Performance Analysis of Co-Phased Combining for Achieving Binary Consensus Over FadingWireless Channels With Imperfect CSI

Abstract

This paper considers the problem of achieving binary consensus among a set of nodes using physical layer communication over noisy wireless links. The channel state information (CSI) available at the nodes is imperfect due to practical estimation errors. Two schemes for updating the majority bit estimates at the nodes are contrasted: a linear minimum mean-squared error (LMMSE) based scheme and a co-phased combining scheme. The evolution of network consensus is modeled as a Markov chain, and the average transition probability matrix (TPM) is analytically derived for the co-phased combining scheme, whereas, for the LMMSE based scheme, the average TPM is computed through Monte Carlo simulations. The co-phased combining scheme is found to perform better at low to intermediate pilot SNRs, in addition to being analytically tractable and having lower computational complexity, compared with the LMMSE-based scheme. Also, to further characterize the consensus behavior, the probability of accurate consensus, the second eigenvalue of the TPM, the average hitting time to the first consensus state, and the average consensus duration are derived for the co-phased combining scheme. The power allocation between the pilot and data symbols is optimized, subject to a total power constraint. It is found that the optimal power allocation can lead to a significant improvement in the consensus performance. Monte Carlo simulation results validate the theoretical results, and provide insights into the complexity and performance tradeoffs involved.