Example 3

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

$\displaystyle {\cal{W}^\bot}={\mbox{\rm {span}}}\{y_j(t)=e^{-100000(t-0.05j)^2}, \;\;t \in [0,1]\}_{j=1}^{200}.$

The chirp after being corrupted by a realization of the noise consisting of

pulses taken randomly from elements of ${\cal{W}^\bot}$ is plotted in the second graph of Fig. 1.

Consider that the signal subspace is well represented by ${\cal{V}}$ given by

$\displaystyle {\cal{V}}= {\mbox{\rm {span}}}\{v_{i+1}(t)=\cos{\pi i t}, \;\;t\in [0,1]\}_{i=0}^{M=99}.$

In order to eliminate the impulsive noise from the chirp we have to compute the measurement vectors $\{w_i\}_{i=1}^{100}$ , here functions of $t\in [0,1]$ , determining the appropriate projector. For this we first need a representation of $\hat{P}_{{\cal{W}^\bot}}$ , which is obtained simply by transforming the set $\{y_j\}_{j=1}^{200}$ into an orthonormal set $\{o_j\}_{j=1}^{200}$ to have

$\displaystyle \hat{P}_{{\cal{W}^\bot}}= \sum_{j=1}^{200} o_j \langle o_j, \cdot \rangle .$

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

With $\hat{P}_{{\cal{W}^\bot}}$ we construct vectors

$\displaystyle u_{i+1}(t)=\cos{\pi i t} - \sum_{j=1}^{200} o_j(t) \langle o_j(t), \cos{\pi i t}\rangle , \;\;\; i=0,\ldots,99, \;\; t\in [0,1].$

The inner products involved in the above equations and in the elements, $g_{i,j}$ , of matrix

$\displaystyle g_{i+1,j+1}= \int_{0}^1 u_{i+1}(t) \cos{\pi j t} dt,\quad i=0,\ldots,99, \;j=0,\ldots,99$

are computed numerically. This matrix has an inverse, which is used to obtain functions $\{w_i(t), t\in [0,1]\}_{i=1}^{100}$ giving rise to the required oblique projector. The chirp filtered by such a projector is depicted in the last graph of Fig. 1.

**Figure 1:** Chirp signal (first graph). Chirp corrupted by 95 randomly taken pulses (middle graph). Chirp denoised by oblique projection (last graph).
$\includegraphics[width=5.6cm]{chi.eps}$ $\includegraphics[width=5.6cm]{chi95.eps}$ $\includegraphics[width=5.6cm]{chifil.eps}$

Highly Nonlinear Approximations for Sparse Signal Representation

Example 3