Robust and tractable multidimensional exponential analysis

TL;DR

This paper introduces a new localized kernel algorithm for multidimensional exponential analysis that accurately recovers signal parameters from limited samples, demonstrating robustness and improvements over existing methods.

Contribution

The authors develop a novel localized kernel method for multidimensional exponential analysis that is stable with low SNR and requires fewer samples than traditional techniques.

Findings

Accurately estimates number of components and parameters from subsampled data.

Demonstrates robustness under low SNR conditions.

Outperforms Prony, MUSIC, and ESPRIT methods in examples.

Abstract

Motivated by a number of applications in signal processing, we study the following question. Given samples of a multidimensional signal of the form $$ f(\boldsymbol\ell)=\sum_{k=1}^K a_k\exp(-i\langle \boldsymbol\ell, \mathbf{w}_k\rangle), \quad \mathbf{w}_1,\cdots,\mathbf{w}_k\in\mathbb{R}^q, \ \boldsymbol\ell\in \mathbb{Z}^q, \ |\boldsymbol\ell| <n, $$ determine the values of the number $K$ of components, and the parameters $a_k$ and $\mathbf{w}_k$'s. We note that the the number of samples of $f$ in the above equation is $(2n-1)^q$. We develop an algorithm to recuperate these quantities accurately using only a subsample of size $\mathcal{O}(qn)$ of this data. For this purpose, we use a novel localized kernel method to identify the parameters, including the number $K$ of signals. Our method is easy to implement, and is shown to be stable under a very low SNR range. We demonstrate the effectiveness of our resulting algorithm using 2 and 3 dimensional examples from the literature, and show substantial improvements over state-of-the-art techniques including Prony based, MUSIC and ESPRIT approaches.