodes. 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. n source and destination pairs randomly located in an area want to communicate with each other. Signals transmitted from one user to another at distance r apart are subject to a power loss of r-alpha as well as a random phase. We identify the scaling laws of the information-theoretic capacity of the network when nodes can relay information for each other. In the case of dense networks, where the area is fixed and the density of nodes increasing, we show that the total capacity of the network scales linearly with n. This improves on the best known achievability result of n2 3 of Aeron and Saligrama. In the case of extended networks, where the density of nodes is fixed and the area increasing linearly with n, we show that this capacity scales as n2-alpha 2 for 2lesalpha 4. Thus, much better scaling than multihop can be achieved in dense networks, as well as in extended networks with low attenuation. The performance gain is achieved by intelligent node cooperation and distributed multiple-input multiple-output (MIMO) communication. The key ingredient is a hierarchical and digital architecture for nodal exchange of information for realizing the cooperation.