Highly Nonlinear Approximations for Sparse Signal Representation

Example 3

Suppose that the chirp signal in the first graph of Fig. 1 is corrupted by impulsive noise belonging to the subspace

The chirp after being corrupted by a realization of the noise consisting of pulses taken randomly from elements of is plotted in the second graph of Fig. 1.

Consider that the signal subspace is well represented by given by

In order to eliminate the impulsive noise from the chirp we have to compute the measurement vectors , here functions of , determining the appropriate projector. For this we first need a representation of , which is obtained simply by transforming the set into an orthonormal set to have

The function for constructing an orthogonal projector in a number of different ways is OrthProj.m.

With we construct vectors

The inner products involved in the above equations and in the elements, , of matrix

are computed numerically. This matrix has an inverse, which is used to obtain functions giving rise to the required oblique projector. The chirp filtered by such a projector is depicted in the last graph of Fig. 1.