On the identification of parametric underspread linear systems

Identification of time-varying linear systems, which introduce both time-shifts (delays) and frequency-shifts (Doppler-shifts), is a central task in many engineering applications. This paper studies the problem of identification of underspread linear systems (ULSs), defined as time-varying linear systems whose responses lie within a unit-area region in the delay-Doppler space, by probing them with a known input signal. The main contribution of the paper is that it characterizes conditions on the bandwidth and temporal support of the input signal that ensure identification of ULSs described by a finite set of delays and Doppler-shifts, and referred to as parametric ULSs, from single observations. In particular, the paper establishes that sufficiently-underspread parametric linear systems are identifiable as long as the time-bandwidth product of the input signal is proportional to the square of the total number of delay-Doppler pairs in the system. In addition, the paper describes a procedure that enables identification of parametric ULSs from an input train of pulses in polynomial time by exploiting recent results on sub-Nyquist sampling for time delay estimation and classical results on recovery of frequencies from a sum of complex exponentials.