  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
as control system applications. Toeplitz matrices are also desirable on account
of their filtering structure, which allows for fast implementation together
with reduced storage requirements.
