Exact Reconstruction of Sparse Non-Harmonic Signals from Fourier\n Coefficients

2020/11/26 by Markus Petz, Petz, Markus, Gerlind Plonka +3
Computer Science · Mathematics · #41A20 #42A16 #42C15 #65D15 #94A12 #A priori and a posteriori #Algorithm #Applied mathematics #Combinatorics #Computer science #Exponential function #FOS: Mathematics #Fourier analysis #Fourier series #Fourier transform #Harmonic #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #Mathematical analysis #Mathematics #Numerical Analysis (math.NA) #Physics #Signal processing #Signal reconstruction #Statistical and numerical algorithms #cs.NA #math.NA #msc:41A20 #msc:42A16 #msc:42C15 #msc:65D15 #msc:94A12

paper · pdf · doi:10.48550/arxiv.2011.13346

28 pages, 2 figures

arxiv created 2020/11/26 · openalex publication_date 2020/11/26 · arxiv updated 2020/11/30 · openalex created_date 2022/07/25 · openalex updated_date 2026/08/06

Abstract

In this paper, we derive a new reconstruction method for real non-harmonic\nFourier sums, i.e., real signals which can be represented as sparse exponential\nsums of the form f(t) = \∑_j=1^K \γ_j , \cos(2\π a_j t +\nb_j), where the frequency parameters a_j \∈ mathbb R (or a_j \∈\n mathrm i mathbb R) are pairwise different. Our method is based on the\nrecently proposed stable iterative rational approximation algorithm in\n citeNST18. For signal reconstruction we use a set of classical Fourier\ncoefficients of f with regard to a fixed interval (0, P) with P>0. Even\nthough all terms of f may be non-P-periodic, our reconstruction method\nrequires at most 2K+2 Fourier coefficients c_n(f) to recover all\nparameters of f. We show that in the case of exact data, the proposed\niterative algorithm terminates after at most K+1 steps. The algorithm can\nalso detect the number K of terms of f, if K is a priori unknown and\nL>2K+2 Fourier coefficients are available. Therefore our method provides a\nnew stable alternative to the known numerical approaches for the recovery of\nexponential sums that are based on Prony's method.\n Keywords: sparse exponential sums, non-harmonic Fourier sums, reconstruction\nof sparse non-periodic signals, rational approximation, AAA algorithm,\nbarycentric representation, Fourier coefficients\n

Citations

• On approximation of functions by exponential sums
• Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool
• ESPRIT-estimation of signal parameters via rotational invariance techniques
• Sampling signals with finite rate of innovation
• Numerical Fourier Analysis

Related

• Exact Reconstruction of Extended Exponential Sums using Rational Approximation of their Fourier Coefficients
• Improved Reconstruction for Fourier-Sparse Signals
• Modifications of Prony's Method for the Recovery and Sparse Approximation of Generalized Exponential Sums
• Recovery of Sparse 1-D Signals from the Magnitudes of their Fourier Transform
• Parameter estimation for multivariate exponential sums via iterative rational approximation