Aperiodic Sequences With Uniformly Decaying Correlations With Applications to Compressed Sensing and System Identification

In this paper, we present a new family of discrete aperiodic sequences having “random like” uniformly decaying autocorrelation properties. The new class of infinite length aperiodic sequences are higher order chirps based on algebraic irrational numbers. We show the uniformly decaying autocorrelation 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.