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Modeling, Estimation and Optimal Filtration in Signal Processing by Mohamed Najim

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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 Ψi(k), as follows:

images

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

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

images

where δij 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:

images

with:

images

where images* denotes the complex conjugate of images.

Moreover, the projection coefficients ki satisfy the following relation:

images

The above equation can be written as follows:

where:

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

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