Compressed / Compressive Sensing - Sensor Placement I have a beam fixed at one end and free at the other. It obviously results in vibrations due to the disturbances in the environment or when induced. This system can be represented using a differential equation. The vibrations are sensed using sensors, analyzed and reduced. Now, the sensing is done through compressed sensing technique.

How is it possible to find the optimal sensor placement (minimal number of sensors along with the details of its spatial position) just with the help of the measurement matrix (that we design in compressed sensing for reconstruction purpose)?

Suppose we want to retrieve $\mathbf x$ (vibrational info) from: $\mathbf{y= Cx}$, where $\mathbf C$ is the fat matrix (measurement matrix * sparse matrix). I want to find a relationship that the measurement matrix can hold with the placement of sensors. It's like an inverse problem, instead of designing a measurement matrix with respect to the sensors placed, I want to place the sensors with the knowledge I have about the measurement matrix.
Suppose your beam has ten vertical degree of freedom. You can only apply CS to compress measurement vector of each DOF. You can not reconstruct all of the responses using limited number of them.
The ten-dof structure would have ten pick- related to ten natural frequency- in frequency domain, so its sparsity has been already guaranteed. Hope this help.