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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.

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  • $\begingroup$ What type of measurement matrix do you have? $\endgroup$ – lp251 Jan 30 '14 at 4:57
  • $\begingroup$ basically a projection matrix, not yet formulated. $\endgroup$ – Preethi shreeya Jan 30 '14 at 5:27
  • $\begingroup$ Gaussian/Bernoulli (i.i.d) probably. $\endgroup$ – Preethi shreeya Jan 30 '14 at 5:35
  • $\begingroup$ How would you propose to implement a Gaussian measurement matrix in your context? The measurement matrix is not independent of sensor position. $\endgroup$ – lp251 Jan 30 '14 at 5:41
  • $\begingroup$ That is actually the question I have. In all the papers I had read through, there were only specific matrices which satisfied the RIP property. Here, I want to try to design a measurement matrix which can ultimately tell me the sensor positions. I want to know the relationship between these two if there is any. $\endgroup$ – Preethi shreeya Jan 30 '14 at 5:46
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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.

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Measurement matrix is generally a Gaussian iid matrix. It is used (afaik),or can be used for several applications of compressive sensing. If you want to aim for a better matrix, which eventually would give you a good reconstruction with less amount of measurements, I believe you should analyse your signal well and come up with a measurement matrix that is incoherent as possible with your signal. I can't directly tell you what kind of a matrix you'll need for your specific application but incoherency between measurement and sparsifying basis is a must, and can be approved for further efficiency. Hope that helps.

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