Abstract

Compressed sensing is by now well-established as an effective tool for
extracting sparsely distributed information, where sparsity is a {\it discrete}
concept, referring to the number of dominant nonzero signal components in some
basis for the signal space. In this paper, we establish a framework for
estimation of {\it continuous-valued} parameters based on compressive
measurements on a signal corrupted by additive white Gaussian noise (AWGN).
While standard compressed sensing based on naive discretization has been shown
to suffer from performance loss due to basis mismatch, we demonstrate that this
is not an inherent property of compressive measurements. Our contributions are
summarized as follows: (a) We identify the isometries required to preserve
fundamental estimation-theoretic quantities such as the Ziv-Zakai bound (ZZB)
and the Cram\'er-Rao bound (CRB). Under such isometries, compressive
projections can be interpreted simply as a reduction in "effective SNR." (b) We
show that the threshold behavior of the ZZB provides a criterion for
determining the minimum number of measurements for "accurate" parameter
estimation. (c) We provide detailed computations of the number of measurements
needed for the isometries in (a) to hold for the problem of frequency
estimation in a mixture of sinusoids. We show via simulations that the design
criterion in (b) is accurate for estimating the frequency of a single sinusoid.