Discrete Stochastic Processes and Optimal Filtering

5.2. Resolution and calculation of the FIR filter

The error is written:

with and Y ∈ (L²)^N.

We have a function : cost to be minimized which is a mapping:

The vector ĥ = h_optimal is such that = 0

given (scalar)

then (vector Nx1).

NOTE.– This is the theorem of projection on Hilbert spaces. Obviously this is the principle of orthogonality again.

This least mean square error will be minimal when:

By using expression ;

all the components of the vector are empty (or ).

Let E(X_K Y) = E(YY^T)ĥ.

We will ...