A direct solution to the interpolative inverse non-uniform fast Fourier transform problem for spectral analyses of non-equidistant time-series data

TL;DR

The paper tackles spectral analysis of irregularly sampled time-series by deriving a direct, non-iterative least-squares solution for interpolative inverse non-uniform FFTs (i-iNFFT). It formulates the problem as minimizing a weighted Fourier-coefficient norm subject to a forward-adjoint constraint, and stabilizes the inversion using the Kailath identity and LU decomposition of $A A^H$, with a Sobolev-style weight kernel controlling frequency emphasis via parameters like $\gamma$. The approach yields a closed-form, convex optimization for $\hat{h}_k$ that can be efficiently computed and applied to real remote-sensing data, outperforming weighted truncated iFFT in many scenarios. The work provides a portable Python implementation, demonstrates robustness to missing data, analyzes time complexity, and offers guidance on time-label alignment, significantly advancing practical spectral analysis under irregular sampling.

Abstract

A simple least-squares optimisation enables the determination of the spectrum for irregularly sampled data that is readily reconstructed using an adjoint transformation of the Non-Uniform Fast Fourier Transform (NFFT). This is an improvement upon previously reported iterative methods for such problems, and is competitive in terms of time complexity with more recently proposed direct NFFT inversions when considering comparable matrix pre-computation steps. The software is highly portable, and available as a convenient Python package using standard libraries. Given its mathematical simplicity however, it can be easily implemented on any platform.

Figures (5)

• Figure 1: Various values for $\gamma$ and their impact on the Sobolev kernel function. For lower values of $\gamma$, the filter weights higher frequencies more strongly.
• Figure 2: Results of the i-iNFFT algorithm using the Sobolev kernel with $\gamma = 1e-2$, and $N=1024$. The mean absolute fractional error: $\frac{1}{M}\sum_{j=0}^M|(y_{pred} - y_{obs}) / y_{obs}|$ is $6.76 \times 10^{-2}$ for the i-iNFFT reconstruction, versus $7.48 \times 10^{-2}$ for the equivalent weighted truncated inverse FFT (t-iFFT) reconstruction. Note the stronger tendency of the t-iFFT to regress towards the mean for larger gaps.
• Figure 3: Reconstruction of irregularly sampled time-domain data by $\gamma \in [1e-1,1e-6]$ and $N \in [2^{4},2^{11}]$
• Figure 4: Reconstruction of irregularly sampled time-domain data by $\gamma \in [1e-1,1e-6]$ and $N \in [2^{4},2^{11}]$ with 20% missing data as one contiguous block
• Figure 5: Calculating optimal time labels using a variation of the SGD algorithm. Results using the naïeve time labels are shown above, and the calculated time labels are shown below. The error in the reconstruction of the data with naïeve time labels was $6.76 \times10^{-2}$, versus $1.41 \times 10^{-2}$ for the calculated time labels via the absolute mean fractional error.

Theorems & Definitions (4)

• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 2.4