I am trying to get a clear understanding of how compressed sensing works.
A continuous signal $x(t)$ is under-sampled (less samples are collected than the numbers required by the Nyquist theorem). The signal is sampled using a random matrix $\Psi$: $$y= \Psi x$$ where $\Psi$ is a random matrix (measurement matrix). The discrete signal $x[n]= \Phi s$, i.e. it has a sparse representation under the specific basis $\Phi$. Technically, there are multiple bases under which the signal is potentially sparse. The above equation becomes $$y=\Psi \Phi s$$ where the matrix $(\Psi \Phi)$ is not invertible since it is not a square matrix.
The system $y=\Psi \Phi s$ represents an underdetermined system of linear equations with zero or infinite solutions. The objective is to find the sparsest solution $s$. The vector $s$ is a vector containing the coefficients for the random basis vector represented by the columns of the matrix $\Psi \Phi$. Is that correct?
By imposing specific and extra conditions, the underdetermined system is solvable if we look for the sparsest of the solutions $s$ via a minimization process of the $L1$-norm. Is that correct?
Do we need to know a priori, before applying CS, the basis in which the signal is sparse?
What characteristics does the random sensing matrix $\Psi$ need to have? Are its rows supposed to be independent vectors?