Linear Estimation of Stationary Autoregressive Processes

Consider a sequence of an th-order Autoregressive (AR) stationary discrete-time process and assume that at least −1 consecutive neighboring samples of an unknown sample are available. It is not important that the neighbors are from one side or are from both the left and right sides. In this paper, we find explicit solutions for the optimal linear estimation of the unknown sample in terms of the neighbors. We write the estimation errors as the linear combination of innovation noises. We also calculate the corresponding mean square errors (MSE). To the best of our knowledge, there is no explicit solution for this problem. The known solutions are the implicit ones through orthogonality equations. Also, there are no explicit solutions when fewer than −1 samples are available. The order of the process () and the feedback coefficients are assumed to be known.