Reconstruction of the scattering function of overspread radar targets

Abstract

In many radar scenarios, the radar target or the medium is assumed to possess randomly varying parts. The properties of a target are described by a random process known as the spreading function. Its second order statistics under the WSSUS assumption are given by the scattering function. Recent developments in the operator identification theory suggest a channel sounding procedure that allows to determine the spreading function given complete statistical knowledge of the operator echo. We show that in a continuous model it is indeed theoretically possible to identify a scattering function of an overspread target given full statistics of a received echo from a single sounding by a custom weighted delta train. Our results apply whenever the scattering function is supported on a set of area less than one. Absent such complete statistics, we construct and analyze an estimator that can be used as a replacement of the averaged periodogram estimator in case of poor geometry of the support set of the scattering function. * O. Oktay, G. E. Pfander and P. Zheltov are with Jacobs University Bremen

Estimation of Overspread Scattering Functions

“…The mean-square error of reconstruction for the scattering functions given by C 2 (τ, ν) and C 3 (τ, ν) are also given on Figure 5. In our tests, the estimator behaved as expected, with error decaying linearly in the number of soundings J, as predicted by Equation 24. For simulated cases C 1 (τ, ν), C 2 (τ, ν), C 3 (τ, ν), the normalized mean square error of the estimator is shown on Figure 5.…”

“…Remark II.6. An alternative way to reconstruct C(τ, ν) is given in the following theorem, proven in a preceding paper [24]. It requires the area of the support set to be less than or equal to one.…”

Identification of stochastic operators

“…Such operators are referred to as wide-sense stationary operators with uncorrelated scattering, or WSSUS. The function C η (t, ν) is then called scattering function [13,14,10]. Our results do not presuppose the stationarity of H, instead, they include it as an interesting special case.…”

“…In [10], we argue that as in the deterministic case, the condition for the spread factor to be less than one is sufficient in the case of identifiable WSSUS channels, and establish the direct applicability of time-frequency analysis techniques of Kozek, Pfander and Walnut in this simplified stochastic setting. In [11], we assume functional analytic results proven here and give a detailed analysis of the general case of stochastic operator sampling with a fully stochastic spreading function.…”

Sampling and Reconstruction of Operators

“…More recently, sampling results for stochastic operators, that is, for operators with stochastic spreading functions, have been obtained [22], [32], [31]. Also, in applications, it is required to replace the identifier considered in this paper by finite time or finite bandwidth, that is, smooth, signals.…”