Reconstruction of the joint 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.

Wideband Extended Range-Doppler Imaging and Waveform Design in the Presence of Clutter and Noise

“…A similar approach was also noted in [19]. In [24], the approach in [21] was extended to include frames as a set of transmitted waveforms. In [36], the use the wavelet transform for the image recovery problem was proposed in a deterministic setting.…”

“…It is well known that the echo from the clutter/interference is waveform dependent [23]. Thus, we model the clutter echo as the affine Fourier transform of the clutter reflectivity density function evaluated at the transmitted waveform, i.e., (24) where is the clutter reflectivity density function. We also assume that the measurements are contaminated with zero mean additive white noise process.…”

“…The wideband echo model has been studied by many researchers [19], [21], [24], [36]. In [21], affine group Fourier theory was utilized and a method of waveform design and wideband range-Doppler target reflectivity reconstruction was introduced.…”

“…However, this approach requires the target reflectivity function to be in the reproducing kernel Hilbert space of the transmitted mother wavelet. In all of these studies, [19], [21], [24], and [36], the received echo is modeled in a deterministic setting without any consideration of noise and clutter in sensor measurements. Our study is closest in spirit to [21], however, our problem is to design clutter rejecting waveforms and to estimate the wideband range-Doppler target reflectivity density function embedded in clutter using a priori statistical information.…”

Estimation of radar target reflectivity in ultrawideband regime - a group theoretic approach

“…The UWB wave propagation model as described above has been studied before (See, [1]- [4], [11], [19]- [20] and references therein). In [9] and [31], Naparst and Miller suggested to use the Fourier theory of the affine group and proposed a method to reconstruct the target reflectivity density function in a deterministic setting.…”