Minimum Mean-Square Error Estimation

When dealing with random signals, least-squares estimation must be reformulated to accommodate the randomness of the signal. As usual, we deal with random signals using averaging over the ensemble of signals. For the deterministic case, we obtain the optimal estimate by minimizing the estimation error. For random signals, we replace the error by its mean square-value. This motivates minimizing the mean-square estimation error. The Kalman filter is a state estimator that minimizes the mean-square error. We examine the Kalman filter and discuss its stability. Some of the material on Kalman filter stability can be skipped by readers who do not have a strong background in linear system theory.