Classical Approach

Abstract

The following will illustrate the classical Wiener approach to a discrete filtering problem followed by the Riccati equation approach in a later chapter. The development here follows [29]; note that in all that follows, t takes on integer values, and t₀ = - ∞:

$$ \begin{gathered} {{x}_{{n + 1}}} = \lambda {{x}_{n}} + {{u}_{n}},\quad \left| \lambda \right| < 1 \hfill \\ {{z}_{n}} = {{x}_{n}} + {{v}_{n}},\quad E{{v}_{i}}{{v}_{j}} = {{\delta }_{{ij}}}r. \hfill \\ \end{gathered} $$

$$ \begin{gathered} E\left\{ {z(t + \tau )z'(t)} \right\} = o(\tau ) = m{{\lambda }^{{|\tau |}}} + r \hfill \\ E\left\{ {x(t + \tau )z'(t)} \right\} = c(\tau ) = m{{\lambda }^{{|\tau |}}} \hfill \\ E\{ x(t + \tau )x'(t)\} = s(\tau ) = m{{\lambda }^{{|\tau |}}} \hfill \\ \end{gathered} $$