Physical layer binary consensus over fading wireless channels and with imperfect CSI

This paper considers the problem of achieving binary consensus among a set of nodes using physical layer communication over noisy wireless links. Starting with initial binary values, the nodes exchange messages over i.i.d. fading channels, detect the majority bit across the sensors, and update their majority bit estimates, over multiple cycles. The bits are updated using either an LMMSE-based scheme or a co-phased combining scheme. The channel state information (CSI) available at the nodes are imperfect due to practical estimation errors. 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. The two schemes are compared in terms of the probability of accurate consensus and the second largest eigen value of the TPM. It is found that the co-phased combining scheme is better at low to intermediate pilot SNRs, in addition to having a lower computational complexity and better analytical tractability, compared to the LMMSE-based scheme.