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

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

The chirp after being corrupted by a realization of the noise consisting of $ 95$ 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 $ G$

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

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}