Reconstruction of the join time-delay Doppler-scale reflectivity density in the wide-band regime. A frame theory based approach

Abstract

The inverse scattering problem concerning the determination of the joint time-delay Doppler-scale reflectivity density characterizing continuous target environments is addressed by recourse to the generalized frame theory. A reconstruction formula, involving the echoes of a frame of outgoing signals and its corresponding reciprocal frame, is developed. A ‘‘realistic’’ situation with respect to the transmission of a finite number of signals is further considered. In such a case, our reconstruction formula is shown to yield the orthogonal projection of the reflectivity density onto a subspace generated by the transmitted signals.