Optimal Time of Arrival Estimation for MIMO Backscatter Channels

In this paper, we propose a novel time of arrival (TOA) estimator for multiple-input-multiple-output (MIMO) backscatter channels in closed form. The proposed estimator refines the estimation precision from the topological structure of the MIMO backscatter channels, and can considerably enhance the estimation accuracy. Particularly, we show that for the general $M \times N$ bistatic topology, the mean square error (MSE) is $\frac{M+N-1}{MN}\sigma^2_0$, and for the general $M \times M$ monostatic topology, it is $\frac{2M-1}{M^2}\sigma^2_0$ for the diagonal subchannels, and $\frac{M-1}{M^2}\sigma^2_0$ for the off-diagonal subchannels, where $\sigma^2_0$ is the MSE of the conventional least square estimator. In addition, we derive the Cramer-Rao lower bound (CRLB) for MIMO backscatter TOA estimation which indicates that the proposed estimator is optimal. Simulation results verify that the proposed TOA estimator can considerably improve both estimation and positioning accuracy, especially when the MIMO scale is large.