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In this paper we present a new family of discrete sequences having "randomlike" uniformly decaying auto-correlation properties. The new class of infinitelength sequences are higher order chirps constructed using irrational numbers.Exploiting results from the theory of continued fractions and diophantineapproximations, we show that the class of sequences so formed has the propertythat the worst-case auto-correlation coefficients for every finite lengthsequence decays at a polynomial rate. These sequences display doppler immunityas well. We also show that Toeplitz matrices formed from such sequences satisfyrestricted-isometry-property (RIP), a concept that has played a central rolerecently in Compressed Sensing applications. Compressed sensing hasconventionally dealt with sensing matrices with arbitrary components.Nevertheless, such arbitrary sensing matrices are not appropriate for linearsystem identification and one must employ Toeplitz structured sensing matrices.Linear system identification plays a central role in a wide variety ofapplications such as channel estimation for multipath wireless systems as wellas control system applications. Toeplitz matrices are also desirable on accountof their filtering structure, which allows for fast implementation togetherwith reduced storage requirements.

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