Minimum mean-square-error expression of LMMSE channel estimation in SISO OFDM systems

An expression of the minimum mean square error (MMSE) of the linear MMSE channel estimation is given in the case of a non-invertible channel covariance matrix, as in single-input single-output (SISO) OFDM system. A matrix expression, already proposed for a multi-input multi-output OFDM system in a previous article, is not valid in SISO. A new proof is then proposed, by deriving a scalar expression of the MMSE, which leads to solve an optimisation problem. Furthermore, we show that the proposed solution is the global minimum. Simulations validate the proposed development.