Eigenfunctions of Underspread Linear Communication Systems

T he knowledge of the eigenfunctions of a linear time-varying (LTV) system is a fundamental issue from both theoretical and applications points of view. Nonetheless, no analytic expressions are available for the eigenfunctions of a general LTV system. However, approximate expressions have been proposed for slowly-varying operators. In particular, in [1] it was shown that an underspread system can be well approximated by a normal operator, so that we can properly talk about eigendecomposition. The interesting result derived in [1] is that the class of eigenfunctions of underspread systems can be approximated by a set of signals obtained as shifted versions, in both time and frequency, of a given pulse waveform g(t), which is well localized in the time-frequency domain. The validity of this approximation depends on the system spread along the delay and Doppler axes. An alternative approach, for Hermitian slowly-varying operators, was proposed in [4], [5] and [6] where the authors used the WKB (Wentzel-Kromers-Brillouin) method to derive a relationship between the instantaneous frequency of the channel eigenfunctions and the contour lines of the Wigner Transform of the operator kernel (or Weyl symbol). In this paper, we will follow an approach similar to [4] and 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 the system time-varying transfer functions, but not all values are possible, as the system exhibits an inherent quantization.