Statistical Analysis of Array Gain for Cooperative MISO Transmitters without CSI

Virtual Multiple-Input Single-Output (MISO) is recently proposed to extend the benefits of transmitter space diversity to networks in which the deployment of antenna arrays on individual nodes is infeasible from a practical point of view. Ad-hoc and sensor networks are examples of these type of networks. In these scenarios, nodes equipped with single antenna can cooperatively transmit to emulate an antenna array. However, cooperative transmissions require knowledge of the channel state either at the transmitter side or the receiver side in order to achieve full performance gains. Several solutions are proposed in the literature under these assumptions, but at the expense of increased overhead and energy consumption. In this paper, the array gain at the receiver from the non-coherent combining of the signals from multiple transmitters is analyzed in statistical sense, under the assumption that the channel knowledge is unavailable. The transmitters are assumed to be randomly spread over a circular region. More specifically, exact or very accurate closed-form expressions for the expectation and variance of the array gain are obtained, and then a complete statistical distribution is postulated and validated by means of heuristic procedures, goodness-of-fit tests and specialized software. The results obtained in this paper can be especially useful for the implementation of two-tiered wide area sensor networks.