A $\mathbb{T} -\text{Proper}$ Karhunen-Loeve Expansion and its Application to the Problem of Simulation

The paper addresses the Karhunen-Loéve series representation in the tessarine domain. Based on augmented statistics, a tessarine widely linear Karhunen-Loève expansion is defined. Then, the impact of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{T}$</tex>-properness on this representation is analyzed, leading to a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{T}$</tex>-proper Karhunen-Loève expansion that means a dimensionality reduction. Furthermore, this series representation serves as a versatile simulation tool, valid for both stationary and non-stationary, Gaussian and non-Gaussian random signals. Finally, the applicability of the simulation technique proposed is examined numerically.