Compressive Parameter Estimation in AWGN

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 Cramér-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.