Pilot Optimization and Channel Estimation for Two-Way Relaying Network Aided by IRS With Finite Discrete Phase Shifters

In this paper, we investigate the problem of pilot optimization and channel estimation of two-way relaying network (TWRN) aided by an intelligent reflecting surface (IRS) with finite discrete phase shifters. In a TWRN, there exists a challenging problem that the two cascading channels from User1-to-IRS-to-relay and User2-to-IRS-to-relay and two direct channels from User1-to-relay and User2-to-relay interfere with each other. Via smartly designing the initial phase shifts of IRS and pilot pattern, the two cascading channels are separated over only four pilot sequences by using simple arithmetic operations like addition and subtraction. Then, the least-squares estimator is adopted to estimate the two cascading channels and two direct channels. The corresponding sum mean square errors (MSE) of channel estimators are derived. By minimizing Sum-MSE, the optimal phase shift matrix of IRS is proved. Then, two special matrices Hadamard and discrete Fourier transform (DFT) matrix are shown to be two optimal training matrices for IRS. Furthermore, the IRS with discrete finite phase shifters is taken into account. The Hadamard matrix requires only one-bit phase shifters to achieve the optimal Sum-MSE performance for specific numbers of IRS elements. Using theoretical derivation and numerical simulations, we find that 3 or 4-bit phase shifters are sufficient for IRS to achieve a negligible MSE performance loss for any number.