{
  "id": "2011.13346",
  "version": "v1",
  "published": "2020-11-26T15:24:40.000Z",
  "updated": "2020-11-26T15:24:40.000Z",
  "title": "Exact Reconstruction of Sparse Non-Harmonic Signals from Fourier Coefficients",
  "authors": [
    "Markus Petz",
    "Gerlind Plonka",
    "Nadiia Derevianko"
  ],
  "comment": "28 pages, 2 figures",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "In this paper, we derive a new reconstruction method for real non-harmonic Fourier sums, i.e., real signals which can be represented as sparse exponential sums of the form $f(t) = \\sum_{j=1}^{K} \\gamma_{j} \\, \\cos(2\\pi a_{j} t + b_{j})$, where the frequency parameters $a_{j} \\in {\\mathbb R}$ (or $a_{j} \\in {\\mathrm i} {\\mathbb R}$) are pairwise different. Our method is based on the recently proposed stable iterative rational approximation algorithm in \\cite{NST18}. For signal reconstruction we use a set of classical Fourier coefficients of $f$ with regard to a fixed interval $(0, P)$ with $P>0$. Even though all terms of $f$ may be non-$P$-periodic, our reconstruction method requires at most $2K+2$ Fourier coefficients $c_{n}(f)$ to recover all parameters of $f$. We show that in the case of exact data, the proposed iterative algorithm terminates after at most $K+1$ steps. The algorithm can also detect the number $K$ of terms of $f$, if $K$ is a priori unknown and $L>2K+2$ Fourier coefficients are available. Therefore our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony's method. Keywords: sparse exponential sums, non-harmonic Fourier sums, reconstruction of sparse non-periodic signals, rational approximation, AAA algorithm, barycentric representation, Fourier coefficients",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-26T15:24:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A20",
      "42A16",
      "42C15",
      "65D15",
      "94A12"
    ],
    "keywords": [
      "fourier coefficients",
      "sparse non-harmonic signals",
      "exact reconstruction",
      "iterative rational approximation algorithm",
      "sparse exponential sums"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}