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Deterministic Designs with Deterministic Guarantees: Toeplitz Compressed Sensing Matrices, Sequence Design and System ID

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1 Author(s)
Saligrama, V. ; Venkatesh Saligrama is with the Department of Electrical and Computer Engineering Boston University, Boston MA 02215 (email: srv@bu.edu)

In this paper we present a new family of discrete aperiodic sequences having random like uniformly decaying auto-correlation properties. The new class of infinite length aperiodic sequences are higher order chirps based on algebraic irrational numbers. We show the uniformly decaying auto-correlation property by exploiting results from the theory of continued fractions and diophantine approximations. Specifically, we demonstrate that every finite n-length truncation of a higher-order chirp has a worst-case autocorrelation that decays as O(n..1=4). Construction of aperiodic sequences with good autocorrelation properties is motivated by the problem of system identification of finite dimensional linear systems with unmodeled dynamics. We also utilize the uniformly decaying autocorrelation property to bound the singular values for finite Toeplitz structured matrices formed from n-length higher-order chirp sequences. These singular value bounds imply restricted isometry property (RIP) and lead to deterministic Toeplitz matrix constructions with RIP property.

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Information Theory, IEEE Transactions on  (Volume:PP ,  Issue: 99 )