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## Eigenfunctions of linear systems 1 0.1 EIGENFUNCTIONS OF UNDERSPREAD LINEAR SYSTEMS: THEORY AND APPLICATIONS TO DIGITAL COMMUNI-

by Unknown Authors

by Unknown Authors

@MISC{_eigenfunctionsof,

author = {},

title = {Eigenfunctions of linear systems 1 0.1 EIGENFUNCTIONS OF UNDERSPREAD LINEAR SYSTEMS: THEORY AND APPLICATIONS TO DIGITAL COMMUNI-},

year = {}

}

The knowledge of the eigenfunctions of a linear system is a fundamental issue both from the theoretical as well as from the applications point of view. Nonetheless, no analytic solution is available for the eigenfunctions of a general linear system. There are two important classes of contributions suggesting analytic expressions for the eigenfunctions of slowly-varying operators: [5], and the references therein, where it was proved that the eigenfunctions of underspread operators can be ap-proximated by signals whose time-frequency distribution (TFD) is well localized in the time-frequency plane, and [7] where a strict relationship between the in-stantaneous frequency of the channel eigenfunctions and the contour lines of the Wigner Transform of the operator kernel (or Weyl symbol) was derived for Hermi-tian slowly-varying operators. In this article, following an approach similar to [7], we will show that the eigenfunctions can be found exactly for systems whose spread function is concentrated along a straight line and they can be found in approximate sense for those systems whose spread function is maximally concentrated in regions of the Doppler-delay plane whose area is smaller than one. 0.1.2 Eigenfunctions of time-varying systems The input/output relationship of a continuous-time (CT) linear system is [3]: y(t) = h(t, τ)x(t − τ)dτ (0.1.1) where h(t, τ) is the system impulse response. Although throughout this section we will use the terminology commonly adopted in the transit of signals through time-varying channels, it is worth pointing out that the mathematical formulation is much more general. For example, (0.1.1) can be used to describe the propagation of waves through non homogeneous media and in such a case the independent variables t and τ are spatial coordinates. Following the same notation introduced by Bello [3], any linear time-varying (LTV) channel can be fully characterized by its impulse response h(t, τ), or equivalently by the delay-Doppler spread function (or simply spread function) S(ν, τ):= − ∞ h(t, τ)e −j2piνtdt, or by the time-varying transfer function H(t, f):= − ∞ h(t, τ)e

linear system eigenfunctions underspread linear system spread function application digital communi underspread operator strict relationship slowly-varying operator mathematical formulation hermi-tian slowly-varying operator time-frequency distribution time-varying channel channel eigenfunctions important class input output relationship doppler-delay plane linear time-varying application point homogeneous medium operator kernel approximate sense time-frequency plane impulse response j2pi tdt analytic expression contour line independent variable general linear system delay-doppler spread function system impulse response weyl symbol fundamental issue in-stantaneous frequency straight line wigner transform analytic solution spatial coordinate time-varying system time-varying transfer function

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