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# Consensus Based Synchronization For Distributed Transmit Beamforming

Carrier synchronization is critical to any implementation of distributed transmit beamforming. For a practical network, the transmit nodes are not tethered and each has its own local oscillator (LO). Due to the low cost of these off-the-shelf LOs, there is substantial frequency offsets which hinder carrier synchronization between cooperating transmitters. A robust algorithm to pre-synchronize the carriers of the transmitters is the so called two node consensus based synchronization. Here, a designated master node transmits a monotone, $A\cos{(\theta_m)}$, to a node $i$, where $\theta$ denotes the phase. Node $i$ synthesizes the received tone, which includes zero mean additive noise $n_i(t)$ following the setup illustrated in Figure 1.

Figure 1: Carrier synchronization at node $i$.

The algorithm implemented at node $i$ follows the following equations:

$\dot{\theta}_i(t)=w_i(t)$

$\dot{w}_i(t)=-\beta A\sin{(\theta_i(t) - \theta_m(t)+\alpha w_i(t))}, \{\alpha, \beta\} >0$

Here, $w_i(t)$ is the instantaneous frequency from the LO of node $i$, $\alpha$ is a stability regularization term with $\beta$ being a descent step size. The above system is locally stable and the algorithm converges asymptotically leading to the paired transmitter coherence. In practice, carrier pre-synchronization is achieved before distributed beamforming to a far end receiver starts.