LMMSE-Optimal Pilot Pattern Design Based on Covariance Matrix Approximation for OFDM Channel Estimation in Doubly Dispersive Channel

This paper investigates the optimal pilot pattern design, in the linear minimum mean square error (LMMSE) estimator sense, for OFDM systems in doubly dispersive channels. To enable analytical tractability, the channel covariance matrix is decomposed into the Kronecker product of two Hermitian Toeplitz matrices corresponding to the delay and Doppler domains. By invoking the Szeg\"{o} limit theorem, these matrices are shown to be approximately diagonalizable by discrete Fourier transform (DFT) matrices. Based on this structure, the LMMSE channel estimation error is reformulated into a compact analytical form, from which a closed-form lower bound is derived. Furthermore, we establish the condition under which this bound is achieved by a lattice-based pilot pattern. Numerical results verify that the proposed matrix approximation introduces negligible error and examples of the proposed lattice design are given.