  Compressed sensing is by now well-established as an effective tool for
extracting sparsely distributed information, where sparsity is a 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 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
Cramer-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.
