On the Problem of Parameter Estimation in Exponential Sums

Let I≥1 be an integer, ω0=0<ω1<⋯<ωI≤π, and for j=0,…,I, aj∈ℂ, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{-j}={\overline{{a_{j}}}}$\end{document}, ω−j=−ωj, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{j}\not=0$\end{document} if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$j\not=0$\end{document}. We consider the following problem: Given finitely many noisy samples of an exponential sum of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{x}(k)= \sum_{j=-I}^I a_j\exp(-i\omega _jk) +\epsilon (k), \quad k=-2N,\ldots,2N,$$\end{document} where ϵ(k) are random variables with mean zero, each in the range [−ϵ,ϵ] for some ϵ>0, determine approximately the frequencies ωj. We combine the features of several recent works to use the available information to construct the moments \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{y}_{N}(k)$\end{document} of a positive measure on the unit circle. In the absence of noise, the support of this measure is exactly \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{\exp(-i\omega _{j}) : a_{j}\not=0\}$\end{document}. This support can be recovered as the zeros of the monic orthogonal polynomial of an appropriate degree on the unit circle with respect to this measure. In the presence of noise, this orthogonal polynomial structure allows us to provide error bounds in terms of ϵ and N. It is not our intention to propose a new algorithm. Instead, we prove that a preprocessing of the raw moments \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{x}(k)$\end{document} to obtain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{y}_{N}(k)$\end{document} enables us to obtain rigorous performance guarantees for existing algorithms. We demonstrate also that the proposed preprocessing enhances the performance of existing algorithms.