An exact solution to the Fourier Transform of band-limited periodic functions with nonequispaced data and application to non-periodic functions

TL;DR

The paper tackles reconstructing the Fourier spectrum of band-limited signals from irregular samples. It derives an exact inversion formula for periodic signals by inverting a Toeplitz matrix C_Γ, giving hat{S}_{Sol} = hat{S}_{Reg} = N Δx^{-1} C_Γ^{-1} hat{S}_Γ, and extends the framework to higher dimensions; for non-periodic signals it yields an excellent approximation with a theoretical error term E_th and practical leakage control via apodization. The method achieves very high dynamic ranges (up to ~10^{13} in double precision) and runs with O(N^2) complexity (Levinson–Durbin), potentially improved to near O(N log^2 N) with advanced solvers. In practice, it offers a fast, accurate alternative to iterative nonuniform-spectrum methods and can serve as a robust initializer for further refinement, with applicability to imaging and multidimensional spectral analysis.

Abstract

The need to Fourier transform data sets with irregular sampling is shared by various domains of science. This is the case for example in astronomy or sismology. Iterative methods have been developed that allow to reach approximate solutions. Here an exact solution to the problem for band-limited periodic signals is presented. The exact spectrum can be deduced from the spectrum of the non-equispaced data through the inversion of a Toeplitz matrix. The result applies to data of any dimension. This method also provides an excellent approximation for non-periodic band-limit signals. The method allows to reach very high dynamic ranges ($10^{13}$ with double-float precision) which depend on the regularity of the samples.

Figures (3)

• Figure 1: Example of periodic band-limited signal: a mix of 4 frequencies (67, 100, 200 and 400 Hz) with respective amplitudes 1, 2, 1, 1. (a) Equispaced (black dots) and non-equispaced samples (red dots) plotted over theoretical signal; 1/8$^{\mathrm{th}}$ of the temporal window is displayed here. The amplitude of the sampling error in units of regular sampling interval is 2. (b) Amplitude of the spectrum $\hat{S}_\mathrm{Reg}$ of the signal with equispaced samples (green), amplitude of the direct DFT $\hat{S}_\Gamma$ of the non-equispaced samples (red) and amplitude of the spectrum $\hat{S}_\mathrm{Sol}$ reconstructed with the method of this paper (blue-dashed line). The condition number is $5.50\times10^{8}$. (c) Histogram of the normalized sampling errors. (d) Log of the spectrum. The black line is the amplitude of the difference $\hat{S}_{\mathrm{Sol}} - \hat{S}_{\mathrm{Reg}}$.
• Figure 2: Example of non-periodic band-limited function: a fringe pattern with 200 Hz temporal frequency and width 44 Hz. (a) Equispaced (black dots) and non-equispaced samples (red dots) plotted over theoretical signal. The amplitude of the sampling error in units of regular sampling interval is 2. (b) Amplitude of the spectrum $\hat{S}_\mathrm{Reg}$ of the signal with equispaced samples (green), amplitude of the direct DFT $\hat{S}_\Gamma$ of the non-equispaced samples (red) and amplitude of the spectrum $\hat{S}_\mathrm{Sol}$ reconstructed with the method of this paper (blue-dashed line). The condition number is $4.43\times10^{10}$. (c) Histogram of the normalized sampling errors. (d) Log of the spectrum. The black line is the amplitude of the difference $\hat{S}_{\mathrm{Sol}} - \hat{S}_{\mathrm{Reg}}$.
• Figure 3: Plot of the normalized errors on the reconstructed spectra vs. the condition number of the Toeplitz matrix for various realizations of sampling grids for the periodic signal of Section \ref{['sec:spectrum_periodic']} (green) and the non-periodic signals of Section \ref{['sec:spectrum_non-periodic']} (interferogram in orange, interferogram with Hann windowing in blue and Ricker wavelet in yellow). The errors have been normalized by the maximum of the spectrum amplitude to compare the impact of relative machine rounding errors on the reconstructed spectra. The continuous lines are the errors $\|\hat{{\cal S}}_{\mathrm{Sol}} - \hat{{\cal S}}_{\mathrm{Reg}}\|$ normalized by the maximum of $|\hat{{S}}_{\mathrm{Reg}}|$. The red-dotted line is a model of the trend of the error due to machine precision as explained in Section \ref{['sec:accuracy']}. The dotted lines with open circles are the theoretical errors ${\cal E}_{\mathrm{th}}$ of equation (\ref{['eq:general_solution']}). Note that the computation of the theoretical errors is also subject to machine precision beyond condition number $10^{15}$.