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

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 {

Start Free Trial

No credit card required