In this first of two lectures on Kalman Filtering, I will introduce the concepts by means of very simple examples. My first approach will be by means of least squares applied to find the optimal estimate and its covariance matrix for several examples with Gaussian noise. These examples include sampling statistics and the random walk, cases with measurement noise and dynamical noise, respectively. I will show how the least-squares results can be put in recursive Kalman Filter form. This form has the advantage that an update to the estimate and its covariance matrix can be made in terms of the previous estimate, the previous covariance matrix, and the new data, and discuss how this form is useful in control and guidance. I will also show how the same results can be obtained by finding the estimate with the minimum variance.
In the second lecture I will show extensions of these basic ideas. These will include 1) generalization to a system consisting of an n'th order linear stochastic system (i.e. with dynamical noise) plus measurement noise, 2) treatment in terms of conditional probabilities, and 3) the nonlinear or Extended Kalman Filter.