In the book Compressed Sensing by Kutyniok et al, the author talks about data separation using sparse representation. In summary, if we have a signal vector
$x = x_1 + x_2$
Then, it would be possible to have vectors $x_1$ and $x_2$ as a result. In order to do that, they use two basis $\Phi_1$ and $\Phi_2$ on which $x_1$ has a sparse representation in $\Phi_1$ but not in $\Phi_2$ and $x_2$ has a sparse representation in $\Phi_2$ and not in $\Phi_1$.
I am working with thin signals (spikes) and thick signals (sinc functions, gaussians, plateaus). There is a paper that intends to do this but uses sparsity in $x_1$ (thin signals) and sparsity in Daubechies 8 (D8) space for thick signals. This last point does not make sense to me since spikes and some thicks signals are sparse in D8 space.
Am I getting wrong something? Do you have any experience separating signals using CS? My mainly concern is what bases should I use to reconstruct thick signals.