This letter considers the task of estimating the norm of an n-dimensional Gaussian random vector given a noisy/perturbed observation of it. In particular, the focus is on the case of additive Gaussian noise perturbation, which is assumed to be independent of the original vector. First, an expression for the optimal estimator is derived, and then the corresponding minimum mean square error (MMSE) is computed. The regime of large vector size is also analyzed, and it is shown that the MMSE normalized by n equals zero when n → ∞.
Bibliographical noteFunding Information:
Manuscript received May 29, 2019; revised July 10, 2019; accepted July 10, 2019. Date of publication July 22, 2019; date of current version July 31, 2019. The work of A. Dytso and H. V. Poor was supported by the U.S. National Science Foundation under Grant CCF-0939370. The work of M. Cardone was supported by the U.S. National Science Foundation under Grant CCF-1849757. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ashish Pandharipande. (Corresponding author: Martina Cardone.) A. Dytso and H. V. Poor are with the Princeton University, Princeton, NJ 08544 USA (e-mail: firstname.lastname@example.org; email@example.com).
- Gaussian noise
- MMSE estimator
- vector norm estimation