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

In this paper we present a new family of discrete sequences having “random like” uniformly decaying auto-correlation properties. The new class of infinite length sequences are higher order chirps constructed using irrational numbers. Exploiting results from the theory of continued fractions and diophantine approximations, we show that the class of sequences so formed has the property that the worst-case auto-correlation coefficients for every finite length sequence decays at a polynomial rate. These sequences display doppler immunity as well. We also show that Toeplitz matrices formed from such sequences satisfy restricted-isometry-property (RIP), a concept that has played a central role recently in Compressed Sensing applications. Compressed sensing has conventionally dealt with sensing matrices with arbitrary components. Nevertheless, such arbitrary sensing matrices are not appropriate for linear system identification and one must employ Toeplitz structured sensing matrices. Linear system identification plays a central role in a wide variety of applications such as channel estimation for multipath wireless systems as well This research was supported by ONR Young Investigator Award N00014-02-100362 and a Presidential Early Career Award (PECASE), NSF CAREER award ECS 0449194, and NSF Grant CCF 0430983 and CNS-0435353