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In this paper, we address the problem of observability of a linear dynamical system from compressive measurements and the knowledge of its external inputs. Observability of a high-dimensional system state in general requires a correspondingly large number of measurements. We show that if the initial state vector admits a sparse representation, the number of measurements can be significantly reduced by using random projections for obtaining the measurements. Our analysis gives sufficient conditions for the restricted isometry property of the observability matrix to hold, which leads to guarantees for the observability of the system. Our results depend only on the properties of system transfer and observation matrices, and are derived using tools from probability theory and compressed sensing. Unlike the prior work in this direction, our results are applicable to systems with an arbitrary nonzero system transfer matrix. Moreover, our results are stronger than the existing results in the regime where they are comparable.
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