Robust and tractable multidimensional exponential analysis

Abstract Motivated by a number of applications in signal processing, we study the following question. Given samples of a multidimensional signal of the form $$f(\varvec{\ell })=\sum _{k=1}^K a_k\exp (-i\langle \varvec{\ell }, \textbf{w}_k\rangle ), \quad \textbf{w}_1,\cdots ,\textbf{w}_k\in \mathbb {R}^q, \ \varvec{\ell }\in \mathbb {Z}^q, \ |\varvec{\ell }| <n,$$ determine the values of the number K of components, and the parameters $$a_k$$ and $$\textbf{w}_k$$ ’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.