  In this paper, we present an approach to the reconstruction of signals
exhibiting sparsity in a transformation domain, having some heavily disturbed
samples. This sparsity-driven signal recovery exploits a carefully suited
random sampling consensus (RANSAC) methodology for the selection of an inlier
subset of samples. To this aim, two fundamental properties are used: a signal
sample represents a linear combination of the sparse coefficients, whereas the
disturbance degrade original signal sparsity. The properly selected samples are
further used as measurements in the sparse signal reconstruction, performed
using algorithms from the compressive sensing framework. Besides the fact that
the disturbance degrades signal sparsity in the transformation domain, no other
disturbance-related assumptions are made -- there are no special requirements
regarding its statistical behavior or the range of its values. As a case study,
the discrete Fourier transform (DFT) is considered as a domain of signal
sparsity, owing to its significance in signal processing theory and
applications. Numerical results strongly support the presented theory. In
addition, exact relation for the signal-to-noise ratio (SNR) of the
reconstructed signal is also presented. This simple result, which conveniently
characterizes the RANSAC-based reconstruction performance, is numerically
confirmed by a set of statistical examples.
