Eigenfunctions of Underspread Linear Communication Systems

TL;DR

This paper derives exact eigenfunctions for certain underspread linear communication systems and approximates them for others, revealing how system characteristics influence eigenfunctions and eigenvalues.

Contribution

It provides explicit eigenfunction solutions for systems with specific spread functions and uncovers the quantization of eigenvalues related to the transfer function.

Findings

Eigenfunctions can be exactly determined for systems with line-concentrated spread functions.

Eigenvalues are bounded and exhibit inherent quantization.

Eigenfunctions' instantaneous frequency relates to the transfer function contour level.

Abstract

In this paper we show that the eigenfunctions can be found exactly for systems whose delay-Doppler spread function is concentrated along a straight line and they can be found in approximate sense for systems having a spread function maximally concentrated in regions of the Doppler-delay plane whose area is smaller than one. The interesting results are that: i) the instantaneous frequency of the eigenfunctions is dictated by the contour level of the time-varying transfer function; ii) the eigenvalues are restricted between the minimum and maximum value of the system time-varying transfer function, but not all values are possible, as the system exhibits an inherent quantization.