Modeling, Estimation and Optimal Filtration in Signal Processing

Appendix H

An Alternative Way of Understanding Kalman Filtering

The orthogonal projection approach is an alternative to the algebraic method presented in Chapter 5 to obtain the equations of the Kalman filter. Given {e₁,...,e_n}, an orthogonal base of the subspace spanned by the observations can be defined as the projection of the state vector (k) on the measurement subspace Y = {y(1),..., y(n)}, see Figure H.1.

Figure H.1. Projection of the state vector on the measurement subspace

(k / n) is thus expressed as follows:

thus:

Note: let υ be orthogonal to . Thus, E[υ e_i] = 0 for i = 1, 2,...,n. Moreover, .

We can also show that (k/k) is the maximum likelihood estimation of the state. If we assume ...