Modeling, Estimation and Optimal Filtration in Signal Processing

Appendix A

Karhunen Loeve Transform

Let us consider N consecutive samples of a random sequence {y(k)}_{k=0,...,N–1}. Our purpose is to decompose this sequence into a set of independent orthonormal functions Ψ_i(k), as follows:

As a preamble, let us define the various notations we will use in the rest of this analysis.

First, the functions Ψ_i(k) satisfy the following condition:

where δ_ij denotes the Kronecker symbol. It is equal to 1 when i = j and 0 otherwise.

The above equation can equivalently be written in the matrix form as follows:

with:

where * denotes the complex conjugate of .

Moreover, the projection coefficients k_i satisfy the following relation:

The above equation can be written as follows:

where:

The coefficients k_i are random variables because the sequence {y(k)}_{k=0,...,N–1}