Reconstruction of wideband reflectivity densities by wavelet transforms

Abstract

This paper is concerned with reconstruction problems arising in the context of radar signal analysis. The goal in radar is to obtain information about objects by emitting certain signals and analyzing the reflected echoes. In this paper, we shall focus on the general wideband model for radar echoes and on the case of continuously distributed objects D (reflectivity density). In this case, the echo is given by an inverse wavelet transform of the density D where the role of the analyzing wavelet is played by the transmitted signal. However, the null space of an inverse wavelet transform is nontrivial, it is described by the corresponding reproducing kernel. Following the approach of Naparst [14] and Rebolla-Neira et al. [16], we suggest to treat this problem by transmitting not just one signal but a family of signals. Indeed, a reconstruction formula for one- and 2-dimensional reflectivity densities can be derived, provided that the set of outgoing signals forms an orthogonal basis or – more general – a frame. We also present some rigorous error estimates for these reconstruction formulas. The theoretical results are confirmed by some numerical examples.