# How to know which type of sensing matrix would do the job?

Compressed sensing refers to the recovery of a high-dimensional but sparse vector $$x\in\mathbb{R}^n$$ from its linear measurement $$y = Ax+\eta$$, where $$A\in\mathbb{R}^{m\times n}$$ $$(m< is a measurement matrix and $$\eta$$ is the measurement noise with a known upper bound $$||\eta||_2\leq \epsilon$$.

Following is the most popular method to recover the $$x$$:

$$\hat{x}= \arg\min ||z||_1 , \mbox{such that}\ ||y-Az||_2\leq\epsilon.$$ In order for the above formulation to work, $$A$$ must satisfy the Restricted Isometry Property (RIP) or Robust Null Space Property (RNSP).

Now, suppose $$x$$ itself is not sparse but it is sparse in some other basis, say $$\Phi\in\mathbb{C}^{n\times n}$$. So, $$x = \Phi \alpha$$, where $$\alpha\in\mathbb{C}^n$$ is a sparse vector. Now, if we measure the $$x$$ using any sampling matrix $$A$$, we would get

$$y= Ax+\eta=A\Phi\alpha +\eta.$$ The corresponding recovery algorithm would be:

$$\hat{\alpha}= \arg\min ||z||_1 , \mbox{such that}\ ||y-A\Phi z||_2\leq\epsilon ,$$

$$\hat{x}= \Phi \hat{\alpha}$$.

Now, my question is how to construct $$A$$ such that the $$A\Phi$$ satisfies the RIP or RNSP? I know a tons of method for designing RIP satisfying measurement matrix $$A$$. But I don't know how to design $$A$$ such that the product $$A\Phi$$ satisfies the RIP?