Convergence condition of simplified information geometry approach for massive MIMO-OFDM channel estimation

In this paper, we prove the convergence of the simplified information geometry approach (SIGA), which was proposed for massive MIMO-OFDM channel estimation. For a general Bayesian inference problem, we first show that the iteration of the common second-order natural parameter (SONP) is separated from that of the common first-order natural parameter (FONP). Hence, the convergence of the common SONP can be checked independently. We show that with the initialization satisfying a specific but large range, the common SONP is convergent regardless of the value of the damping factor. For the common FONP, we establish a sufficient condition of its convergence and prove that the convergence of the common FONP relies on the spectral radius of a particular matrix related to the damping factor. We give the range of the damping factor that guarantees the convergence in the worst case. Further, we determine the range of the damping factor for massive MIMOOFDM channel estimation by using the specific properties of the measurement matrices. Simulation results are provided to confirm the theoretical results.