Robust and tractable multidimensional exponential analysis

Motivated by a number of applications in signal processing, we study the following question. Given samples of a multidimensional signal of the form f(boldsymbolℓ)=sumk=1K akexp(-ilangle boldsymbolℓ, wkrangle), quad w1,cdots,wk∈ℝq, boldsymbolℓ∈ mathbbZq, |boldsymbolℓ| <n, determine the values of the number K of components, and the parameters ak and wk'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 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.